基于空间局部间断伽辽金近似和时间谱延迟校正演化的黏性Burgers方程的高阶方法

Higher-Order Method for the Viscous Burgers Equation Based on Spatial Local Discontinuous Galerkin Approximation and Temporal Spectral Deferred Correction Evolution

摘要

本文构造和分析了一种求解黏性Burgers方程的时空高阶方法．该方法的主要思想是通过选取合适的数值通量，构造基于空间局部间断伽辽金（LDG）近似与时间谱延迟校正（SDC）方法相结合的时空高阶演化方法．在理论上，首先证明了空间半离散LDG方法的L^{\mathrm{2}}稳定性和最优误差估计，接着给出了时空全离散方法的稳定性，最后通过数值算例验证了所提方法的有效性．

Abstract

In this paper, a higher-order method to solve the viscous Burgers equation is constructed and analysed. The main idea of the method is based on a proper selection of the numerical fluxes, a higher-order spatiotemporal evolution method which combines the local discontinuous Galerkin (LDG) method in the spatial direction and the spectral deferred correction (SDC) method in the temporal direction is proposed. Theoretically, the L^{\mathrm{2}} stability and the optimal error estimation for the spatial semi-discrete LDG method are proved, and the stability of the fully discrete method is given. Numerical examples are displayed to show the effectiveness of the proposed method.

Graphical abstract

关键词

黏性Burgers方程 / 局部间断伽辽金方法 / 谱延迟校正方法 / 稳定性 / 误差估计

Key words

viscous Burgers equation / local discontinuous Galerkin method / spectral deferred correction method / stability / error estimation

0 引 言

Bateman在1915年研究黏性流体中非线性波的传播与耗散机制时，特别关注激波在黏性系数趋近于0时的演化行为^［1］，首次推导出一个描述此类现象的偏微分方程．1948年，Burgers将其作为湍流模型进行系统分析与推广^［2］，因其在流体力学领域的杰出贡献，该方程以他的名字命名．关于Burgers方程数学理论的研究，已有多项成果，如Hopf和Cole引入一种变换，后来被称为Hopf-Cole变换^［3‒4］，将Burgers方程转换成线性热方程，从而可以求解带有任意初值问题的Burgers方程．

许多学者研究了Burgers方程的数值方法．对于无黏的Burgers方程，Yang等研究了TVD和FCT方法，以及MUSCL、ENO和PPM等几何方法^［5］．Mittal等提出了一种时间方向上采用Crank-Nicolson格式、空间方向上采用三次B样条函数方法，用于求解耦合黏性Burgers方程^［6］．需要注意的是，上述数值研究主要集中在低阶方法的构造分析，但时间和空间的阶数可能不匹配．为了设计一个更高阶的时空逼近方法，空间高阶插值和时间演化的校正方法可能是合适的选择．一般来说，空间上全局的更高阶插值将导致离散系统的模板扩展或带宽增加．此外，更高阶的时间插值需要提供许多预定义的值，时间演化的校正方法通常需要求解一系列低阶格式以获得更高阶的校正解．

基于上述分析，本文围绕LDG方法并结合Minion提出的半隐式SDC方法^［7］，设计了时空高阶演化方法．核心思想是通过选取合适的单元交界面数值通量，结合LDG方法的空间局部性质及SDC方法的时间低阶格式多次校正，最终达到时空高阶精度，从而有效避免直接构造时间多步高阶格式的复杂性，同时每一时间层只需逐单元求解规模极低的离散代数方程组．本文对空间半离散LDG方法的稳定性和最优误差估计进行证明，进一步分析全离散方法的稳定性．

1 一维黏性Burgers方程的LDG变分问题

为简化叙述，以一维黏性Burgers方程为例，相关算法和理论可以扩展到高维和复杂情形^［8］．

1.1 控制方程

假设一维区域(a,b)是有界的，考虑如下右端齐次黏性Burgers方程

\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}-\frac{{\partial }^{\mathrm{2}}u}{\partial x^{\mathrm{2}}}=\mathrm{0},\forall \left(x,t\right)\in \left(a,b\right)\times \left(\mathrm{0},T\right],(1)

式中：未知变量u(t)=u(x,t)；补充周期边界条件和初始条件u(x,\mathrm{0})=u_{\mathrm{0}}(x)，这里周期边界条件的要求不是本质的．下文中，符号c或c_j(j=\mathrm{0},\mathrm{1},\dots )用于表示正常数，其依赖于区域(a,b)、Sobolev不等式等，在不同的位置可能取值不同．

1.2 LDG变分问题

下面给出方程（1）基于空间LDG方法的变分问题．首先，考虑任意有界区域I，记H^m(I),m=\mathrm{0},\mathrm{1},\mathrm{2},\dots，是标准的Sobolev空间，并赋予通常的范数{‖·‖}_{m,I}．当m=\mathrm{0}时，定义L^{\mathrm{2}}(I)=H^{\mathrm{0}}(I)，范数为{‖·‖}_I．记{‖·‖}_{\mathrm{\infty },I}是L^{\mathrm{\infty }}(I)空间的最大模范数．特别地，当I=(a,b)时，上述范数中的下标I可以省略．其次，假设为区域(a,b)的一个剖分，记单元I_i=\left[x_{i-\frac{\mathrm{1}}{\mathrm{2}}},x_{i+\frac{\mathrm{1}}{\mathrm{2}}}\right]的长度为h_i=x_{i+\frac{\mathrm{1}}{\mathrm{2}}}-x_{i-\frac{\mathrm{1}}{\mathrm{2}}},h=\underset{i}{\mathrm{m}\mathrm{a}\mathrm{x}} h_i,i=\mathrm{1},\mathrm{2},\dots ,M．假设剖分是正则的，即存在正常数c_{\mathrm{1}},c_{\mathrm{2}}满足c_{\mathrm{1}}h\le h_i\le c_{\mathrm{2}}h．假设\mathrm{\Gamma }为单元交界面点的集合，对任意的v\in L^{\mathrm{2}}(a,b)，定义在\mathrm{\Gamma }上的范数为{‖v‖}_{\mathrm{\Gamma }}={\left({\sum }_{i=\mathrm{1}}^M \left({\left(v_{i-\frac{\mathrm{1}}{\mathrm{2}}}^{+}\right)}^{\mathrm{2}}+{\left(v_{i+\frac{\mathrm{1}}{\mathrm{2}}}^{-}\right)}^{\mathrm{2}}\right)\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}，上标“-”和“+”代表对应点的左右极限．另外，记通过点x_{i+\frac{\mathrm{1}}{\mathrm{2}}}的跃度为\left[v_{i+\frac{\mathrm{1}}{\mathrm{2}}}\right]=v_{i+\frac{\mathrm{1}}{\mathrm{2}}}^{+}-v_{i+\frac{\mathrm{1}}{\mathrm{2}}}^{-}，并记全局跃度为|[v]|={\left({\sum }_{i=\mathrm{1}}^M {[v]}_{{\mathrm{x}}_{\mathrm{i}-\frac{\mathrm{1}}{\mathrm{2}}}^{}}^{\mathrm{2}}\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}．进一步引入区域(a,b)上的分片Sobolev空间V=\left\{v\in L^{\mathrm{2}}(a,b):{\left.v\right|}_{I_i}\in H^{\mathrm{1}}\left(I_i\right),i=\mathrm{1},\mathrm{2},\dots ,M\right\}．定义单元I_i上的k次多项式空间P^k\left(I_i\right)，k=\mathrm{1},\mathrm{2},\cdots ,K，并引入有限维离散Sobolev空间V_h=\left\{v\in L^{\mathrm{2}}(a,b):{\left.v\right|}_{I_i}\in P^k\left(I_i\right),i=\mathrm{1},\mathrm{2},\dots ,M\right\}．

基于LDG方法的基本思想，方程（1）可等价为以下一阶方程组

\frac{\partial u}{\partial t}+\frac{\partial (f(u)-q)}{\partial x}=\mathrm{0},(2)

q-\frac{\partial u}{\partial x}=\mathrm{0},(3)

式中：非线性函数f(u)=\frac{u^{\mathrm{2}}}{\mathrm{2}}并足够光滑，如f(u)和f^{\text{'}}(u)是有界的．

方程组（2~3）的变分问题为：对任意t\in (\mathrm{0},T]，求(u,q)=(u(·,t),q(·,t))\in V\times V，使得对任意(\rho ,\phi )\in V\times V，成立

{\int }_{I_i} \frac{\partial u}{\partial t}\rho \mathrm{d}x-{\int }_{I_i} (f(u)-q)\frac{\partial \rho }{\partial x}\mathrm{d}x+{\left((\widehat{f}(u)-\widehat{q}){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}-{\left((\widehat{f}(u)-\widehat{q}){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}=\mathrm{0},(4)

{\int }_{I_i} q\phi \mathrm{d}x+{\int }_{I_i} u\frac{\partial \phi }{\partial x}\mathrm{d}x-{\left(\widehat{u}{\phi }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{u}{\phi }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}=\mathrm{0},(5)

式中：i=\mathrm{1},\mathrm{2},\cdots ,M．变分问题（4~5）中的符号“\widehat{}”代表数值通量．

本文黏性项对应的数值通量选取交替通量形式，即

\widehat{q}=q^{+},\widehat{u}=u^{-}.(6)

数值通量f(\widehat{u})=\widehat{f}\left(u^{-},u^{+}\right)的选取满足如下条件：（a）一致性．\widehat{f}(u,u)=f(u)． （b）连续性．\widehat{f} 满足局部Lipschitz连续条件，即\mathrm{􀰘}\widehat{f}(u_{\mathrm{1}}^{-},u_{\mathrm{1}}^{+})-\widehat{f}(u_{\mathrm{2}}^{-},u_{\mathrm{2}}^{+})\mathrm{􀰘}\le c_{\mathrm{0}}\left(\mathrm{􀰘}{u}_{\mathrm{1}}^{-}-u_{\mathrm{2}}^{-}\mathrm{􀰘}+\mathrm{􀰘}{u}_{\mathrm{1}}^{+}-u_{\mathrm{2}}^{+}\mathrm{􀰘}\right),\mathrm{􀰘}\widehat{f}\left(u^{-},u^{+}\right)-\widehat{f}\left(u^{+},u^{+}\right)\mathrm{􀰘}\le c_{\mathrm{0}}\mathrm{􀰘}[u]\mathrm{􀰘}，其中c_{\mathrm{0}}为Lipschitz系数． （c）单调性．\widehat{f}\left(u^{-},u^{+}\right)函数的单调性是关于第一个变量非递减、第二个变量非递增．由文献［9］可知，这样的数值通量形式是容易构造的，如文献［10］提到的满足局部Lipschitz连续的单调通量．

变分问题（4~5）对应的空间半离散变分问题为：对任意t\in (\mathrm{0},T]，求\left(u_h,q_h\right)=\left(u_h(·,t),q_h(·,t)\right)\in V_h\times V_h，使得对任意(\rho ,\phi )\in V_h\times V_h及i=\mathrm{1},\mathrm{2},\dots ,M，成立

{\int }_{I_i} \frac{\partial u_h}{\partial t}\rho \mathrm{d}x-{\int }_{I_i} \left(f\left(u_h\right)-q_h\right)\frac{\partial \rho }{\partial x}\mathrm{d}x+{\left(\left(\widehat{f}\left(u_h\right)-{\widehat{q}}_h\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}-{\left(\left(\widehat{f}\left(u_h\right)-{\widehat{q}}_h\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}=\mathrm{0},(7)

{\int }_{I_i} q_h\phi \mathrm{d}x+{\int }_{I_i} u_h\frac{\partial \phi }{\partial x}\mathrm{d}x-{\left({\widehat{u}}_h{\phi }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left({\widehat{u}}_h{\phi }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}=\mathrm{0}.(8)

为方便表述，进一步将空间半离散变分问题（7~8）写成算子形式

{\left(\frac{\partial u_h}{\partial t},\rho \right)}_{I_i}=H_i\left(f\left(u_h\right);u_h;\rho \right)-{\overline{B}}_i\left(q_h;\rho \right),(9)

{\left(q_h,\phi \right)}_{I_i}=-{\tilde{B}}_i\left(u_h;\phi \right),(10)

对非线性算子H_i、线性算子{\overline{B}}_i和{\tilde{B}}_i定义如下

H_i(f(\xi );\xi ;\rho )={\left(f(\xi ),\frac{\partial \rho }{\partial x}\right)}_i-{\left(\widehat{f}(\xi ){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}(\xi ){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}},(11)

{\overline{B}}_i(\eta ;\rho )={\left(\eta ,\frac{\partial \rho }{\partial x}\right)}_i-{\left(\widehat{\eta }{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{\eta }{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}},(12)

{\tilde{B}}_i(\xi ;\phi )={\left(\xi ,\frac{\partial \phi }{\partial x}\right)}_i-{\left(\widehat{\xi }{\phi }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{\xi }{\phi }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}},(13)

式中：\xi ,\eta ,\rho ,\phi \in V_h，并记H\left(f\left(\xi \right);\xi ;\rho \right)={\sum }_{i=\mathrm{1}}^M H_i\left(f\left(\xi \right);\xi ;\rho \right),\overline{B}\left(\eta ;\rho \right)={\sum }_{i=\mathrm{1}}^M {\overline{B}}_i\left(\eta ;\rho \right),\tilde{B}(\xi ;\phi )={\sum }_{i=\mathrm{1}}^M{\tilde{B}}_i(\xi ;\phi )．

依据数值通量的选取形式，可得

{\overline{B}}^{+}\left(q_h;\rho \right)={\sum }_{i=\mathrm{1}}^M {\overline{B}}_i^{+}\left(q_h;\rho \right)={\sum }_{i=\mathrm{1}}^M \left({\left(q_h,\frac{\partial \rho }{\partial x}\right)}_i-{\left(q_h^{+}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(q_h^{+}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right),(14)

{\tilde{B}}^{-}\left(u_h;\phi \right)={\sum }_{i=\mathrm{1}}^M {\tilde{B}}_i^{-}\left(u_h;\phi \right)={\sum }_{i=\mathrm{1}}^M \left({\left(u_h,\frac{\partial \phi }{\partial x}\right)}_i-{\left(u_h^{-}{\phi }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(u_h^{-}{\phi }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right),(15)

若（u_h,q_h）是空间半离散变分问题（9~10）的解，则算子\overline{B}和\tilde{B}满足如下关系

{\tilde{B}}^{-}\left(u_h;u_h\right)=-{\left[u_h\right]}^{\mathrm{2}},(16)

{\tilde{B}}^{-}\left(u_h;q_h\right)=-{\overline{B}}^{+}\left(q_h;u_h\right).(17)

1.3 预备知识

为推导误差估计，引入Gauss-Radau投影^［11］．对任意u\in V，定义投影P^{\pm }:V\to V_h满足如下关系

\begin{array}{lll}{\left(P^{-}u-u,v\right)}_{I_i}=\mathrm{0},& \forall v\in P^{k-\mathrm{1}}\left(I_i\right),& {\left(P^{-}u\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}^{-}=u_{i+\frac{\mathrm{1}}{\mathrm{2}}}^{-}\end{array},(18)

\begin{array}{ccc}{\left(P^{+}u-u,v\right)}_{I_i}=\mathrm{0},& \forall v\in P^{k-\mathrm{1}}\left(I_i\right),& {\left(P^{+}u\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}^{+}=u_{i-\frac{\mathrm{1}}{\mathrm{2}}}^{-}\end{array}.(19)

进一步引入数值误差w\mathrm{、}z和投影误差w^e\mathrm{、}{z}^e

w=P^{-}u-u_h,z=P^{+}q-q_h,w^e=P^{-}u-u,z^e=P^{+}q-q,(20)

由文献［12‒14］可知，如下结论成立．

引理 1 假设(u,q)\mathrm{、}(u_h,q_h)分别为变分问题（4~5）和半离散变分问题（7~8）的解，成立

‖\frac{\partial w}{\partial x}‖\le c\left(‖z‖+‖z^e‖\right),(21)

h^{-\frac{\mathrm{1}}{\mathrm{2}}}|[w]|\le c\left(‖z‖+‖z^e‖\right),(22)

‖\frac{\partial u_h}{\partial x}‖+c_{\mathrm{1}}{h}^{-\frac{\mathrm{1}}{\mathrm{2}}}|[u_h]|\le c_{\mathrm{2}}‖q_h‖,(23)

‖w^e‖+h{‖w^e‖}_{\mathrm{\infty }}+h^{\frac{\mathrm{1}}{\mathrm{2}}}{‖w^e‖}_{\mathrm{\Gamma }}\le c{h}^{k+\mathrm{1}}.(24)

引理 2 对任意u,v,\rho \in V_h，成立

H(f(u);u;u)\le \mathrm{0},(25)

|H(f(u);u;\rho )|\le c_{\mathrm{0}}\left(‖\frac{\partial u}{\partial x}‖+c_{\mathrm{1}}{h}^{-\frac{\mathrm{1}}{\mathrm{2}}}|[u]|\right)‖\rho ‖,(26)

|H(f(u);u;\rho )-H(f(v);v;\rho )|\le c_{\mathrm{0}}\Vert u-v\Vert \left(‖\frac{\partial \rho }{\partial x}‖+c_{\mathrm{1}}{h}^{-\frac{\mathrm{1}}{\mathrm{2}}}|[\rho ]|\right),(27)

进一步在空间V_h中成立不等式：‖\frac{\partial w_h}{\partial x}‖\le c_{\mathrm{1}}{h}^{-\mathrm{1}}‖w_h‖,{‖w_h‖}_{\mathrm{\Gamma }}\le \sqrt[]{c_{\mathrm{1}}{h}^{-\mathrm{1}}}‖w_h‖,{‖w_h‖}_{\mathrm{\infty }}\le c_{\mathrm{1}}{h}^{-\frac{\mathrm{1}}{\mathrm{2}}}‖w_h‖,\forall w_h\in V_h，以及对任意a,b\in R^d，成立

\mathrm{2}\left(a,b\right)=\left(a,a\right)+\left(b,b\right)-(a-b,a-b).(28)

2 一维离散方法的数值分析

本节考虑一维黏性Burgers方程空间半离散方法的稳定性和误差估计，以及全离散方法的稳定性．

2.1 空间半离散方法的稳定性分析

命题 1 假设（u_h,q_h）是半离散变分问题（9~10）的解，则存在熵通量\widehat{F}，见式（32~33），成立熵不等式

\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{I_i} {\left|u_h\left(·,t\right)\right|}^{\mathrm{2}}\mathrm{d}x+{\int }_{I_i} {\left|q_h\left(·,t\right)\right|}^{\mathrm{2}}\mathrm{d}x+\widehat{F}{(·,t)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}-\widehat{F}{(·,t)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\le \mathrm{0},i=\mathrm{1},\mathrm{2},\dots ,M.(29)

证明 考虑半离散变分问题（9~10），将其写为求和形式

{\left(\frac{\partial u_h}{\partial t},\rho \right)}_{I_i}+{\left(q_h,\phi \right)}_{I_i}-H_i\left(f\left(u_h\right);u_h;\rho \right)+{\overline{B}}_i\left(q_h;\rho \right)+{\tilde{B}}_i\left(u_h;\phi \right)=\mathrm{0},(30)

取检验函数\rho =u_h\mathrm{、}\phi =q_h，定义\tilde{F}={\int }_{}^{u}f(\xi )\mathrm{d}\xi，利用数值通量表达式的定义，由式（30）可知

\begin{array}{rrr}& \mathrm{0}={\left(\frac{\partial u_h}{\partial t},u_h\right)}_{I_i}+{\left(q_h,q_h\right)}_{I_i}-H_i\left(f\left(u_h\right);u_h;u_h\right)+{\overline{B}}_i^{+}\left(q_h;u_h\right)+{\tilde{B}}_i^{-}\left(u_h;q_h\right)& \\ & =\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{I_i} {\left|u_h\right|}^{\mathrm{2}}\mathrm{d}x+{\int }_{I_i} {\left|q_h\right|}^{\mathrm{2}}\mathrm{d}x-{\int }_{I_i} f\left(u_h\right)\frac{\partial u_h}{\partial x}\mathrm{d}x+{\int }_{I_i} q_h\frac{\partial u_h}{\partial x}+\frac{\partial q_h}{\partial x}{u}_h\mathrm{d}x& \\ & +{\left(\widehat{f}\left(u_h\right)u_h^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}-{\left({\widehat{q}}_h{u}_h^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}-{\left({\widehat{u}}_h{q}_h^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}& \\ & -{\left(\widehat{f}\left(u_h\right)u_h^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}+{\left({\widehat{q}}_h{u}_h^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}+{\left({\widehat{u}}_h{q}_h^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}& \\ & =\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{I_i} {\left|u_h\right|}^{\mathrm{2}}\mathrm{d}x+{\int }_{I_i} {\left|q_h\right|}^{\mathrm{2}}\mathrm{d}x+{\widehat{F}}_{i+\frac{\mathrm{1}}{\mathrm{2}}}-{\widehat{F}}_{i-\frac{\mathrm{1}}{\mathrm{2}}}+{\mathrm{\Theta }}_{i-\frac{\mathrm{1}}{\mathrm{2}}},& \end{array}(31)

其中熵通量{\widehat{F}}_{i+\frac{\mathrm{1}}{\mathrm{2}}}\mathrm{、}{\widehat{F}}_{i-\frac{\mathrm{1}}{\mathrm{2}}}和函数{\mathrm{\Theta }}_{i-\frac{\mathrm{1}}{\mathrm{2}}}定义为

{\widehat{F}}_{i+\frac{\mathrm{1}}{\mathrm{2}}}=-\tilde{F}{\left(u_h^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_h\right)u_h^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}-{\left(q_h^{+}{u}_h^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}},(32)

{\widehat{F}}_{i-\frac{\mathrm{1}}{\mathrm{2}}}=-\tilde{F}{\left(u_h^{-}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_h\right)u_h^{-}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}-{\left(q_h^{+}{u}_h^{-}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}},(33)

{\mathrm{\Theta }}_{i-\frac{\mathrm{1}}{\mathrm{2}}}={\left[-\tilde{F}\left(u_h^{-}\right)+\widehat{f}\left(u_h\right)u_h^{-}-q_h^{+}{u}_h^{-}-q_h^{+}{u}_h^{+}-\widehat{f}\left(u_h\right)u_h^{+}+q_h^{+}{u}_h^{+}+u_h^{-}{q}_h^{+}+\tilde{F}\left(u_h^{+}\right)\right]}_{i-\frac{\mathrm{1}}{\mathrm{2}}}.(34)

由数值通量\widehat{f}\left(u^{-},u^{+}\right)的单调性可得{\mathrm{\Theta }}_{i-\frac{\mathrm{1}}{\mathrm{2}}}=-\tilde{F}{\left(u_h^{-}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}+\tilde{F}{\left(u_h^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}+\widehat{f}\left(u_h\right){\left[u_h\right]}_{i-\frac{\mathrm{1}}{\mathrm{2}}}={\int }_{u^{+}}^{u^{-}} \left(f\left(s\right)-\widehat{f}\left(u^{-},u^{+}\right)\right)\mathrm{d}s\ge \mathrm{0}．利用式（31）可得单元熵不等式成立．对单元熵不等式（29）中的下标i进行累加求和，可得如下L^{\mathrm{2}}稳定性结论．

推论 1 半离散变分问题（9~10）的解满足不等式

\frac{\mathrm{d}}{\mathrm{d}t}{‖u_h(·,t)‖}^{\mathrm{2}}+\mathrm{2}{‖q_h(·,t)‖}^{\mathrm{2}}\le \mathrm{0},(35)

即对任意t\in (\mathrm{0},T]成立，{‖u_h(·,t)‖}^{\mathrm{2}}+\mathrm{2}{\int }_{\mathrm{0}}^t {‖q_h(·,\tau )‖}^{\mathrm{2}}\mathrm{d}\tau \le {‖u_h(·,\mathrm{0})‖}^{\mathrm{2}}．

2.2 空间半离散方法的误差估计

本节考虑一维黏性Burgers方程空间半离散方法的最优误差估计．

定理 1 设(u,q)是变分问题（4~5）的解、\left(u_h,q_h\right)是半离散变分问题（9~10）的解，则成立最优误差估计

‖u(·,t)-u_h(·,t)‖\le c{h}^{k+\mathrm{1}}.(36)

证明 空间半离散方法的最优误差估计证明分为以下三步：首先构造能量方程，其次分别对能量方程中的线性部分和非线性部分进行估计，最后可得最优误差估计．

首先构造能量方程．定义误差e_u=u-u_h\mathrm{、}{e}_q=q-q_h，由变分问题（4~5）和半离散变分问题（9~10），可得单元上的误差方程

{\left(\frac{\partial e_u}{\partial t},\rho \right)}_{I_i}+{\left(e_q,\phi \right)}_{I_i}-H_i\left(f\left(u\right);u;\rho \right)+H_i\left(f\left(u_h\right);u_h;\rho \right)+{\overline{B}}_i^{+}\left(e_q;\rho \right)+{\tilde{B}}_i^{-}\left(e_u;\phi \right)=\mathrm{0},(37)

引入算子形式

{\mathrm{ℬ}}_i\left(e_q,e_u;\rho ,\phi \right)={\overline{B}}_i^{+}\left(e_q;\rho \right)+{\tilde{B}}_i^{-}\left(e_u;\phi \right),(38)

可将单元上的误差方程（37）改写为

{\left(\frac{\partial e_u}{\partial t},\rho \right)}_{I_i}+{\left(e_q,\phi \right)}_{I_i}=-{\mathrm{ℬ}}_i\left(e_q,e_u;\rho ,\phi \right)+H_i(f(u);u;\rho )-H_i\left(f\left(u_h\right);u_h;\rho \right),(39)

对方程（39）关于指标i累加求和，可得误差方程

\begin{array}{l}{\sum }_{i=\mathrm{1}}^M {\left(\frac{\partial e_u}{\partial t},\rho \right)}_{I_i}+{\sum }_{i=\mathrm{1}}^M {\left(e_q,\phi \right)}_{I_i}\\ =-{\sum }_{i=\mathrm{1}}^M {\mathrm{ℬ}}_i\left(e_q,e_u;\rho ,\phi \right)+{\sum }_{i=\mathrm{1}}^M H_i(f(u);u;\rho )-{\sum }_{i=\mathrm{1}}^M H_i\left(f\left(u_h\right);u_h;\rho \right),\end{array}(40)

记e_u=w-w^e\mathrm{、}{e}_q=z-z^e，在式（40）中选取检验函数\rho =w\mathrm{、}\phi =z，可得能量方程

{\sum }_{i=\mathrm{1}}^M {\left(\frac{\partial \left(w-w^e\right)}{\partial t},w\right)}_{I_i}+{\sum }_{i=\mathrm{1}}^M {\left(z-z^e,z\right)}_{I_i}=-\mathrm{ℬ}+\mathfrak{H},(41)

式中：\mathrm{ℬ}={\sum }_{i=\mathrm{1}}^M {\mathrm{ℬ}}_i\left(e_q,e_u;w,z\right);\mathfrak{H}={\sum }_{i=\mathrm{1}}^M H_i(f(u);u;w)-{\sum }_{i=\mathrm{1}}^M H_i\left(f\left(u_h\right);u_h;w\right)．

接下来，对能量方程（41）中的线性部分进行估计， 即

\begin{array}{rrr}& \mathrm{ℬ}={\sum }_{i=\mathrm{1}}^M {\mathrm{ℬ}}_i\left(e_q,e_u;w,z\right)={\sum }_{i=\mathrm{1}}^M {\overline{B}}_i^{+}\left(e_q;w\right)+{\sum }_{i=\mathrm{1}}^M {\tilde{B}}_i^{-}\left(e_u;z\right)& \\ & ={\sum }_{i=\mathrm{1}}^M \left({\int }_{I_i} \left(z-z^e\right)\frac{\partial w}{\partial x}\mathrm{d}x-{\left(\left(z^{+}-z^e\right)w^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\left(z^{+}-z^e\right)w^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right.& \\ & \left.+{\int }_{I_i} \left(w-w^e\right)\frac{\partial z}{\partial x}\mathrm{d}x-{\left(\left(w^{-}-w^e\right)z^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\left(w^{-}-w^e\right)z^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & ={\sum }_{i=\mathrm{1}}^M \left({\int }_{I_i} z\frac{\partial w}{\partial x}\mathrm{d}x-{\left(z^{+}{w}^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(z^{+}{w}^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +{\sum }_{i=\mathrm{1}}^M {\int }_{I_i} w\frac{\partial z}{\partial x}\mathrm{d}x-{\left(w^{-}{z}^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(w^{-}{z}^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}& \\ & ={\int }_a^b z\frac{\partial w}{\partial x}\mathrm{d}x+{\sum }_{i=\mathrm{1}}^M ({[zw])}_{i-\frac{\mathrm{1}}{\mathrm{2}}}+{\int }_a^b w\frac{\partial z}{\partial x}\mathrm{d}x,& \end{array}(42)

然后，对能量方程（41）中的非线性部分进行估计，即

\mathfrak{H}={\sum }_{i=\mathrm{1}}^M {\int }_{I_i} \left(f(u)-f\left(u_h\right)\right)\frac{\partial w}{\partial x}\mathrm{d}x+{\sum }_{i=\mathrm{1}}^M {\left(\left(\widehat{f}(u)-\widehat{f}\left(u_h\right)\right)[w]\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}},(43)

对其进行分部积分，可得

\begin{array}{rrr}& \mathfrak{H}=-{\sum }_{i=\mathrm{1}}^M {\int }_{I_i} \frac{\partial \left(f(u)-f\left(u_h\right)\right)}{\partial x}w\mathrm{d}x+{\sum }_{i=\mathrm{1}}^M {\left(\left(f\left(u^{-}\right)-f\left(u_h^{-}\right)\right)w^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}& \\ & -{\sum }_{i=\mathrm{1}}^M {\left(\left(f\left(u^{+}\right)-f\left(u_h^{+}\right)\right)w^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}+{\sum }_{i=\mathrm{1}}^M {\left(\left(\widehat{f}(u)-\widehat{f}\left(u_h\right)\right)[w]\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}& \\ & ={\sum }_{i=\mathrm{1}}^M \left({\mathrm{Ⅰ}}_i+{\mathrm{Ⅱ}}_i\right),& \end{array}(44)

为方便分析，忽略指标i-\frac{\mathrm{1}}{\mathrm{2}}．由引理1、Cauchy-Schwarz不等式及Young不等式，可推得

\begin{array}{c}{\sum }_{i=\mathrm{1}}^M \left|{\mathrm{I}}_i\right|={\sum }_{i=\mathrm{1}}^M {\int }_{I_i} \left|\left(f^{\text{'}}\left(u\right)\frac{\partial u}{\partial x}-f^{\text{'}}\left(u_h\right)\frac{\partial u_h}{\partial x}\right)w\right|\mathrm{d}x\\ ={\sum }_{i=\mathrm{1}}^M {\int }_{I_i} c\left|\frac{\partial u}{\partial x}-\frac{\partial u_h}{\partial x}\right|\left|w\right|\mathrm{d}x\le {\sum }_{i=\mathrm{1}}^M {\int }_{I_i} c\left|\frac{\partial \left(w-w^e\right)}{\partial x}\right|\left|w\right|\mathrm{d}x\\ \le \frac{\mathrm{1}}{\mathrm{4}}{‖\frac{\partial w}{\partial x}‖}^{\mathrm{2}}+{‖\frac{\partial w^e}{\partial x}‖}^{\mathrm{2}}+c\Vert w{\Vert }^{\mathrm{2}}\le \frac{\mathrm{1}}{\mathrm{2}}\Vert z{\Vert }^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}}{‖z^e‖}^{\mathrm{2}}+c\Vert w{\Vert }^{\mathrm{2}}+{‖\frac{\partial w^e}{\partial x}‖}^{\mathrm{2}},\end{array}(45)

{\mathrm{Ⅱ}}_i=\left(\left(f\left(u^{-}\right)-f\left(u^{+}\right)\right)-\left(f\left(u_h^{-}\right)-f\left(u_h^{+}\right)\right)\right)w^{-}+\left(\widehat{f}\left(u\right)-f\left(u^{+}\right)\right)\left[w\right]-\left(\widehat{f}\left(u_h\right)-f\left(u_h^{+}\right)\right)\left[w\right]

\le -f^{\text{'}}\left(u\right)\left[u\right]w^{-}+f^{\text{'}}\left(u_h\right)\left[u_h\right]w^{-}+\left(\widehat{f}\left(u^{-},u^{+}\right)-f\left(u^{+},u^{+}\right)\right)\left[w\right]-\left(\widehat{f}\left(u_h^{-},u_h^{+}\right)-f\left(u_h^{+},u_h^{+}\right)\right)\left[w\right],(46)

由\widehat{f}的Lipschitz连续性质、引理1及方程（46），可推得

\begin{array}{rrr}{\sum }_{i=\mathrm{1}}^M \left|{\mathrm{Ⅱ}}_i\right|\le {\sum }_{i=\mathrm{1}}^M \left|{\left(-f^{\text{'}}(u)[u]w^{-}+f^{\text{'}}\left(u_h\right)\left[u_h\right]w^{-}+c_{\mathrm{0}}|[u]|[w]-c_{\mathrm{0}}\left|\left[u_h\right]\right|[w]\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right|& & \\ \le c_{\mathrm{0}}{c}_{\mathrm{1}}{h}^{-\mathrm{1}}\left|\left[w\right]\right|‖w‖+c_{\mathrm{0}}{c}_{\mathrm{1}}{h}^{-\mathrm{1}}\left|\left[w^e\right]\right|‖w‖& & \\ \le c\left(‖z‖+‖z^e‖\right)‖w‖+c_{\mathrm{0}}{c}_{\mathrm{1}}{h}^{-\mathrm{1}}\left|\left[w^e\right]\right|‖w‖& & \\ \le \frac{\mathrm{1}}{\mathrm{16}}\Vert z{\Vert }^{\mathrm{2}}+c\Vert z^e{\Vert }^{\mathrm{2}}+c\Vert w{\Vert }^{\mathrm{2}}+{\left|\left[w^e\right]\right|}^{\mathrm{2}},& & \end{array}(47)

联立方程（45）和（47），运用Cauchy-Schwarz不等式，可得非线性部分\mathfrak{H}的估计

\mathfrak{H}\le \frac{\mathrm{9}}{\mathrm{16}}{‖z‖}^{\mathrm{2}}+\left(\frac{\mathrm{1}}{\mathrm{4}}+c\right){‖z^e‖}^{\mathrm{2}}+c{‖w‖}^{\mathrm{2}}+{‖\frac{\partial w^e}{\partial x}‖}^{\mathrm{2}}+{\left|\left[w^e\right]\right|}^{\mathrm{2}},(48)

结合能量方程（41）、线性部分的估计（42）、非线性部分的估计（48），可得

\begin{array}{rrr}& \frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_a^b |w{|}^{\mathrm{2}}\mathrm{d}x={\int }_a^b \frac{\partial w}{\partial t}w\mathrm{d}x& \\ & \le {\int }_a^b \frac{\partial w^e}{\partial t}w\mathrm{d}x-{\int }_a^b z^{\mathrm{2}}\mathrm{d}x+{\int }_a^b z^{e}z\mathrm{d}x+{\int }_a^b z\frac{\partial w}{\partial x}\mathrm{d}x+{\sum }_{i=\mathrm{1}}^M({[zw])}_{i-\frac{\mathrm{1}}{\mathrm{2}}} & \\ & +{\int }_a^b w\frac{\partial z}{\partial x}\mathrm{d}x+\frac{\mathrm{9}}{\mathrm{16}}{‖z‖}^{\mathrm{2}}+\left(\frac{\mathrm{1}}{\mathrm{4}}+c\right){‖z^e‖}^{\mathrm{2}}+c{‖w‖}^{\mathrm{2}}+{‖\frac{\partial w^e}{\partial x}‖}^{\mathrm{2}}+{\left|\left[w^e\right]\right|}^{\mathrm{2}}& \\ & \le {\int }_a^b \frac{\partial w^e}{\partial t}w\mathrm{d}x-\frac{\mathrm{1}}{\mathrm{4}}{‖z‖}^{\mathrm{2}}+\left(\frac{\mathrm{1}}{\mathrm{2}}+c\right){‖z^e‖}^{\mathrm{2}}+(\mathrm{4}+c){‖w‖}^{\mathrm{2}}+{‖\frac{\partial w^e}{\partial x}‖}^{\mathrm{2}}+{\left|\left[w^e\right]\right|}^{\mathrm{2}},& \end{array}(49)

对式（49）运用引理1和Cauchy-Schwarz不等式，得到

\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_a^b {\left|w\right|}^{\mathrm{2}}\mathrm{d}x\le c{‖w‖}^{\mathrm{2}}+c{h}^{\mathrm{2}k+\mathrm{2}},(50)

最后，可由三角不等式和引理1得到最优误差估计式（36）．

2.3 全离散方法及其稳定性分析

本节采用LDG方法结合SDC方法的策略，构造全离散方法，并给出时间二阶和三阶全离散方法的稳定性分析．

2.3.1 SDC方法

考虑如下一阶常微分方程模型

\frac{\mathrm{d}\psi }{\mathrm{d}t}=F_E\left(t,\psi \left(t\right)\right)+F_I\left(t,\psi \left(t\right)\right),t\in \left(\mathrm{0},T\right],(51)

记[\mathrm{0},T]={\bigcup }_{n=\mathrm{0}}^N \left[t_n,t_{n+\mathrm{1}}\right]，进一步通过在时间子区间\left[t_n,t_{n+\mathrm{1}}\right]上引入Chebyshev节点，将其剖分成p个小区间，即，并记\mathrm{\Delta }{t}_n=t_{n+\mathrm{1}}-t_n\mathrm{、}\mathrm{\Delta }{t}_{n,r}=t_{n,r+\mathrm{1}}-t_{n,r}\left(r=\mathrm{0},\mathrm{1},\dots ,p\right)．经过l次校正后的离散误差近似方程为

\begin{array}{ccc}& {\delta }_{n,r+\mathrm{1}}^{(l)}={\delta }_{n,r}^{(l)}+\mathrm{\Delta }{t}_{n,r}\left(F_E\left(t_{n,r},{\psi }_{n,r}^{(l)}+{\delta }_{n,r}^{(l)}\right)-F_E\left(t_{n,r},{\psi }_{n,r}^{(l)}\right)\right.& \\ & \left.+F_I\left(t_{n,r+\mathrm{1}},{\psi }_{n,r+\mathrm{1}}^{(l)}+{\delta }_{n,r+\mathrm{1}}^{(l)}\right)-F_I\left(t_{n,r+\mathrm{1}},{\psi }_{n,r+\mathrm{1}}^{(l)}\right)\right)& \\ & +I_r^{r+\mathrm{1}}\left({\psi }_n^{(l)}\right)-{\psi }_{n,r+\mathrm{1}}^{(l)}+{\psi }_{n,r}^{(l)},& \end{array}(52)

通过简单的代数运算，可得校正方程

\begin{array}{ccc}& {\psi }_{n,r+\mathrm{1}}^{(l+\mathrm{1})}={\psi }_{n,r}^{(l+\mathrm{1})}+\mathrm{\Delta }{t}_{n,r}\left(F_E\left(t_{n,r},{\psi }_{n,r}^{(l+\mathrm{1})}\right)-F_E\left(t_{n,r},{\psi }_{n,r}^{(l)}\right)\right.& \\ & \left.+F_I\left(t_{n,r+\mathrm{1}},{\psi }_{n,r+\mathrm{1}}^{(l+\mathrm{1})}\right)-F_I\left(t_{n,r+\mathrm{1}},{\psi }_{n,r+\mathrm{1}}^{(l)}\right)\right)+I_r^{r+\mathrm{1}}\left({\psi }_n^{(l)}\right),& \end{array}(53)

式中：I_r^{r+\mathrm{1}}\left({\psi }_n^{(l)}\right)为{\int }_{t_{n,r}}^{t_{n,r+\mathrm{1}}} F\left(\tau ,{\psi }^{(l)}(\tau )\right)\mathrm{d}\tau数值积分的近似，具体算法实现中可通过采用不同精度的数值积分公式进行数值近似．

选取p=\mathrm{1}，可得二阶SDC方法

I_{\mathrm{0}}^{\mathrm{1}}\left({\psi }_n^{(\mathrm{1})}\right)=\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{2}}F\left(t_{n,\mathrm{0}},{\psi }_{n,\mathrm{0}}^{(\mathrm{1})}\right)+\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{2}}F\left(t_{n,\mathrm{1}},{\psi }_{n,\mathrm{1}}^{(\mathrm{1})}\right),(54)

选取p=\mathrm{2}，可得三阶SDC方法

I_{\mathrm{0}}^{\mathrm{1}}\left({\psi }_n^{(l)}\right)=\frac{\mathrm{5}\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{24}}F\left(t_{n,\mathrm{0}},{\psi }_{n,\mathrm{0}}^{(l)}\right)+\frac{\mathrm{8}\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{24}}F\left(t_{n,\mathrm{1}},{\psi }_{n,\mathrm{1}}^{(l)}\right)-\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{24}}F\left(t_{n,\mathrm{2}},{\psi }_{n,\mathrm{2}}^{(l)}\right),l=\mathrm{1},\mathrm{2},(55)

I_{\mathrm{1}}^{\mathrm{2}}\left({\psi }_n^{(l)}\right)=-\frac{\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{24}}F\left(t_{n,\mathrm{0}},{\psi }_{n,\mathrm{0}}^{(l)}\right)+\frac{\mathrm{8}\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{24}}F\left(t_{n,\mathrm{1}},{\psi }_{n,\mathrm{1}}^{(l)}\right)+\frac{\mathrm{5}\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{24}}F\left(t_{n,\mathrm{2}},{\psi }_{n,\mathrm{2}}^{(l)}\right),l=\mathrm{1},\mathrm{2},(56)

式（54~56）中对应系数至p=\mathrm{5}的情形已在文献［8］中详细给出，不再赘述．

2.3.2 全离散方法

基于上述空间LDG及时间方向SDC方法，可以构造时空全离散方法，并使时间精度和空间精度相匹配．全离散算法的基本思想是在每个时间区间\left[t_n,t_{n+\mathrm{1}}\right]上通过采用一阶Euler方法进行SDC校正，达到对应精度后，开始下一个时间区间的计算．本文分别以时间二阶和三阶SDC给出对应的全离散方法，对应的思想可以推广到更高阶的时间全离散方法．

时间二阶全离散方法：已知u_{n,\mathrm{0}}^{(\mathrm{1})}=u_{n,\mathrm{0}}^{(\mathrm{2})}=u_n，求\left(u_{n,r}^{(l)},q_{n,r}^{(l)}\right)\in V_h\times V_h,l=\mathrm{1},\mathrm{2}，使得任意(\rho ,\phi )\in V_h\times V_h，成立

\begin{array}{c}{\left(u_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}-u_{n,\mathrm{0}}^{\left(\mathrm{1}\right)},\rho \right)}_{I_i}=-\mathrm{\Delta }{t}_{n,\mathrm{0}}\left({\left(q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)\\ +\mathrm{\Delta }{t}_{n,\mathrm{0}}\left({\left(f\left(u_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}\right),\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right),\end{array}(57)

\begin{array}{ccc}& {\left(u_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}-u_{n,\mathrm{0}}^{\left(\mathrm{2}\right)},\rho \right)}_{I_i}=-\mathrm{\Delta }{t}_{n,\mathrm{0}}\left({\left(q_{n,\mathrm{1}}^{\left(\mathrm{2}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{2}}\left({\left(q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & -\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{2}}\left({\left(q_{n,\mathrm{0}}^{\left(\mathrm{1}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{2}}\left({\left(f\left(u_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}\right),{\rho }_x\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{2}}\left({\left(f\left(u_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}\right),{\rho }_x\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right),& \end{array}(58)

{\left(q_{n,\mathrm{1}}^{(\mathrm{1})},\phi \right)}_{I_i}=-{\left(u_{n,\mathrm{1}}^{(\mathrm{1})},\frac{\partial \phi }{\partial x}\right)}_{I_i}+{\left(\widehat{u_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}}{\phi }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}-{\left(\widehat{u_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}}{\phi }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}},(59)

{\left(q_{n,\mathrm{1}}^{(\mathrm{2})},\phi \right)}_{I_i}=-{\left(u_{n,\mathrm{1}}^{(\mathrm{2})},\frac{\partial \phi }{\partial x}\right)}_{I_i}+{\left(\widehat{u_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}}{\phi }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}-{\left(\widehat{u_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}}{\phi }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}},(60)

进一步令u_{n+\mathrm{1}}=u_{n,\mathrm{1}}^{(\mathrm{2})}\mathrm{、}{q}_{n+\mathrm{1}}=q_{n,\mathrm{1}}^{(\mathrm{2})}．

时间三阶全离散方法：已知u_{n,\mathrm{0}}^{(\mathrm{1})}=u_{n,\mathrm{0}}^{(\mathrm{2})}=u_{n,\mathrm{0}}^{(\mathrm{3})}=u_n，求\left(u_{n,r}^{(l)},q_{n,r}^{(l)}\right)\in V_h\times V_h,l=\mathrm{1},\mathrm{2},\mathrm{3}，使得任意(\rho ,\phi )\in V_h\times V_h，成立

\begin{array}{ccc}& {\left(u_{n,\mathrm{1}}^{(\mathrm{1})}-u_{n,\mathrm{0}}^{(\mathrm{1})},\rho \right)}_{I_i}=-\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{2}}\left({\left(q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{2}}\left({\left(f\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right),\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right),& \end{array}(61)

\begin{array}{ccc}& {\left(u_{n,\mathrm{2}}^{(\mathrm{1})}-u_{n,\mathrm{1}}^{(\mathrm{1})},\rho \right)}_{I_i}=-\frac{\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{2}}\left({\left(q_{n,\mathrm{2}}^{\left(\mathrm{1}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{2}}^{\left(\mathrm{1}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{2}}^{\left(\mathrm{1}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{2}}\left({\left(f\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right),\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right),& \end{array}(62)

\begin{array}{ccc}& {\left(u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{0}}^{(\mathrm{2})},\rho \right)}_{I_i}=-\frac{\mathrm{5}\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{24}}\left({\left(q_{n,\mathrm{0}}^{\left(\mathrm{1}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{6}}\left({\left(q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{24}}\left({\left(q_{n,\mathrm{2}}^{\left(\mathrm{1}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{2}}^{\left(\mathrm{1}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{2}}^{\left(\mathrm{1}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & -\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{3}}\left({\left(q_{n,\mathrm{1}}^{\left(\mathrm{2}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{5}\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{24}}\left({\left(f\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right),\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{3}}\left({\left(f\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right),\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & -\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{24}}\left({\left(f\left(u_{n,\mathrm{2}}^{(\mathrm{1})}\right),\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{2}}^{(\mathrm{1})}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{2}}^{(\mathrm{1})}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right),& (\mathrm{63})\end{array}

\begin{array}{ccc}& {\left(u_{n,\mathrm{2}}^{(\mathrm{2})}-u_{n,\mathrm{1}}^{(\mathrm{2})},\rho \right)}_{I_i}=\frac{\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{24}}\left({\left(q_{n,\mathrm{0}}^{\left(\mathrm{1}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & -\frac{\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{3}}\left({\left(q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{7}\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{24}}\left({\left(q_{n,\mathrm{2}}^{\left(\mathrm{1}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{2}}^{\left(\mathrm{1}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{2}}^{\left(\mathrm{1}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & -\frac{\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{24}}\left({\left(f\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right),\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & -\frac{\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{6}}\left({\left(f\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right),\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{5}\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{24}}\left({\left(f\left(u_{n,\mathrm{2}}^{(\mathrm{1})}\right),\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{2}}^{(\mathrm{1})}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{2}}^{(\mathrm{1})}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & -\frac{\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{2}}\left({\left(q_{n,\mathrm{2}}^{\left(\mathrm{2}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{2}}^{\left(\mathrm{2}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{2}}^{\left(\mathrm{2}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{2}}\left({\left(f\left(u_{n,\mathrm{1}}^{(\mathrm{2})}\right),\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{1}}^{(\mathrm{2})}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{1}}^{(\mathrm{2})}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right),& (\mathrm{64})\end{array}

\begin{array}{ccc}& {\left(u_{n,\mathrm{1}}^{\left(\mathrm{3}\right)}-u_{n,\mathrm{0}}^{\left(\mathrm{3}\right)},\rho \right)}_{I_i}=-\frac{\mathrm{5}\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{24}}\left({\left(q_{n,\mathrm{0}}^{\left(\mathrm{2}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{0}}^{\left(\mathrm{2}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{0}}^{\left(\mathrm{2}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{6}}\left({\left(q_{n,\mathrm{1}}^{\left(\mathrm{2}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{24}}\left({\left(q_{n,\mathrm{2}}^{\left(\mathrm{2}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{2}}^{\left(\mathrm{2}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{2}}^{\left(\mathrm{2}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & -\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{2}}\left({\left(q_{n,\mathrm{1}}^{\left(\mathrm{3}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{3}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{3}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{5}\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{24}}\left({\left(f\left(u_{n,\mathrm{0}}^{\left(\mathrm{2}\right)}\right),\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{0}}^{\left(\mathrm{2}\right)}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{0}}^{\left(\mathrm{2}\right)}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{3}}\left({\left(f\left(u_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}\right),\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & -\frac{\mathrm{\Delta }{t}_{n,\mathrm{0}}}{\mathrm{24}}\left({\left(f\left(u_{n,\mathrm{2}}^{\left(\mathrm{2}\right)}\right),\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{2}}^{\left(\mathrm{2}\right)}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{2}}^{\left(\mathrm{2}\right)}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right),& (\mathrm{65})\end{array}

\begin{array}{ccc}& {\left(u_{n,\mathrm{2}}^{(\mathrm{3})}-u_{n,\mathrm{1}}^{(\mathrm{3})},\rho \right)}_{I_i}=\frac{\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{24}}\left({\left(q_{n,\mathrm{0}}^{\left(\mathrm{2}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{0}}^{\left(\mathrm{2}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{0}}^{\left(\mathrm{2}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & -\frac{\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{3}}\left({\left(q_{n,\mathrm{1}}^{\left(\mathrm{2}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{7}\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{24}}\left({\left(q_{n,\mathrm{2}}^{\left(\mathrm{2}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{2}}^{\left(\mathrm{2}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{2}}^{\left(\mathrm{2}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & -\frac{\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{24}}\left({\left(f\left(u_{n,\mathrm{0}}^{(\mathrm{2})}\right),\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{0}}^{(\mathrm{2})}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{0}}^{(\mathrm{2})}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & -\frac{\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{6}}\left({\left(f\left(u_{n,\mathrm{1}}^{(\mathrm{2})}\right),\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{1}}^{(\mathrm{2})}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{1}}^{(\mathrm{2})}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{5}\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{24}}\left({\left(f\left(u_{n,\mathrm{2}}^{(\mathrm{2})}\right),\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{2}}^{(\mathrm{2})}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{2}}^{(\mathrm{2})}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & -\frac{\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{2}}\left({\left(q_{n,\mathrm{2}}^{\left(\mathrm{3}\right)},\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{q_{n,\mathrm{2}}^{\left(\mathrm{3}\right)}}{\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{q_{n,\mathrm{2}}^{\left(\mathrm{3}\right)}}{\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right)& \\ & +\frac{\mathrm{\Delta }{t}_{n,\mathrm{1}}}{\mathrm{2}}\left({\left(f\left(u_{n,\mathrm{1}}^{(\mathrm{3})}\right),\frac{\partial \rho }{\partial x}\right)}_{I_i}-{\left(\widehat{f}\left(u_{n,\mathrm{1}}^{(\mathrm{3})}\right){\rho }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}+{\left(\widehat{f}\left(u_{n,\mathrm{1}}^{(\mathrm{3})}\right){\rho }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}}\right),& (\mathrm{66})\end{array}

{\left(q_{n,r}^{(l)},\phi \right)}_{I_i}=-{\left(u_{n,r}^{\left(l\right)},\frac{\partial \phi }{\partial x}\right)}_{I_i}+{\left(\widehat{u_{n,r}^{\left(l\right)}}{\phi }^{-}\right)}_{i+\frac{\mathrm{1}}{\mathrm{2}}}-{\left(\widehat{u_{n,r}^{\left(l\right)}}{\phi }^{+}\right)}_{i-\frac{\mathrm{1}}{\mathrm{2}}},l=\mathrm{1},\mathrm{2},\mathrm{3},r=\mathrm{1},\mathrm{2},(67)

进一步令u_{n+\mathrm{1}}=u_{n,\mathrm{2}}^{(\mathrm{3})}\mathrm{、}{q}_{n+\mathrm{1}}=q_{n,\mathrm{2}}^{(\mathrm{3})}．

2.3.3 稳定性分析

下面给出二阶全离散方法的稳定性分析，三阶全离散方法的稳定性同理可得，由于证明过程繁琐，故不再列出，类似的推导可见文献［8］．

为方便表述，将二阶全离散方法写为算子形式

\left(u_{n,\mathrm{1}}^{(\mathrm{1})}-u_{n,\mathrm{0}}^{(\mathrm{1})},\rho \right)=\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(f\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};\rho \right)-\mathrm{\Delta }{t}_{n,\mathrm{0}}{\overline{B}}^{+}\left(q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)};\rho \right),(68)

\begin{array}{l}\left(u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{0}}^{(\mathrm{2})},\rho \right)=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(f\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right);u_{n,\mathrm{1}}^{(\mathrm{1})};\rho \right)+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(f;u_{n,\mathrm{0}}^{(\mathrm{2})};\rho \right)\\ +\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{\overline{B}}^{+}\left(q_{n,\mathrm{1}}^{(\mathrm{1})};\rho \right)-\mathrm{\Delta }{t}_{n,\mathrm{0}}{\overline{B}}^{+}\left(q_{n,\mathrm{1}}^{(\mathrm{2})};\rho \right)-\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{\overline{B}}^{+}\left(q_{n,\mathrm{0}}^{(\mathrm{1})};\rho \right),\end{array}(69)

\left(q_{n,\mathrm{1}}^{(\mathrm{1})},\phi \right)=-{\tilde{B}}^{-}\left(u_{n,\mathrm{1}}^{\left(\mathrm{1}\right)};\phi \right),(70)

\left(q_{n,\mathrm{1}}^{(\mathrm{2})},\phi \right)=-{\tilde{B}}^{-}\left(u_{n,\mathrm{1}}^{\left(\mathrm{2}\right)};\phi \right),(71)

由式（16~17）的性质，可推出{\overline{B}}^{+}\left(\phi ;u_{n,\mathrm{1}}^{(l)}\right)=-{\tilde{B}}^{-}\left(u_{n,\mathrm{1}}^{(l)};\phi \right)=\left(q_{n,\mathrm{1}}^{(l)},\phi \right),{\overline{B}}^{+}\left(q_{n,\mathrm{1}}^{(l)};u_{n,\mathrm{1}}^{(l)}\right)={‖q_{n,\mathrm{1}}^{(l)}‖}^{\mathrm{2}},l=\mathrm{1},\mathrm{2}．

定理 2 假设u_{n+\mathrm{1}}=u_{n,\mathrm{1}}^{(\mathrm{2})}是全离散方法（68~71）的解，可得

‖u_{n+\mathrm{1}}‖\le ‖u_n‖,n=\mathrm{0},\mathrm{1},\cdots ,N,(72)

其中\mathrm{\Delta }{t}_{n,\mathrm{0}}=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_n\left(\mathrm{c}\mathrm{o}\mathrm{s}\frac{p-\mathrm{1}}{p}\pi +\mathrm{1}\right)\le \frac{\mathrm{1}}{\mathrm{8}}\frac{\mathrm{1}}{c_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}}，即\mathrm{\Delta }{t}_n\le \frac{\mathrm{1}}{\mathrm{4}\left(\mathrm{c}\mathrm{o}\mathrm{s}\frac{p-\mathrm{1}}{p}\pi +\mathrm{1}\right)}\frac{\mathrm{1}}{c_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}}．

证明式（68）中取检验函数\rho =u_{n,\mathrm{1}}^{(\mathrm{1})}，式（69）中取检验函数\rho =u_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{u}_{n,\mathrm{0}}^{(\mathrm{1})}，利用式（28）及算子{\overline{B}}^{+}的性质，可得

\frac{\mathrm{1}}{\mathrm{2}}{‖u_{n,\mathrm{1}}^{(\mathrm{1})}-u_{n,\mathrm{0}}^{(\mathrm{1})}‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{‖u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}-\frac{\mathrm{1}}{\mathrm{2}}{‖u_{n,\mathrm{0}}^{(\mathrm{1})}‖}^{\mathrm{2}}=\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(f\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{1})}\right)-\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖q_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}},(73)

\begin{array}{c}\frac{\mathrm{9}}{\mathrm{4}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}‖}^{\mathrm{2}}-\frac{\mathrm{3}}{\mathrm{2}}{‖u_{n,\mathrm{0}}^{(\mathrm{2})}‖}^{\mathrm{2}}-\frac{\mathrm{3}}{\mathrm{4}}{‖u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}+E_{\mathrm{1}}=-\mathrm{3}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖q_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{n,\mathrm{0}}^{(\mathrm{1})}‖}^{\mathrm{2}}\\ +\frac{\mathrm{5}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\left(q_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{n,\mathrm{0}}^{(\mathrm{1})},q_{n,\mathrm{1}}^{(\mathrm{1})}\right)+\frac{\mathrm{3}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(f\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right);u_{n,\mathrm{1}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{u}_{n,\mathrm{0}}^{(\mathrm{1})}\right)\\ +\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(f\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{u}_{n,\mathrm{0}}^{(\mathrm{1})}\right),\end{array}(74)

其中

E_{\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{4}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{0}}^{(\mathrm{2})}‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}}{‖u_{n,\mathrm{1}}^{(\mathrm{1})}-u_{n,\mathrm{0}}^{(\mathrm{1})}‖}^{\mathrm{2}}.(75)

进一步，式（73）乘\frac{\mathrm{3}}{\mathrm{2}}加式（74），消去{‖u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}，可得

\frac{\mathrm{9}}{\mathrm{4}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}‖}^{\mathrm{2}}-\frac{\mathrm{9}}{\mathrm{4}}{‖u_{n,\mathrm{0}}^{(\mathrm{2})}‖}^{\mathrm{2}}+E=R_B+R_H,(76)

其中

E=\frac{\mathrm{1}}{\mathrm{4}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{0}}^{(\mathrm{2})}‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}+{‖u_{n,\mathrm{1}}^{(\mathrm{1})}-u_{n,\mathrm{0}}^{(\mathrm{1})}‖}^{\mathrm{2}},(77)

R_B=-\mathrm{3}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖q_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{n,\mathrm{0}}^{(\mathrm{1})}‖}^{\mathrm{2}}+\frac{\mathrm{5}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\left(q_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{n,\mathrm{0}}^{(\mathrm{1})},q_{n,\mathrm{1}}^{(\mathrm{1})}\right)-\frac{\mathrm{3}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖q_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}},(78)

\begin{array}{c}{R}_H=\frac{\mathrm{3}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(f\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right);u_{n,\mathrm{1}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{u}_{n,\mathrm{0}}^{(\mathrm{1})}\right)+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(f\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{u}_{n,\mathrm{0}}^{(\mathrm{1})}\right)\\ +\frac{\mathrm{3}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(f\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{1})}\right),\end{array}(79)

E是一个非负的量，对式（76）的能量分析没有影响．利用式（28），R_B可转换为

R_B=-\mathrm{3}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖q_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}‖}^{\mathrm{2}}+\frac{\mathrm{5}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\left(q_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{n,\mathrm{0}}^{\left(\mathrm{1}\right)},q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}\right)-\frac{\mathrm{3}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖q_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}‖}^{\mathrm{2}}

=-\mathrm{\Delta }{t}_{n,\mathrm{0}}{\int }_a^b {\alpha }^T\mathit{A}\alpha \mathrm{d}x，(80)

式中：\alpha ={\left(q_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{n,\mathrm{0}}^{(\mathrm{1})},q_{n,\mathrm{1}}^{(\mathrm{1})}\right)}^T;\mathit{A}=\left(\begin{array}{cc}\mathrm{3}& -\frac{\mathrm{5}}{\mathrm{4}}\\ -\frac{\mathrm{5}}{\mathrm{4}}& \frac{\mathrm{3}}{\mathrm{2}}\end{array}\right) ．易知\mathit{A}为正定矩阵，且R_B\le \mathrm{0}，对式（76）的能量分析也没有影响，故全离散方法的稳定性分析主要针对R_H进行

\begin{array}{lll}& R_H=\frac{\mathrm{3}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\left(H\left(f\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{1})}\right)-H\left(f\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right);u_{n,\mathrm{1}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{1})}\right)\right)& \\ & +\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\left(H\left(f\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{u}_{n,\mathrm{0}}^{(\mathrm{1})}\right)-H\left(f\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right);u_{n,\mathrm{1}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{u}_{n,\mathrm{0}}^{(\mathrm{1})}\right)\right)& \\ & +\mathrm{2}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(f\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right);u_{n,\mathrm{1}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{1}}^{(\mathrm{1})}\right)+\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(f\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right);u_{n,\mathrm{1}}^{(\mathrm{1})};u_{n,\mathrm{0}}^{(\mathrm{1})}-u_{n,\mathrm{1}}^{(\mathrm{1})}\right)& \\ & +\frac{\mathrm{9}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(f\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right);u_{n,\mathrm{1}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{1})}\right)& \\ & \le \frac{\mathrm{51}}{\mathrm{8}}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖u_{n,\mathrm{0}}^{(\mathrm{1})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}+\mathrm{4}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}& \\ & +\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\left({‖q_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}+{‖q_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{q}_{n,\mathrm{0}}^{(\mathrm{1})}‖}^{\mathrm{2}}\right)& \\ & =\frac{\mathrm{51}}{\mathrm{8}}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖u_{n,\mathrm{0}}^{(\mathrm{1})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}+\mathrm{4}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}& \\ & +\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{\int }_a^b {\alpha }^T\alpha \mathrm{d}x,& \end{array}(81)

因\mathit{A}-\frac{\mathrm{1}}{\mathrm{2}}\mathit{I}为正定矩阵，由Young不等式和引理2，可得

-\mathrm{\Delta }{t}_{n,\mathrm{0}}{\int }_a^b {\alpha }^T\left(\mathit{A}-\frac{\mathrm{1}}{\mathrm{2}}\mathit{I}\right)\alpha \mathrm{d}x\le \mathrm{0},(82)

结合式（76~77）、式（80~81）可得

\begin{array}{ccc}& \left.\frac{\mathrm{9}}{\mathrm{4}}\left({‖u_{n,\mathrm{1}}^{(\mathrm{2})}‖}^{\mathrm{2}}\right.-{‖u_{n,\mathrm{0}}^{(\mathrm{1})}‖}^{\mathrm{2}}\right)+\frac{\mathrm{1}}{\mathrm{4}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{0}}^{(\mathrm{2})}‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}+{‖u_{n,\mathrm{1}}^{(\mathrm{1})}-u_{n,\mathrm{0}}^{(\mathrm{1})}‖}^{\mathrm{2}}& \\ & \le \frac{\mathrm{51}}{\mathrm{8}}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖u_{n,\mathrm{0}}^{(\mathrm{1})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}+\mathrm{4}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}},& \end{array}(83)

若\frac{\mathrm{51}}{\mathrm{8}}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\le \mathrm{1}且\mathrm{4}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\le \frac{\mathrm{1}}{\mathrm{2}}，即\mathrm{\Delta }{t}_{n,\mathrm{0}}\le \frac{\mathrm{1}}{\mathrm{8}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}}，则可得‖u_{n,\mathrm{1}}^{(\mathrm{2})}‖\le ‖u_{n,\mathrm{0}}^{(\mathrm{1})}‖．注意到u_{n+\mathrm{1}}=u_{n,\mathrm{1}}^{(\mathrm{2})}\mathrm{、}{u}_n=u_{n,\mathrm{0}}^{(\mathrm{1})}，即‖u_{n+\mathrm{1}}‖\le ‖u_n‖．

3 二维离散方法的数值分析

本节将上述方法推广到空间二维情形，并给出对应的全离散方法及稳定性分析．

3.1 二维黏性Burgers方程及空间半离散变分问题

给定二维有界凸区域\mathrm{\Omega }，其边界充分光滑．考虑二维黏性Burgers方程

u_t+\nabla ·\mathbf{F}(u)-\mathrm{\Delta }u=\mathrm{0},(x,t)\in \mathrm{\Omega }\times (\mathrm{0},T],(84)

式中：未知量为u(x,t)=u\left(x_{\mathrm{1}},x_{\mathrm{2}},t\right)；通量函数\mathbf{F}(u)=\left(f_{\mathrm{1}}(u),f_{\mathrm{2}}(u)\right)已知．假设{\mathrm{\Omega }}_h={\bigcup }_{m=\mathrm{1}}^M {\mathrm{ℐ}}_m是对区域\mathrm{\Omega }的一致三角形单元的离散化，{\mathrm{ℐ}}_m为剖分单元，这里假设\mathrm{\Omega }与{\mathrm{\Omega }}_h之间的几何逼近误差可以忽略不计，e为单元上的边，\mathrm{\Gamma }为所有单元内部边界的集合．进一步引入分片Sobolev空间，V_{\mathrm{1}}=\left\{v\in L^{\mathrm{2}}(\mathrm{\Omega })|v{|}_{{\mathrm{ℐ}}_m}\in H^{\mathrm{1}}\left({\mathrm{ℐ}}_m\right),m=\mathrm{1},\mathrm{2},\dots ,M\right\}，{\mathit{V}}_{\mathrm{2}}=\left\{v\in L^{\mathrm{2}}{(\mathrm{\Omega })}^{\mathrm{2}}|v{|}_{{\mathrm{ℐ}}_m}\in H^{\mathrm{1}}{\left({\mathrm{ℐ}}_m\right)}^{\mathrm{2}},m=\mathrm{1},\mathrm{2},\dots ,M\right\}；对应的有限维逼近空间V_{\mathrm{1}h}=\left\{v\in L^{\mathrm{2}}(\mathrm{\Omega })\right.|v{|}_{{\mathrm{ℐ}}_m}\in \left.P^k\left({\mathrm{ℐ}}_m\right),m=\mathrm{1},\mathrm{2},\dots ,M\right\},{\mathit{V}}_{\mathrm{2}h}=\left\{v\in L^{\mathrm{2}}{(\mathrm{\Omega })}^{\mathrm{2}}|v{|}_{{\mathrm{ℐ}}_m}\in P^k{\left({\mathrm{ℐ}}_m\right)}^{\mathrm{2}},m=\mathrm{1},\mathrm{2},\cdots ,M\right\}．

接下来，基于空间半离散LDG方法的思想，引入辅助变量\mathit{q}，将式（84）转换为等价的一阶系统

u_t+\nabla ·(\mathbf{F}(u)-\mathit{q})=\mathrm{0},(85)

\mathit{q}-\nabla u=\mathrm{0},(86)

对应的空间半离散LDG变分问题为：求\left(u_h,{\mathit{q}}_h\right)=\left(u_h(\cdot ,t),{\mathit{q}}_h(\cdot ,t)\right)\in V_{\mathrm{1}h}\times {\mathit{V}}_{\mathrm{2}h}，使得对任意(\rho ,\mathit{\phi })\in V_{\mathrm{1}h}\times {\mathit{V}}_{\mathrm{2}h}，成立

{\left({\left(u_h\right)}_t,\rho \right)}_{\mathrm{\Omega }}=H\left(\mathbf{F}\left(u_h\right);u_h;\rho \right)+\overline{B}\left({\mathit{q}}_h;\rho \right),(87)

{\left({\mathit{q}}_h,\mathit{\phi }\right)}_{\mathrm{\Omega }}=\tilde{B}\left(u_h;\mathit{\phi }\right),(88)

其中

H\left(\mathbf{F}\left(u_h\right);u_h;\rho \right)={\sum }_{m=\mathrm{1}}^M H_{{\mathrm{ℐ}}_m}\left(\mathbf{F}\left(u_h\right);u_h;\rho \right)={\sum }_{m=\mathrm{1}}^M \left({\left(\mathbf{F}\left(u_h\right),\nabla \rho \right)}_{{\mathrm{ℐ}}_m}-{〈\widehat{\mathbf{F}}\left(u_h\right)\cdot {\mathit{n}}_{{\mathrm{ℐ}}_m},\rho 〉}_{\partial {\mathrm{ℐ}}_m}\right),(89)

\overline{B}\left({\mathit{q}}_h;\rho \right)={\sum }_{m=\mathrm{1}}^M {\overline{B}}_{{\mathrm{ℐ}}_m}\left({\mathit{q}}_h;\rho \right)={\sum }_{m=\mathrm{1}}^M \left(-{\left({\mathit{q}}_h,\nabla \rho \right)}_{{\mathrm{ℐ}}_m}+{〈{\widehat{\mathit{q}}}_h\cdot {\mathit{n}}_{{\mathrm{ℐ}}_m},\rho 〉}_{\partial {\mathrm{ℐ}}_m}\right),(90)

\tilde{B}\left(u_h;\mathit{\phi }\right)={\sum }_{m=\mathrm{1}}^M {\tilde{B}}_{{\mathrm{ℐ}}_m}\left(u_h;\mathit{\phi }\right)={\sum }_{m=\mathrm{1}}^M \left(-{\left(u_h,\nabla \cdot \mathit{\phi }\right)}_{{\mathrm{ℐ}}_m}+{〈{\widehat{u}}_h,\mathit{\phi }\cdot {\mathit{n}}_{{\mathrm{ℐ}}_m}〉}_{\partial {\mathrm{ℐ}}_m}\right),(91)

式中：\widehat{\mathbf{F}}\left(u_h\right)\mathrm{、}{\widehat{\mathit{q}}}_h和{\widehat{u}}_h为数值通量；{\mathit{n}}_{{\mathrm{ℐ}}_m}为单元{\mathrm{ℐ}}_m边界上的外法向量；{(u,\rho )}_{{\mathrm{ℐ}}_m}={\int }_{{\mathit{ℐ}}_m} u\rho \mathrm{d}x\mathrm{d}y\mathrm{、}{(\mathit{q},\nabla \rho )}_{{\mathrm{ℐ}}_m}={\int }_{{\mathit{ℐ}}_m} \mathit{q}\cdot \nabla \rho \mathrm{d}x\mathrm{d}y\mathrm{、}{〈{\mathit{q}}_h\cdot {\mathit{n}}_{{\mathrm{ℐ}}_m},\rho 〉}_{\partial {\mathrm{ℐ}}_m}={\int }_{\partial {\mathit{ℐ}}_m} {\mathit{q}}_h\cdot {\mathit{n}}_{{\mathrm{ℐ}}_m}\rho \mathrm{d}s分别表示单元和单元边界上的内积．

3.2 全离散方法

空间半离散LDG变分问题是一个微分代数问题，可使用前述SDC方法开展进一步的时间方向离散，结合本文提出的二阶和三阶SDC方法，二维黏性Burgers方程的全离散方法构造如下．

时间二阶全离散方法：已知u_{n,\mathrm{0}}^{(\mathrm{1})}=u_{n,\mathrm{0}}^{(\mathrm{2})}=u_n，求\left(u_{n,r}^{(l)},{\mathit{q}}_{n,r}^{(l)}\right)\in V_{\mathrm{1}h}\times {\mathit{V}}_{\mathrm{2}h},l=\mathrm{1},\mathrm{2}，使得任意(\rho ,\mathit{\phi })\in V_{\mathrm{1}h}\times {\mathit{V}}_{\mathrm{2}h}，成立

{\left(u_{n,\mathrm{1}}^{(\mathrm{1})}-u_{n,\mathrm{0}}^{(\mathrm{1})},\rho \right)}_{\mathrm{\Omega }}=\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};\rho \right)+\mathrm{\Delta }{t}_{n,\mathrm{0}}\overline{B}\left({\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{1})};\rho \right),(92)

\begin{array}{c}{\left(u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{0}}^{(\mathrm{2})},\rho \right)}_{\mathrm{\Omega }}=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right);u_{n,\mathrm{1}}^{(\mathrm{1})};\rho \right)+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};\rho \right)\\ +\mathrm{\Delta }{t}_{n,\mathrm{0}}\overline{B}\left({\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{2})};\rho \right)-\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\overline{B}\left({\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{1})};\rho \right)+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\overline{B}\left({\mathit{q}}_{n,\mathrm{0}}^{(\mathrm{1})};\rho \right),\end{array}(93)

{\left({\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{1})},\mathit{\phi }\right)}_{\mathrm{\Omega }}=\tilde{B}\left(u_{n,\mathrm{1}}^{(\mathrm{1})};\mathit{\phi }\right),(94)

{\left({\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{2})},\mathit{\phi }\right)}_{\mathrm{\Omega }}=\tilde{B}\left(u_{n,\mathrm{1}}^{(\mathrm{2})};\mathit{\phi }\right),(95)

这里u_{n+\mathrm{1}}=u_{n,\mathrm{1}}^{(\mathrm{2})}\mathrm{、}{\mathit{q}}_{n+\mathrm{1}}={\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{2})}．

时间三阶全离散方法：已知u_{n,\mathrm{0}}^{(\mathrm{1})}=u_{n,\mathrm{0}}^{(\mathrm{2})}=u_{n,\mathrm{0}}^{(\mathrm{3})}=u_n，求\left(u_{n,r}^{(l)},{\mathit{q}}_{n,r}^{(l)}\right)\in V_{\mathrm{1}h}\times {\mathit{V}}_{\mathrm{2}h},l=\mathrm{1},\mathrm{2},\mathrm{3}，使得任意(\rho ,\mathit{\phi })\in V_{\mathrm{1}h}\times {\mathit{V}}_{\mathrm{2}h}，成立

{\left(u_{n,\mathrm{1}}^{(\mathrm{1})}-u_{n,\mathrm{0}}^{(\mathrm{1})},\rho \right)}_{\mathrm{\Omega }}=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};\rho \right)+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\overline{B}\left({\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{1})};\rho \right),(96)

{\left(u_{n,\mathrm{2}}^{(\mathrm{1})}-u_{n,\mathrm{1}}^{(\mathrm{1})},\rho \right)}_{\mathrm{\Omega }}=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right);u_{n,\mathrm{1}}^{(\mathrm{1})};\rho \right)+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{1}}\overline{B}\left({\mathit{q}}_{n,\mathrm{2}}^{(\mathrm{1})};\rho \right),(97)

\begin{array}{c}{\left(u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{0}}^{(\mathrm{2})},\rho \right)}_{\mathrm{\Omega }}=\frac{\mathrm{5}}{\mathrm{24}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};\rho \right)+\frac{\mathrm{1}}{\mathrm{3}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right);u_{n,\mathrm{1}}^{(\mathrm{1})};\rho \right)\\ -\frac{\mathrm{1}}{\mathrm{24}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{2}}^{(\mathrm{1})}\right);u_{n,\mathrm{2}}^{(\mathrm{1})};\rho \right)+\frac{\mathrm{5}}{\mathrm{24}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\overline{B}\left({\mathit{q}}_{n,\mathrm{0}}^{(\mathrm{1})};\rho \right)\\ -\frac{\mathrm{1}}{\mathrm{6}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\overline{B}\left({\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{1})};\rho \right)-\frac{\mathrm{1}}{\mathrm{24}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\overline{B}\left({\mathit{q}}_{n,\mathrm{2}}^{(\mathrm{1})};\rho \right)+\frac{\mathrm{1}}{\mathrm{3}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\overline{B}\left({\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{2})};\rho \right),\end{array}(98)

\begin{array}{c}{\left(u_{n,\mathrm{2}}^{(\mathrm{2})}-u_{n,\mathrm{1}}^{(\mathrm{2})},\rho \right)}_{\mathrm{\Omega }}=-\frac{\mathrm{1}}{\mathrm{24}}\mathrm{\Delta }{t}_{n,\mathrm{1}}H\left(\mathbf{F}\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};\rho \right)-\frac{\mathrm{1}}{\mathrm{6}}\mathrm{\Delta }{t}_{n,\mathrm{1}}H\left(\mathbf{F}\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right);u_{n,\mathrm{1}}^{(\mathrm{1})};\rho \right)\\ +\frac{\mathrm{5}}{\mathrm{24}}\mathrm{\Delta }{t}_{n,\mathrm{1}}H\left(\mathbf{F}\left(u_{n,\mathrm{2}}^{(\mathrm{1})}\right);u_{n,\mathrm{2}}^{(\mathrm{1})};\rho \right)+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{1}}H\left(\mathbf{F}\left(u_{n,\mathrm{1}}^{(\mathrm{2})}\right);u_{n,\mathrm{1}}^{(\mathrm{2})};\rho \right)\\ -\frac{\mathrm{1}}{\mathrm{24}}\mathrm{\Delta }{t}_{n,\mathrm{1}}\overline{B}\left({\mathit{q}}_{n,\mathrm{0}}^{(\mathrm{1})};\rho \right)+\frac{\mathrm{1}}{\mathrm{3}}\mathrm{\Delta }{t}_{n,\mathrm{1}}\overline{B}\left({\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{1})};\rho \right)\\ -\frac{\mathrm{7}}{\mathrm{24}}\mathrm{\Delta }{t}_{n,\mathrm{1}}\overline{B}\left({\mathit{q}}_{n,\mathrm{2}}^{(\mathrm{1})};\rho \right)+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{1}}\overline{B}\left({\mathit{q}}_{n,\mathrm{2}}^{(\mathrm{2})};\rho \right),\end{array}(99)

\begin{array}{c}{\left(u_{n,\mathrm{1}}^{(\mathrm{3})}-u_{n,\mathrm{0}}^{(\mathrm{3})},\rho \right)}_{\mathrm{\Omega }}=\frac{\mathrm{5}}{\mathrm{24}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{0}}^{(\mathrm{2})}\right);u_{n,\mathrm{0}}^{(\mathrm{2})};\rho \right)+\frac{\mathrm{1}}{\mathrm{3}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{1}}^{(\mathrm{2})}\right);u_{n,\mathrm{1}}^{(\mathrm{2})};\rho \right)\\ -\frac{\mathrm{1}}{\mathrm{24}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{2}}^{(\mathrm{2})}\right);u_{n,\mathrm{2}}^{(\mathrm{2})};\rho \right)+\frac{\mathrm{5}}{\mathrm{24}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\overline{B}\left({\mathit{q}}_{n,\mathrm{0}}^{(\mathrm{2})};\rho \right)\\ -\frac{\mathrm{1}}{\mathrm{6}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\overline{B}\left({\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{2})};\rho \right)-\frac{\mathrm{1}}{\mathrm{24}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\overline{B}\left({\mathit{q}}_{n,\mathrm{2}}^{(\mathrm{2})};\rho \right)+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\overline{B}\left({\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{3})};\rho \right),\end{array}(100)

\begin{array}{c}{\left(u_{n,\mathrm{2}}^{(\mathrm{3})}-u_{n,\mathrm{1}}^{(\mathrm{3})},\rho \right)}_{\mathrm{\Omega }}=-\frac{\mathrm{1}}{\mathrm{24}}\mathrm{\Delta }{t}_{n,\mathrm{1}}H\left(\mathbf{F}\left(u_{n,\mathrm{0}}^{(\mathrm{2})}\right);u_{n,\mathrm{0}}^{(\mathrm{2})};\rho \right)-\frac{\mathrm{1}}{\mathrm{6}}\mathrm{\Delta }{t}_{n,\mathrm{1}}H\left(\mathbf{F}\left(u_{n,\mathrm{1}}^{(\mathrm{2})}\right);u_{n,\mathrm{1}}^{(\mathrm{2})};\rho \right)\\ +\frac{\mathrm{5}}{\mathrm{24}}\mathrm{\Delta }{t}_{n,\mathrm{1}}H\left(\mathbf{F}\left(u_{n,\mathrm{2}}^{(\mathrm{2})}\right);u_{n,\mathrm{2}}^{(\mathrm{2})};\rho \right)+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{1}}H\left(\mathbf{F}\left(u_{n,\mathrm{1}}^{(\mathrm{3})}\right);u_{n,\mathrm{1}}^{(\mathrm{3})};\rho \right)\\ -\frac{\mathrm{1}}{\mathrm{24}}\mathrm{\Delta }{t}_{n,\mathrm{1}}\overline{B}\left({\mathit{q}}_{n,\mathrm{0}}^{(\mathrm{2})};\rho \right)+\frac{\mathrm{1}}{\mathrm{3}}\mathrm{\Delta }{t}_{n,\mathrm{1}}\overline{B}\left({\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{2})};\rho \right)\\ -\frac{\mathrm{7}}{\mathrm{24}}\mathrm{\Delta }{t}_{n,\mathrm{1}}\overline{B}\left({\mathit{q}}_{n,\mathrm{2}}^{(\mathrm{2})};\rho \right)+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{1}}\overline{B}\left({\mathit{q}}_{n,\mathrm{2}}^{(\mathrm{3})};\rho \right),\end{array}(101)

{\left({\mathit{q}}_{n,r}^{(l)},\mathit{\phi }\right)}_{\mathrm{\Omega }}=\tilde{B}\left(u_{n,r}^{(l)};\mathit{\phi }\right),l=\mathrm{1},\mathrm{2},\mathrm{3},r=\mathrm{1},\mathrm{2},(102)

这里u_{n+\mathrm{1}}=u_{n,\mathrm{2}}^{(\mathrm{3})}、{\mathit{q}}_{n+\mathrm{1}}={\mathit{q}}_{n,\mathrm{2}}^{(\mathrm{3})}．

3.3 二阶全离散方法的稳定性分析

为开展二维黏性Burgers方程的全离散方法稳定性分析，首先将一维情形的引理1和引理2推广到二维情形，即引理3和引理4，类似的推导可见文献[\mathrm{8},\mathrm{15}]，本文只给出证明梗概．

引理 3 若u,v,\rho \in V_{\mathrm{1}h}且\mathbf{F}(u),\mathbf{F}(v)\in {\mathit{V}}_{\mathrm{2}h}，则满足如下关系

H(\mathbf{F}(u);u;u)\le \mathrm{0},(103)

|H(\mathbf{F}(u);u;\rho )|\le c_{\mathrm{0}}\left(\Vert \nabla u\Vert +c_{\mathrm{1}}{h}^{-\frac{\mathrm{1}}{\mathrm{2}}}|[u]{|}_{\mathrm{\Gamma }}\right)\Vert \rho \Vert,(104)

|H(\mathbf{F}(u);u;\rho )-H(\mathbf{F}(v);v;\rho )|\le c_{\mathrm{0}}\Vert u-v\Vert \left(\Vert \nabla \rho \Vert +c_{\mathrm{1}}{h}^{-\frac{\mathrm{1}}{\mathrm{2}}}|[\rho ]{|}_{\mathrm{\Gamma }}\right).(105)

证明 式（103）的证明可由式（85）的变分问题推出．为完成式（104~105）的证明，首先对非线性算子H进行分部积分，得出关于变量\nabla u\mathrm{、}[u]与H的等式；其次根据数值通量的一致性和Lipschitz连续性，并运用逆不等式和Cauchy-Schwarz不等式，对变量\nabla u\mathrm{、}[u]与H的等式进行放缩，可证得式（104~105）．

引理 4 假设\left(u_h,{\mathit{q}}_h\right)为半离散LDG变分问题（87~88）的解，即

{\left({\mathit{q}}_h,\mathit{\phi }\right)}_{\mathrm{\Omega }}=-\tilde{B}\left(u_h;\mathit{\phi }\right),\forall \mathit{\phi }\in {\mathit{V}}_{\mathrm{2}h},(106)

则下式成立

‖\nabla u_h‖+c_{\mathrm{1}}{h}^{-\frac{\mathrm{1}}{\mathrm{2}}}{\left|\left[u_h\right]\right|}_{\mathrm{\Gamma }}\le c_{\mathrm{2}}‖{\mathit{q}}_h‖,(107)

式中： {\left|\left[u_h\right]\right|}_{\mathrm{\Gamma }}={\left({\sum }_{e\in \Gamma } {‖u_h‖}_e^{\mathrm{2}}\right)}^{\frac{\mathrm{1}}{\mathrm{2}}} ．

证明 对{\left({\mathit{q}}_h,\mathit{\phi }\right)}_{{\mathrm{ℐ}}_m}进行分部积分，即

\begin{array}{ll}{\left({\mathit{q}}_h,\mathit{\phi }\right)}_{{\mathrm{ℐ}}_m}& =-{\left(u_h,\nabla \cdot \mathit{\phi }\right)}_{{\mathrm{ℐ}}_m}+{〈{\widehat{u}}_h,\mathit{\phi }\cdot {\mathit{n}}_{{\mathrm{ℐ}}_m}〉}_{\partial {\mathrm{ℐ}}_m}={\left(\nabla u_h,\mathit{\phi }\right)}_{{\mathrm{ℐ}}_m}-{〈u_h,\mathit{\phi }\cdot {\mathit{n}}_{{\mathrm{ℐ}}_m}〉}_{\partial {\mathrm{ℐ}}_m}+{〈{\widehat{u}}_h,\mathit{\phi }\cdot {\mathit{n}}_{{\mathrm{ℐ}}_m}〉}_{\partial {\mathrm{ℐ}}_m}\\ & ={\left(\nabla u_h,\mathit{\phi }\right)}_{{\mathrm{ℐ}}_m}-{〈\left[u_h\right],\mathit{\phi }\cdot {\mathit{n}}_{{\mathrm{ℐ}}_m}〉}_{\partial {\mathrm{ℐ}}_m^{-}},\end{array}(108)

由此得到\nabla u_h\mathrm{、}\left[u_h\right]与中间变量{\mathit{q}}_h的关系式．其次由参考单元到实体单元的仿射变换，取检验函数\phi为Legendre多项式，利用Legendre多项式的正交性质，对{\left({\mathit{q}}_h,\mathit{\phi }\right)}_{{\mathrm{ℐ}}_m}进行分析，即可得到引理4．下面给出二阶全离散方法（92~95）的稳定性．

定理 3 设u_{n+\mathrm{1}}=u_{n,\mathrm{1}}^{(\mathrm{2})}为全离散方法（92~95）的解，可得

‖u_{n+\mathrm{1}}‖\le ‖u_n‖,n=\mathrm{0},\mathrm{1},\dots ,N,(109)

其中\mathrm{\Delta }{t}_{n,\mathrm{0}}=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_n\left(\mathrm{c}\mathrm{o}\mathrm{s}\frac{p-\mathrm{1}}{p}\pi +\mathrm{1}\right)\le \frac{\mathrm{1}}{\mathrm{8}}\frac{\mathrm{1}}{c_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}}，即\mathrm{\Delta }{t}_n\le \frac{\mathrm{1}}{\mathrm{4}\left(\mathrm{c}\mathrm{o}\mathrm{s}\frac{p-\mathrm{1}}{p}\pi +\mathrm{1}\right)}\frac{\mathrm{1}}{c_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}}．

证明 在式（92~93）中取检验函数分别为\rho =u_{n,\mathrm{1}}^{(\mathrm{1})}\mathrm{、}\rho =u_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{u}_{n,\mathrm{0}}^{(\mathrm{1})}，并利用式（28）及算子\overline{B}的性质，可得

\frac{\mathrm{1}}{\mathrm{2}}{‖u_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}-u_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{‖u_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}‖}^{\mathrm{2}}-\frac{\mathrm{1}}{\mathrm{2}}{‖u_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}‖}^{\mathrm{2}}

=\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{1})}\right)-\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖{\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}},(110)

\begin{array}{c}\frac{\mathrm{9}}{\mathrm{4}}{‖u_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}‖}^{\mathrm{2}}-\frac{\mathrm{3}}{\mathrm{2}}{‖u_{n,\mathrm{0}}^{\left(\mathrm{2}\right)}‖}^{\mathrm{2}}-\frac{\mathrm{3}}{\mathrm{4}}{‖u_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}‖}^{\mathrm{2}}+E_{\mathrm{1}}\\ =-\mathrm{3}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖{\mathit{q}}_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}+\frac{\mathrm{1}}{\mathrm{2}}{\mathit{q}}_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}‖}^{\mathrm{2}}+\frac{\mathrm{5}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\left({\mathit{q}}_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}+\frac{\mathrm{1}}{\mathrm{2}}{\mathit{q}}_{n,\mathrm{0}}^{\left(\mathrm{1}\right)},{\mathit{q}}_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}\right)\\ +\frac{\mathrm{3}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}\right);u_{n,\mathrm{1}}^{\left(\mathrm{1}\right)};u_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}+\frac{\mathrm{1}}{\mathrm{2}}{u}_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}\right)\\ +\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{u}_{n,\mathrm{0}}^{(\mathrm{1})}\right),\end{array}(111)

其中

E_{\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{4}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{0}}^{(\mathrm{2})}‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}}{‖u_{n,\mathrm{1}}^{(\mathrm{1})}-u_{n,\mathrm{0}}^{(\mathrm{1})}‖}^{\mathrm{2}}.(112)

为构造能量方程，式（110）乘\frac{\mathrm{3}}{\mathrm{2}}加上式（111）消去{‖u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}，得到

\frac{\mathrm{9}}{\mathrm{4}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}‖}^{\mathrm{2}}-\frac{\mathrm{9}}{\mathrm{4}}{‖u_{n,\mathrm{0}}^{(\mathrm{2})}‖}^{\mathrm{2}}+E=R_{\mathbb{B}}+R_{\mathbb{H}},(113)

其中

E=\frac{\mathrm{1}}{\mathrm{4}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{0}}^{(\mathrm{2})}‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}+{‖u_{n,\mathrm{1}}^{(\mathrm{1})}-u_{n,\mathrm{0}}^{(\mathrm{1})}‖}^{\mathrm{2}},(114)

R_{\mathbb{B}}=-\mathrm{3}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖{\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{\mathit{q}}_{n,\mathrm{0}}^{(\mathrm{1})}‖}^{\mathrm{2}}+\frac{\mathrm{5}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\left({\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{\mathit{q}}_{n,\mathrm{0}}^{(\mathrm{1})},{\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{1})}\right)-\frac{\mathrm{3}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖{\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}},(115)

\begin{array}{c}{R}_{\mathbb{H}}=\frac{\mathrm{3}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}\right);u_{n,\mathrm{1}}^{\left(\mathrm{1}\right)};u_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}+\frac{\mathrm{1}}{\mathrm{2}}{u}_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}\right)\\ +\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{u}_{n,\mathrm{0}}^{(\mathrm{1})}\right)\\ +\frac{\mathrm{3}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{1})}\right),\end{array}(116)

E为非负项，对式（113）的能量分析没有影响，利用式（28），可将R_{\mathbb{B}}转换为

\begin{array}{lll}& R_{\mathbb{B}}=-\mathrm{3}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖{\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{\mathit{q}}_{n,\mathrm{0}}^{(\mathrm{1})}‖}^{\mathrm{2}}+\frac{\mathrm{5}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\left({\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{\mathit{q}}_{n,\mathrm{0}}^{(\mathrm{1})},{\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{1})}\right)-\frac{\mathrm{3}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖{\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}& \\ & =-\mathrm{\Delta }{t}_{n,\mathrm{0}}{\int }_{\mathrm{\Omega }} {\alpha }^T\mathit{A}\alpha \mathrm{d}x\mathrm{d}y,& \end{array}(117)

式中：\alpha ={\left({\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{\mathit{q}}_{n,\mathrm{0}}^{(\mathrm{1})},{\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{1})}\right)}^T;\mathit{A}=\left(\begin{array}{cc}\mathrm{3}& -\frac{\mathrm{5}}{\mathrm{4}}\\ -\frac{\mathrm{5}}{\mathrm{4}}& \frac{\mathrm{3}}{\mathrm{2}}\end{array}\right)．可知\mathit{A}为正定矩阵，即R_{\mathbb{B}}\le \mathrm{0}，对式（113）的能量分析同样没有影响，故全离散方法的稳定性分析主要针对R_{\mathbb{H}}进行

\begin{array}{lll}& R_{\mathbb{H}}=\frac{\mathrm{3}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\left(H\left(\mathbf{F}\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{1})}\right)-H\left(\mathbf{F}\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right);u_{n,\mathrm{1}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{1})}\right)\right)& \\ & +\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\left(H\left(\mathbf{F}\left(u_{n,\mathrm{0}}^{(\mathrm{1})}\right);u_{n,\mathrm{0}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{u}_{n,\mathrm{0}}^{(\mathrm{1})}\right)-H\left(\mathbf{F}\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right);u_{n,\mathrm{1}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{u}_{n,\mathrm{0}}^{(\mathrm{1})}\right)\right)& \\ & +\mathrm{2}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right);u_{n,\mathrm{1}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{1}}^{(\mathrm{1})}\right)+\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right);u_{n,\mathrm{1}}^{(\mathrm{1})};u_{n,\mathrm{0}}^{(\mathrm{1})}-u_{n,\mathrm{1}}^{(\mathrm{1})}\right)& \\ & +\frac{\mathrm{9}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}H\left(\mathbf{F}\left(u_{n,\mathrm{1}}^{(\mathrm{1})}\right);u_{n,\mathrm{1}}^{(\mathrm{1})};u_{n,\mathrm{1}}^{(\mathrm{1})}\right)& \\ & \le \frac{\mathrm{3}}{\mathrm{2}}{c}_{\mathrm{0}}{c}_{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}‖{\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{1})}‖‖u_{n,\mathrm{0}}^{(\mathrm{1})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖+\frac{\mathrm{1}}{\mathrm{2}}{c}_{\mathrm{0}}{c}_{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}‖{\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{\mathit{q}}_{n,\mathrm{0}}^{(\mathrm{1})}‖‖u_{n,\mathrm{0}}^{(\mathrm{1})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖& \\ & +\mathrm{2}{c}_{\mathrm{0}}{c}_{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}‖{\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{1})}‖‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖+c_{\mathrm{0}}{c}_{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}‖{\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{1})}‖‖u_{n,\mathrm{0}}^{(\mathrm{1})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖& \\ & \le \frac{\mathrm{51}}{\mathrm{8}}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖u_{n,\mathrm{0}}^{\left(\mathrm{1}\right)}-u_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}‖}^{\mathrm{2}}+\mathrm{4}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖u_{n,\mathrm{1}}^{\left(\mathrm{2}\right)}-u_{n,\mathrm{1}}^{\left(\mathrm{1}\right)}‖}^{\mathrm{2}}& \\ & +\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\left({‖{\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}+{‖{\mathit{q}}_{n,\mathrm{1}}^{(\mathrm{2})}+\frac{\mathrm{1}}{\mathrm{2}}{\mathit{q}}_{n,\mathrm{0}}^{(\mathrm{1})}‖}^{\mathrm{2}}\right)& \\ & =\frac{\mathrm{51}}{\mathrm{8}}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖u_{n,\mathrm{0}}^{(\mathrm{1})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}+\mathrm{4}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{\int }_{\mathrm{\Omega }} {\alpha }^T\alpha \mathrm{d}x\mathrm{d}y,& \end{array}(118)

因\left(\begin{array}{cc}\mathrm{3}& -\frac{\mathrm{5}}{\mathrm{4}}\\ -\frac{\mathrm{5}}{\mathrm{4}}& \frac{\mathrm{3}}{\mathrm{2}}\end{array}\right)-\frac{\mathrm{1}}{\mathrm{2}}\mathit{I}为正定矩阵，由Young不等式和引理4，可得

-\mathrm{\Delta }{t}_{n,\mathrm{0}}{\int }_{\Omega } {\alpha }^T\left(\left(\begin{array}{cc}\mathrm{3}& -\frac{\mathrm{5}}{\mathrm{4}}\\ -\frac{\mathrm{5}}{\mathrm{4}}& \frac{\mathrm{3}}{\mathrm{2}}\end{array}\right)-\frac{\mathrm{1}}{\mathrm{2}}\mathit{I}\right)\alpha \mathrm{d}x\mathrm{d}y\le \mathrm{0},(119)

联立方程（113~114）、方程（117~118），可得

\begin{array}{ccc}& \left.\frac{\mathrm{9}}{\mathrm{4}}\left({‖u_{n,\mathrm{1}}^{(\mathrm{2})}‖}^{\mathrm{2}}\right.-{‖u_{n,\mathrm{0}}^{(\mathrm{1})}‖}^{\mathrm{2}}\right)+\frac{\mathrm{1}}{\mathrm{4}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{0}}^{(\mathrm{2})}‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}+{‖u_{n,\mathrm{1}}^{(\mathrm{1})}-u_{n,\mathrm{0}}^{(\mathrm{1})}‖}^{\mathrm{2}}& \\ & \le \frac{\mathrm{51}}{\mathrm{8}}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖u_{n,\mathrm{0}}^{(\mathrm{1})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}}+\mathrm{4}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}{‖u_{n,\mathrm{1}}^{(\mathrm{2})}-u_{n,\mathrm{1}}^{(\mathrm{1})}‖}^{\mathrm{2}},& \end{array}(120)

若满足\frac{\mathrm{51}}{\mathrm{8}}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\le \mathrm{1}且\mathrm{4}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}\mathrm{\Delta }{t}_{n,\mathrm{0}}\le \frac{\mathrm{1}}{\mathrm{2}}，即\mathrm{\Delta }{t}_{n,\mathrm{0}}\le \frac{\mathrm{1}}{\mathrm{8}{c}_{\mathrm{0}}^{\mathrm{2}}{c}_{\mathrm{2}}^{\mathrm{2}}}，则可得‖u_{n,\mathrm{1}}^{(\mathrm{2})}‖\le ‖u_{n,\mathrm{0}}^{(\mathrm{1})}‖．注意到u_{n+\mathrm{1}}=u_{n,\mathrm{1}}^{(\mathrm{2})}、u_n=u_{n,\mathrm{0}}^{(\mathrm{1})}，即证得‖u_{n+\mathrm{1}}‖\le ‖u_n‖．

4 数值算例

本节针对所提时空高阶数值方法，给出具体的数值验证算例．由SDC方法的经典结论可知，若时间方向采用一阶推进格式，则时间方向校正l步可以达到时间误差阶l+\mathrm{1}．同时，空间方向采用k次插值，其空间L^{\mathrm{2}}误差阶为k+\mathrm{1}，由此得到理论上期待的全离散最优误差为‖u\left(t_n\right)-u_{n,l}^{(l)}‖=c\left(h^{k+\mathrm{1}}+{\tau }^{l+\mathrm{1}}\right)．以下算例选取不同黏性系数、不同插值次数k及校正次数l的匹配情形进行数值实验．

算例 1 一维解析解问题：考虑一维黏性Burgers方程

\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu \frac{{\partial }^{\mathrm{2}}u}{\partial x^{\mathrm{2}}},\forall (x,t)\in (\mathrm{0},\mathrm{1})\times (\mathrm{0},T],(121)

方程的精确解为u(x,t)=\frac{\mathrm{2}\nu \pi e^{-{\pi }^{\mathrm{2}}\nu t}\mathrm{s}\mathrm{i}\mathrm{n}(\pi x)}{\mathrm{2}+e^{-{\pi }^{\mathrm{2}}\nu t}\mathrm{c}\mathrm{o}\mathrm{s}(\pi x)}，初始条件和边界条件可由精确解得到．

为达到较好的时空误差阶数匹配，算例选取k=l，分别取值为1、2、3、4进行验证．表1和图1给出了不同黏性系数下采用LDG方法结合SDC方法求解黏性Burgers方程的数值结果．数值算例表明，对于不同的黏性系数取值，在单元插值次数与校正次数匹配的情形下，算法均能达到对应的误差阶数，该结果正是该算法设计的基本思想体现．

算例 2 小初值能量耗散问题：考虑一维黏性Burgers方程（121）在给定初值条件及齐次边界条件下的能量耗散问题，算例将针对不同黏性系数情形给出数值结果

u(x,\mathrm{0})=\pi \mathrm{s}\mathrm{i}{\mathrm{n}}^{\mathrm{2}}(\pi x),x\in (\mathrm{0},\mathrm{1}).(122)

图2展示了设定单元插值次数k=\mathrm{2}、时间校正次数l=\mathrm{2}情形下，不同时间步长和不同黏性系数的能量耗散演化过程，数值结果表明，对于齐次边界的小扰动初值问题，数值格式展现出良好的长时间数值耗散性与算法稳定性．

算例 3 二维解析解问题：考虑二维区域\mathrm{\Omega }={(\mathrm{0},\mathrm{3})}^{\mathrm{2}}上的黏性Burgers方程

\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+u\frac{\partial u}{\partial y}-\nu \left(\frac{{\partial }^{\mathrm{2}}u}{\partial x^{\mathrm{2}}}+\frac{{\partial }^{\mathrm{2}}u}{\partial y^{\mathrm{2}}}\right)=f,\forall (x,y)\in {(\mathrm{0},\pi )}^{\mathrm{2}},t\in (\mathrm{0},T],(123)

方程的精确解为u(x,t)={\mathrm{e}}^{-t}\mathrm{s}\mathrm{i}\mathrm{n}(x)\mathrm{s}\mathrm{i}\mathrm{n}(y)，初始条件和边界条件可由精确解得到．算例利用商业网格生成软件Gambit对求解区域\mathrm{\Omega }进行非结构化Delaunay三角网格的生成．通过设定4条边界最大网格边长参数h=\frac{\mathrm{1}}{N},N=\mathrm{8},\mathrm{20},\mathrm{32}，Gambit软件可生成拟均匀网格．图3展示了设定单元插值次数k=\mathrm{1}、时间校正次数l=\mathrm{1}情形下，不同空间网格配比N\times N及黏性系数时的数值解演化情况．该问题针对具有解析解的算例，可知在不同黏性系数情形下，随着网格加密，数值解均能达到很好的光滑性，在单元交界面处未出现非物理的振荡，表明数值通量选取的正确性及算法随着空间网格加密的数值收敛性．

表2~4展示了时间T=\mathrm{0.1}和黏性系数\nu =\mathrm{0.1,0.01,0.001}下，二维黏性Burgers方程的数值模拟情况及对应的L^{\mathrm{2}}误差和收敛阶．可知随着黏性系数的增加，当k=l分别取1、2、3时，均达到最优收敛阶．

表5展示了时间T=\mathrm{0.1}、黏性系数\nu =\mathrm{0.000}\mathrm{1}下，二维黏性Burgers方程对应的L^{\mathrm{2}}误差和收敛阶．当黏性系数\nu =\mathrm{0.000}\mathrm{1}时，误差阶达到k+\frac{\mathrm{1}}{\mathrm{2}}，且出现掉阶现象，可能原因为：1）当黏性系数很小时，选取交替通量可能导致数值解的误差增大；2）当黏性系数很小时，可能在数值解中引入额外误差，导致精度降低．

表6~8展示了黏性系数\nu =\mathrm{0.001}、时间T=\mathrm{0.1}下，通过增加校正次数，使时空误差阶数达到匹配的过程．数值结果表明，选取\mathrm{4}\times \mathrm{4}的空间网格时，对应不同的空间插值次数，时间上使用SDC方法可以匹配对应的空间阶数．

5 结 论

本文采用空间LDG方法结合时间SDC方法的策略，通过扩散项选取交替通量、对流项选取满足局部Lipschitz连续的单调通量，构造和分析了求解黏性Burgers方程的时空高阶演化方法，可以实现逐单元求解，从而有效降低离散问题求解规模，通过时间方向的反复校正达到时间高阶．对于一维问题，由单元熵不等式证明了空间半离散方法的L^{\mathrm{2}}稳定性，进一步使用能量分析技巧证明了空间半离散方法的误差估计，并对一维和二维黏性Burgers方程的全离散方法的稳定性进行了分析．数值算例验证了所提方法的有效性．

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基金资助

国家自然科学基金“磁微极流体动力学方程的降维方法研究”(12561075)

新疆维吾尔自治区重点实验室开放课题“两相不可压磁流体动力学方程高效算法研究”(2022D04014)