﻿ 正则长波Burgers方程的混合有限体积元方法 Mixed Finite Volume Element Method for Regularized Long Wave Burgers Equation

Vol.07 No.03(2018), Article ID:24201,12 pages
10.12677/AAM.2018.73032

Mixed Finite Volume Element Method for Regularized Long Wave Burgers Equation

Xue Bai, Hong Li, Zhichao Fang

School of Mathematical Science, Inner Mongolia University, Hohhot Inner Mongolia

Received: Mar. 1st, 2018; accepted: Mar. 20th, 2018; published: Mar. 27th, 2018

ABSTRACT

The mixed finite volume element method for the regularized long wave Burgers equation is developed and studied. By introducing a transfer operator which maps the trial function space into the test function space, the semi-discrete and linear backward Euler fully discrete mixed finite volume element schemes are constructed. Stability analysis for semi-discrete scheme is given, the existence and uniqueness of the solutions are proved, and optimal error estimates for these schemes are obtained. Finally, numerical experiments are given to verify the theoretical results and the effectiveness of the proposed schemes.

Keywords:Mixed Finite Volume Element Method, Regularized Long Wave Burgers Equation, Optimal Error Estimate

1. 引言

$\left\{\begin{array}{l}\left(a\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{t}+\sigma {u}_{x}+\delta u{u}_{x}-\alpha {u}_{xx}-\beta {u}_{xxt}=0,\text{\hspace{0.17em}}\left(x,t\right)\in \Omega ×\Gamma ,\\ \left(b\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(a,t\right)=u\left(b,t\right)=0,\text{\hspace{0.17em}}t\in \overline{\Gamma },\\ \left(c\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(x,0\right)={u}_{0}\left(x\right),\text{\hspace{0.17em}}x\in \Omega ,\end{array}$ (1)

RLW-Burgers方程(1)又称为Benjamin-Bona-Mahony-Burgers方程，是带有耗散项 $-\alpha {u}_{xx}$ 的正则长波方程，这类数学物理方程在非线性波问题中有着十分重要的地位。因此，求解此类问题的方法受到广泛的重视，文献 [1] 利用未知函数变换的思想，得到了RLW-Burgers方程的解析解，文献 [2] 运用一类含参数的G展开法得到了多种函数形式的显式行波解。这些解析解是在限定了初边值条件下给出的，因此研究该方程的数值计算方法是很有意义的。文献 [3] 研究了RLW-Burgers方程的有限差分法，给出了收敛性和稳定性分析。文献 [4] 构造了RLW-Burgers方程的H1-Galerkin混合有限元格式，给出了半离散和全离散格式的最优阶误差估计，并给出数值实验算例。文献 [5] 运用径向基函数法数值求解了高维广义Benjamin-Bona-Mahony-Burgers方程。

2. 半离散混合有限体积元形式

$\left\{\begin{array}{l}\left(a\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{t}+{p}_{x}=0,\text{\hspace{0.17em}}\left(x,t\right)\in \Omega ×\Gamma ,\\ \left(b\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}p=\sigma u+\frac{\delta }{2}f\left(u\right)-\alpha {u}_{x}-\beta {u}_{xt},\text{\hspace{0.17em}}\left(x,t\right)\in \Omega ×\Gamma ,\\ \left(c\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(a,t\right)=u\left(b,t\right)=0,\text{\hspace{0.17em}}t\in \overline{\Gamma },\\ \left(d\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(x,0\right)={u}_{0}\left(x\right),\text{\hspace{0.17em}}x\in \Omega .\end{array}$ (2)

$\left\{\begin{array}{l}\left(a\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({u}_{t},v\right)-\left({v}_{x},p\right)=0,\text{\hspace{0.17em}}\forall v\in {H}_{0}^{1}\left(\Omega \right),\\ \left(b\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(p,q\right)-\sigma \left(u,q\right)+\alpha \left({u}_{x},q\right)+\beta \left({u}_{xt},q\right)=\frac{\delta }{2}\left(f\left(u\right),q\right),\text{\hspace{0.17em}}\forall q\in {L}^{2}\left(\Omega \right),\\ \left(c\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(x,0\right)={u}_{0}\left(x\right),\text{\hspace{0.17em}}x\in \Omega .\end{array}$ (3)

${H}_{0h}=\left\{{\omega }_{h}\in {H}_{0}^{1}\left(\Omega \right):{\omega }_{h}\in {P}_{1}\left(A\right),\forall A\in {T}_{h}\right\},\text{\hspace{0.17em}}{L}_{h}=\left\{{v}_{h}\in {L}^{2}\left(\Omega \right):{{v}_{h}|}_{A}\in {P}_{0}\left(A\right),\forall A\in {T}_{h}\right\}.$

$\left\{{\phi }_{i}:i=1,2,\cdots ,N-1\right\}$$\left\{{\chi }_{{A}_{i}}:i=0,1,\cdots ,N-1\right\}$ 分别为有限元空间 ${H}_{0h}$${L}_{h}$ 的基，其中 ${\varphi }_{i}$ 是分段线性多项式(见文献 [12] )， ${\chi }_{{A}_{i}}$ 是集合 ${A}_{i}$ 的特征函数。

${\gamma }_{h}{\omega }_{h}=\sum _{i=1}^{N-1}{\omega }_{h}\left({x}_{i}\right){\chi }_{{A}_{i}^{*}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall {\omega }_{h}\in {H}_{0h}.$

$\left\{\begin{array}{l}\left(a\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\int }_{{A}_{i}^{*}}{u}_{t}\text{d}x+p\left({x}_{i+1/2},t\right)-p\left({x}_{i-1/2},t\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \Gamma ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(1\le i\le N-1\right),\\ \left(b\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\int }_{{A}_{i}}\left(p-\sigma u+\alpha {u}_{x}+\beta {u}_{xt}\right)\text{d}t=\frac{\delta }{2}{\int }_{{A}_{i}}f\left(u\right)\text{d}x,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \Gamma ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(0\le i\le N-1\right).\end{array}$ (4)

$\left\{\begin{array}{l}\left(a\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({u}_{t}+{p}_{x},{\gamma }_{h}{v}_{h}\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall {v}_{h}\in {H}_{0h},\\ \left(b\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(p-\sigma u+\alpha {u}_{x}+\beta {u}_{xt},{q}_{h}\right)=\frac{\delta }{2}\left(f\left(u\right),{q}_{h}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall {q}_{h}\in {L}_{h}.\end{array}$ (5)

$\left({p}_{x},{\gamma }_{h}{v}_{h}\right)=\sum _{i=1}^{N-1}{\int }_{{A}_{i}^{*}}\left({p}_{x}{v}_{h}\left({x}_{i}\right)\right)\text{d}x=\sum _{i=1}^{N-1}{v}_{h}\left({x}_{i}\right)\left(p\left({x}_{i+1/2}\right)-p\left({x}_{i-1/2}\right)\right)\text{d}x=b\left({\gamma }_{h}{v}_{h},p\right).$ (6)

$\left\{\begin{array}{l}\left(a\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({u}_{ht},{\sigma }_{h}{v}_{h}\right)-\left({v}_{hx},{p}_{h}\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall {v}_{h}\in {H}_{0h},\\ \left(b\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({p}_{h},{q}_{h}\right)-\sigma \left({u}_{h},{q}_{h}\right)+\alpha \left({u}_{hx},{q}_{h}\right)+\beta \left({u}_{hxt},{q}_{h}\right)=\frac{\delta }{2}\left(f\left({u}_{h}\right),{q}_{h}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall {q}_{h}\in {L}_{h},\\ \left(c\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{h}\left(0\right)={\pi }_{h}{u}_{0},\end{array}$ (7)

3. 半离散混合有限体积元形式的误差估计

$‖{\gamma }_{h}{\omega }_{h}‖\le \sqrt{3}‖{\omega }_{h}‖,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall {\omega }_{h}\in {H}_{0h}.$ (8)

$\left({\omega }_{h},{\gamma }_{h}{v}_{h}\right)=\left({\gamma }_{h}{\omega }_{h},{v}_{h}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall {v}_{h},{\omega }_{h}\in {H}_{0h}.$ (9)

$\left({\gamma }_{h}{\omega }_{h},{\omega }_{h}\right)\ge \frac{3}{4}{‖{\omega }_{h}‖}^{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall {\omega }_{h}\in {H}_{0h}.$ (10)

$‖\left(I-{\gamma }_{h}\right){\omega }_{h}‖\le \frac{\sqrt{12}}{12}h{|{\omega }_{h}|}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall {\omega }_{h}\in {H}_{0h}.$ (11)

$|\left({v}_{h},\left(I-{\gamma }_{h}\right){\omega }_{h}\right)|\le \frac{\sqrt{12}}{12}h{|{p}_{h}|}_{1}‖{\omega }_{h}‖,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall {p}_{h},{\omega }_{h}\in {H}_{0h}.$ (12)

$\left({\left(\omega -{\pi }_{h}\omega \right)}_{x},{v}_{h}\right)=0,\text{\hspace{0.17em}}\forall {v}_{h}\in {L}_{h},$ 其中 $\omega \in {H}_{0}^{1}\left(\Omega \right).$ (13)

$\left(\phi -{R}_{h}\phi ,{v}_{h}\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall {v}_{h}\in {L}_{h},$ 其中 $\phi \in {L}^{2}\left(\Omega \right).$ (14)

$‖\frac{{\partial }^{i}\omega }{\partial {t}^{i}}-\frac{{\partial }^{i}{\pi }_{h}\omega }{\partial {t}^{i}}‖\le Ch{‖\frac{{\partial }^{i}\omega }{\partial {t}^{i}}‖}_{1},$ 其中 $\frac{{\partial }^{i}\omega }{\partial {t}^{i}}\in {H}_{0}^{1}\left(\Omega \right),$ (15)

${‖\frac{{\partial }^{i}\omega }{\partial {t}^{i}}-\frac{{\partial }^{i}{\pi }_{h}\omega }{\partial {t}^{i}}‖}_{1}\le Ch{‖\frac{{\partial }^{i}\omega }{\partial {t}^{i}}‖}_{2},$ 其中 $\frac{{\partial }^{i}\omega }{\partial {t}^{i}}\in {H}^{2}\left(\Omega \right)\cap {H}_{0}^{1}\left(\Omega \right),$ (16)

$‖\frac{{\partial }^{i}\phi }{\partial {t}^{i}}-\frac{{\partial }^{i}{R}_{h}\phi }{\partial {t}^{i}}‖\le Ch{‖\frac{{\partial }^{i}\phi }{\partial {t}^{i}}‖}_{1},$ 其中 $\frac{{\partial }^{i}\phi }{\partial {t}^{i}}\in {H}^{1}\left(\Omega \right).$ (17)

$‖v-{\gamma }_{h}{\pi }_{h}v‖\le Ch{‖v‖}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall v\in {H}_{0}^{1}\left(\Omega \right),$ (18)

$|\left(v,\left(I-{\gamma }_{h}\right){\omega }_{h}\right)|\le Ch{‖v‖}_{1}‖{\omega }_{h}‖,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall v\in {H}_{0}^{1}\left(\Omega \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall {\omega }_{h}\in {H}_{0h}.$ (19)

$u-{u}_{h}=u-{\pi }_{h}u+{\pi }_{h}u-{u}_{h}=\xi +\eta ,$

$p-{p}_{h}=p-{R}_{h}p+{R}_{h}p-{p}_{h}=\rho +\epsilon .$

$\left\{\begin{array}{l}\left(a\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({\eta }_{t},{\gamma }_{h}{v}_{h}\right)-\left({v}_{hx},\epsilon \right)=-\left({\xi }_{t},{\gamma }_{h}{v}_{h}\right)-\left({u}_{t},\left(I-{\gamma }_{h}\right){v}_{h}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall {v}_{h}\in {H}_{0h},\\ \left(b\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha \left({\eta }_{x},{q}_{h}\right)+\beta \left({\eta }_{xt},{q}_{h}\right)+\left(\epsilon ,{q}_{h}\right)-\sigma \left(\eta ,{q}_{h}\right)=\left(\xi ,{q}_{h}\right)-\frac{\delta }{2}\left(f\left({u}_{h}\right)-f\left(u\right),{q}_{h}\right),\text{\hspace{0.17em}}\forall {q}_{h}\in {L}_{h},\\ \left(c\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\eta \left(0\right)={\pi }_{h}u\left(0\right)-{u}_{h}\left(0\right)=0.\end{array}$ (20)

${‖{u}_{h}\left(t\right)‖}_{1}\le C{‖{u}_{0}‖}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{‖{u}_{h}\left(t\right)‖}_{\infty }\le C{‖{u}_{0}‖}_{1},$

${‖{u}_{ht}\left(t\right)‖}_{1}+‖{p}_{h}\left(t\right)‖\le C\left({‖{u}_{0}‖}_{1}+{‖{u}_{0}‖}_{1}^{2}+{‖{u}_{0}‖}_{1}^{3}\right).$

${‖u\left(t\right)-{u}_{h}\left(t\right)‖}_{1}\le Ch\left({‖u\left(t\right)‖}_{2}+{\left({\int }_{0}^{t}\left({‖u‖}_{1}^{2}+{‖{u}_{t}‖}_{1}^{2}\right)\text{d}s\right)}^{\frac{1}{2}}\right),$

${‖{u}_{t}\left(t\right)-{u}_{ht}\left(t\right)‖}_{1}\le Ch\left({‖{u}_{t}\left(t\right)‖}_{2}+{‖u\left(t\right)‖}_{1}+{\left({\int }_{0}^{t}\left({‖u‖}_{1}^{2}+{‖{u}_{t}‖}_{1}^{2}\right)\text{d}s\right)}^{\frac{1}{2}}\right),$

$‖p\left(t\right)-{p}_{h}\left(t\right)‖\le Ch\left({‖p\left(t\right)‖}_{1}+{‖u\left(t\right)‖}_{1}+{‖{u}_{t}‖}_{1}+{\left({\int }_{0}^{t}\left({‖u‖}_{1}^{2}+{‖{u}_{t}‖}_{1}^{2}\right)\text{d}s\right)}^{\frac{1}{2}}\right).$

$\begin{array}{l}\alpha \left({\eta }_{x},{\eta }_{x}\right)+\beta \left({\eta }_{xt},{\eta }_{x}\right)+\left({\eta }_{t},{\gamma }_{h}\eta \right)-\sigma \left(\eta ,{\eta }_{x}\right)\\ =-\left({\xi }_{t},{\gamma }_{h}\eta \right)-\left({u}_{t},\left(I-{\gamma }_{h}\right)\eta \right)+\left(\xi ,{\eta }_{x}\right)-\frac{\delta }{2}\left(f\left({u}_{h}\right)-f\left(u\right),{\eta }_{x}\right).\end{array}$ (21)

$\begin{array}{l}\alpha {‖{\eta }_{x}‖}^{2}+\frac{1}{2}\frac{\text{d}}{\text{d}t}\left(\beta {‖{\eta }_{x}‖}^{2}+\left(\eta ,{\gamma }_{h}\eta \right)\right)-\sigma \left(\eta ,{\eta }_{x}\right)\\ =-\left({\xi }_{t},{\gamma }_{h}\eta \right)-\left({u}_{t},\left(I-{\gamma }_{h}\right)\eta \right)+\left(\xi ,{\eta }_{x}\right)-\frac{\delta }{2}\left(f\left({u}_{h}\right)-f\left(u\right),{\eta }_{x}\right).\end{array}$ (22)

$\alpha {‖{\eta }_{x}‖}^{2}+\frac{1}{2}\frac{\text{d}}{\text{d}t}\left(\beta {‖{\eta }_{x}‖}^{2}+\left(\eta ,{\gamma }_{h}\eta \right)\right)\le \frac{\alpha }{2}{‖{\eta }_{x}‖}^{2}+C\left({‖\xi ‖}^{2}+{‖{\xi }_{t}‖}^{2}+{h}^{2}{‖{u}_{t}‖}_{1}^{2}+{‖\eta ‖}^{2}\right).$ (23)

$\alpha {\int }_{0}^{t}{‖{\eta }_{x}‖}^{2}\text{d}t+\beta {‖{\eta }_{x}‖}^{2}+\frac{3}{4}{‖\eta ‖}^{2}\le C{\int }_{0}^{t}\left({‖\xi ‖}^{2}+{‖{\xi }_{t}‖}^{2}+{h}^{2}{‖{u}_{t}‖}_{1}^{2}+{‖\eta ‖}^{2}\right)\text{d}s\text{ }.$ (24)

${‖\eta ‖}_{1}^{2}+{\int }_{0}^{t}{‖{\eta }_{x}‖}^{2}\text{d}s\le C{\int }_{0}^{t}\left({‖\xi ‖}^{2}+{‖{\xi }_{t}‖}^{2}+{h}^{2}{‖{u}_{t}‖}_{1}^{2}\right)\text{d}s\text{ }.$ (25)

$\begin{array}{l}\left({\eta }_{t},{\gamma }_{h}{\eta }_{t}\right)+\alpha \left({\eta }_{x},{\eta }_{xt}\right)+\beta \left({\eta }_{xt},{\eta }_{xt}\right)-\sigma \left(\eta ,{\eta }_{xt}\right)\\ =-\left({\xi }_{t},{\gamma }_{h}{\eta }_{t}\right)-\left({u}_{t},\left(I-{\gamma }_{h}\right){\eta }_{t}\right)+\left(\xi ,{\eta }_{xt}\right)-\frac{\delta }{2}\left(f\left({u}_{h}\right)-f\left(u\right),{\eta }_{xt}\right).\end{array}$ (26)

$\frac{3}{4}{‖{\eta }_{t}‖}^{2}+\beta {‖{\eta }_{xt}‖}^{2}\le C\left({‖{\xi }_{t}‖}^{2}+{‖\eta ‖}^{2}+{h}^{2}{‖{u}_{t}‖}_{1}^{2}+{‖\xi ‖}^{2}+{‖{\eta }_{x}‖}^{2}\right)+\frac{3}{8}{‖{\eta }_{t}‖}^{2}+\frac{\beta }{2}{‖{\eta }_{xt}‖}^{2}.$ (27)

$\frac{3}{8}{‖{\eta }_{t}‖}^{2}+\frac{\beta }{2}{‖{\eta }_{xt}‖}^{2}\le C\left({‖\eta ‖}_{1}^{2}+{‖\xi ‖}^{2}+{‖{\xi }_{t}‖}^{2}+{h}^{2}{‖{u}_{t}‖}_{1}^{2}\right).$ (28)

${‖{\eta }_{t}‖}_{1}^{2}\le C\left({‖\xi ‖}^{2}+{‖{\xi }_{t}‖}^{2}+{h}^{2}{‖{u}_{t}‖}_{1}^{2}\right)+C{\int }_{0}^{t}\left({‖\xi ‖}^{2}+{‖{\xi }_{t}‖}^{2}+{h}^{2}{‖{u}_{t}‖}_{1}^{2}\right)\text{d}s\text{ }.$ (29)

${‖\epsilon ‖}^{2}\le C\left({‖\xi ‖}^{2}+{‖\eta ‖}^{2}+{‖{\eta }_{x}‖}^{2}+{‖{\eta }_{xt}‖}^{2}\right)+\frac{1}{2}{‖\epsilon ‖}^{2}.$ (30)

${‖\epsilon ‖}^{2}\le C\left({‖\xi ‖}^{2}+{‖{\xi }_{t}‖}^{2}+{h}^{2}{‖{u}_{t}‖}_{1}^{2}\right)+C{\int }_{0}^{t}\left({‖\xi ‖}^{2}+{‖{\xi }_{t}‖}^{2}+{h}^{2}{‖{u}_{t}‖}_{1}^{2}\right)\text{d}s\text{ }.$ (31)

4. 全离散混合有限体积元格式的误差估计

$0={t}_{0}<{t}_{1}<\cdots <{t}_{M}=T$ 是时间区域 $\left[0,T\right]$ 的剖分， ${t}_{n}=n\Delta t$ ，其中时间步长 $\Delta t=T/M$ ，M是某一正整数。为了构造全离散格式，对于 $\left[0,T\right]$ 上函数j，记 ${\phi }^{n}=\phi \left({t}_{n}\right)$${\partial }_{t}{\phi }^{n}=\left({\phi }^{n}-{\phi }^{n-1}\right)/\Delta t$ 。并记 ${u}_{h}^{n}$${p}_{h}^{n}$ 分别是u和p在 $t={t}_{n}$ 处的全离散解，现在给出线性向后Euler全离散混合有限体积元格式：求 $\left\{{u}_{h}^{n},{p}_{h}^{n}\right\}\in {H}_{0h}×{L}_{h},\left(n=1,2,\cdots ,M\right)$ ，满足

$\left\{\begin{array}{l}\left(a\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({\partial }_{t}{u}_{h}^{n},{\gamma }_{h}{v}_{h}\right)-\left({v}_{hx},{p}_{h}^{n}\right)=0,\text{\hspace{0.17em}}\forall {v}_{h}\in {H}_{0h},\\ \left(b\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({p}_{h}^{n},{q}_{h}\right)-\sigma \left({u}_{h}^{n},{q}_{h}\right)+\alpha \left({u}_{hx}^{n},{q}_{h}\right)+\beta \left({\partial }_{t}{u}_{hx}^{n},{q}_{h}\right)=\frac{\delta }{2}\left(f\left({u}_{h}^{n-1}\right),{q}_{h}\right),\text{\hspace{0.17em}}\forall {q}_{h}\in {L}_{h},\\ \left(c\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{h}^{n}={\pi }_{h}{u}_{0}.\end{array}$ (32)

$\Delta t\left({p}_{h}^{n},{q}_{h}\right)-\Delta t\sigma \left({u}_{h}^{n},{q}_{h}\right)+\alpha \Delta t\left({u}_{hx}^{n},{q}_{h}\right)+\beta \left({u}_{hx}^{n},{q}_{h}\right)=\beta \left({u}_{hx}^{n-1},{q}_{h}\right)+\frac{\delta \Delta t}{2}\left(f\left({u}_{h}^{n-1}\right),{q}_{h}\right),$ (33)

$\left({u}_{h}^{n},{\gamma }_{h}{v}_{h}\right)-\Delta t\left({v}_{hx},{p}_{h}^{n}\right)=\left({u}_{h}^{n-1},{\gamma }_{h}{v}_{h}\right).$ (34)

${u}_{h}^{n}\left(x\right)=\sum _{j=1}^{N-1}{u}_{hj}^{n}{\phi }_{j}\left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{p}_{h}^{n}\left(x\right)=\sum _{j=1}^{N-1}{p}_{hj}^{n}{\chi }_{{A}_{j}}.$

$\left(\begin{array}{cc}\Delta tA& \left(\beta +\alpha \Delta t\right)B-\sigma \Delta tC\\ -\Delta t{B}^{T}& D\end{array}\right)\left(\begin{array}{c}{\stackrel{˜}{p}}_{h}^{n}\\ {\stackrel{˜}{u}}_{h}^{n}\end{array}\right)=\left(\begin{array}{c}\beta B{\stackrel{˜}{u}}_{h}^{n-1}+0.5\delta \Delta tF\left({\stackrel{˜}{u}}_{h}^{n-1}\right)\\ D{\stackrel{˜}{u}}_{h}^{n-1}\end{array}\right),$ (35)

${\stackrel{˜}{u}}_{h}^{n}={\left({u}_{h1}^{n},{u}_{h2}^{n},\cdots ,{u}_{hN-1}^{n}\right)}^{\text{T}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{˜}{P}}_{h}^{n}={\left({P}_{h1}^{n},{P}_{h2}^{n},\cdots ,{P}_{hN-1}^{n}\right)}^{\text{T}},$

$A={\left({\chi }_{{A}_{i}},{\chi }_{Aj}\right)}_{i=0,\cdots ,N-1;j=0,\cdots ,N-1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}B={\left({\varphi }_{jx},{\chi }_{{A}_{i}}\right)}_{i=0,\cdots ,N-1;j=0,\cdots ,N-1},$

$C={\left({\varphi }_{j},{\chi }_{Ai}\right)}_{i=0,\cdots ,N-1;j=0,\cdots ,N-1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}D={\left({\varphi }_{j},{\gamma }_{h}{\varphi }_{i}\right)}_{i=1,\cdots ,N-1;j=1,\cdots ,N-1},$

$F\left({\stackrel{˜}{u}}_{h}^{n-1}\right)={\left(f\left({u}_{h}^{n-1}\right),{\chi }_{{A}_{i}}\right)}_{i=0,\cdots ,N-1}^{\text{T}}$

$\left(\begin{array}{cc}\Delta tA& \left(\beta +\alpha \Delta t\right)B-\sigma \Delta tC\\ 0& D+\left(\beta +\alpha \Delta t\right){B}^{\text{T}}{A}^{-1}B-\sigma \Delta t{B}^{\text{T}}{A}^{-1}C\end{array}\right)\left(\begin{array}{c}{\stackrel{˜}{p}}_{h}^{n}\\ {\stackrel{˜}{u}}_{h}^{n}\end{array}\right)=\left(\begin{array}{c}\beta B{\stackrel{˜}{u}}_{h}^{n-1}+0.5\delta \Delta tF\left({\stackrel{˜}{u}}_{h}^{n-1}\right)\\ H\end{array}\right),$ (36)

$u\left({t}_{n}\right)-{u}_{h}^{n}=u\left({t}_{n}\right)-{\pi }_{h}u\left({t}_{n}\right)+{\pi }_{h}u\left({t}_{n}\right)-{u}_{h}^{n}={\xi }^{n}+{\eta }^{n},$

$p\left({t}_{n}\right)-{p}_{h}^{n}=p\left({t}_{n}\right)-{R}_{h}p\left({t}_{n}\right)+{R}_{h}p\left({t}_{n}\right)-{p}_{h}^{n}={\rho }^{n}+{\epsilon }^{n}.$

(37)

$u,{u}_{t}\in {L}^{\infty }\left({H}^{2}\left(\Omega \right)\cap {H}_{0}^{1}\left(\Omega \right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}P\in {L}^{\infty }\left({H}^{1}\left(\Omega \right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{tt}\in {L}^{\infty }\left({H}_{0}^{1}\left(\Omega \right)\right),$

$\underset{1\le n\le M}{\mathrm{max}}\left({‖u\left({t}_{n}\right)-{u}_{h}^{n}‖}_{1}+{‖{u}_{t}\left({t}_{n}\right)-{\partial }_{t}{u}_{h}^{n}‖}_{1}+‖p\left({t}_{n}\right)-{p}_{h}^{n}‖\right)\le C\left(h+\Delta t\right).$ (38)

$\begin{array}{l}\left({\partial }_{t}{\eta }^{n},{\gamma }_{h}{\eta }^{n}\right)-\sigma \left({\eta }^{n},{\eta }_{x}^{n}\right)+\alpha \left({\eta }_{x}^{n},{\eta }_{x}^{n}\right)+\beta \left({\partial }_{t}{\eta }_{x}^{n},{\eta }_{x}^{n}\right)\\ =-\left({u}_{t}^{n},\left(I-{\gamma }_{h}\right){\eta }^{n}\right)-\left({\tau }^{n},{\gamma }_{h}{\eta }^{n}\right)-\beta \left({\lambda }^{n},{\eta }_{x}^{n}\right)-\left({\partial }_{t}{\xi }^{n},{\gamma }_{h}{\eta }^{n}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\sigma \left({\xi }^{n},{\eta }_{x}^{n}\right)-\frac{\delta }{2}\left(f\left({u}_{h}^{n-1}\right)-f\left({u}^{n}\right),{\eta }_{x}^{n}\right).\end{array}$ (39)

$\left({\partial }_{t}{\eta }_{x}^{n},{\eta }_{x}^{n}\right)\ge \frac{1}{2\Delta t}\left[{‖{\eta }_{x}^{n}‖}^{2}-{‖{\eta }_{x}^{n-1}‖}^{2}\right],$ (40)

$\left({\partial }_{t}{\eta }^{n},{\gamma }_{h}{\eta }^{n}\right)\ge \frac{1}{2\Delta t}\left[\left({\eta }^{n},{\gamma }_{h}{\eta }^{n}\right)-\left({\eta }^{n-1},{\gamma }_{h}{\eta }^{n-1}\right)\right].$ (41)

$\begin{array}{l}\frac{1}{2\Delta t}\left[\left({\eta }^{n},{\gamma }_{h}{\eta }^{n}\right)-\left({\eta }^{n-1},{\gamma }_{h}{\eta }^{n-1}\right)\right]+\frac{1}{2\Delta t}\left({‖{\eta }_{x}^{n}‖}^{2}-{‖{\eta }_{x}^{n-1}‖}^{2}\right)+\alpha {‖{\eta }_{x}^{n}‖}^{2}\\ \le \frac{\alpha }{2}{‖{\eta }_{x}^{n}‖}^{2}+C\left({h}^{2}{‖{u}_{t}^{n}‖}_{1}^{2}+{‖{\tau }^{n}‖}^{2}+{‖{\lambda }^{n}‖}^{2}+{‖{\eta }^{n}‖}^{2}+{‖{\xi }^{n}‖}^{2}+{‖{\partial }_{t}{\xi }^{n}‖}^{2}+{‖f\left({u}_{h}^{n-1}\right)-f\left({u}^{n}\right)‖}^{2}\right).\end{array}$ (42)

${‖f\left({u}_{h}^{n-1}\right)-f\left({u}^{n}\right)‖}^{2}\le C\left({‖{\xi }^{n-1}‖}^{2}+{‖{\eta }^{n-1}‖}^{2}\right)+C\Delta t{\int }_{{t}^{n-1}}^{{t}^{n}}{‖{u}_{t}‖}^{2}\text{d}t$ (43)

$\begin{array}{l}\left({\eta }^{m},{\gamma }_{h}{\eta }^{m}\right)+{‖{\eta }_{x}^{m}‖}^{2}+\alpha \Delta t\sum _{n=1}^{m}{‖{\eta }_{x}^{n}‖}^{2}\\ \le C\left({‖{\eta }^{0}‖}_{1}^{2}+{‖{\xi }^{0}‖}^{2}\right)+C\Delta t\sum _{n=1}^{m}\left({‖{\eta }^{n}‖}_{1}^{2}+{h}^{2}{‖{u}_{t}^{n}‖}_{1}^{2}+{‖{\tau }^{n}‖}^{2}+{‖{\lambda }^{n}‖}^{2}+‖{\xi }^{n}‖+{‖{\partial }_{t}{\xi }^{n}‖}^{2}+\Delta t{\int }_{{t}^{n-1}}^{{t}^{n}}‖{u}_{t}‖\text{d}t\right)\text{ }\text{ }.\end{array}$ (44)

${‖{\eta }^{m}‖}_{1}^{2}\le C\Delta t\sum _{n=1}^{m}\left({h}^{2}{‖{u}_{t}^{n}‖}_{1}^{2}+{‖{\tau }^{n}‖}^{2}+{‖{\lambda }^{n}‖}^{2}+‖{\xi }^{n}‖+{‖{\partial }_{t}{\xi }^{n}‖}^{2}+\Delta t{\int }_{{t}^{n-1}}^{{t}^{n}}‖{u}_{t}‖\text{d}t\right)\text{ }\text{ }.$ (45)

$\begin{array}{l}\left({\partial }_{t}{\eta }^{n},{\gamma }_{h}{\partial }_{t}{\eta }^{n}\right)-\sigma \left({\eta }^{n},{\partial }_{t}{\eta }_{x}^{n}\right)+\alpha \left({\eta }_{x}^{n},{\partial }_{t}{\eta }_{x}^{n}\right)+\beta \left({\partial }_{t}{\eta }_{x}^{n},{\partial }_{t}{\eta }_{x}^{n}\right)\\ =-\left({u}_{t}^{n},\left(I-{\gamma }_{h}\right){\partial }_{t}{\eta }^{n}\right)-\left({\tau }^{n},{\gamma }_{h}{\partial }_{t}{\eta }^{n}\right)-\left({\partial }_{t}{\xi }^{n},{\gamma }_{h}{\partial }_{t}{\eta }^{n}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\sigma \left({\xi }^{n},{\partial }_{t}{\eta }_{x}^{n}\right)-\frac{\delta }{2}\left(f\left({u}_{h}^{n-1}\right)-f\left({u}^{n}\right),{\partial }_{t}{\eta }_{x}^{n}\right)-\left({\lambda }^{n},{\partial }_{t}{\eta }_{x}^{n}\right).\end{array}$ (46)

$\begin{array}{l}\frac{3}{4}{‖{\partial }_{t}{\eta }^{n}‖}^{2}+\beta {‖{\partial }_{t}{\eta }_{x}^{n}‖}^{2}\\ \le C\left({‖{\eta }^{n}‖}_{1}^{2}+{h}^{2}{‖{u}_{t}^{n}‖}_{1}^{2}+‖{\xi }^{n}‖+{‖{\partial }_{t}{\xi }^{n}‖}^{2}+{‖{\tau }^{n}‖}^{2}+{‖{\lambda }^{n}‖}^{2}\\ \text{\hspace{0.17em}}+\Delta t{\int }_{{t}^{n-1}}^{{t}^{n}}{‖{u}_{t}‖}^{2}\text{d}t+{‖{\eta }^{n-1}‖}^{2}+{‖{\xi }^{n-1}‖}^{2}\right)+\frac{3}{8}{‖{\partial }_{t}{\eta }^{n}‖}^{2}+\beta {‖{\partial }_{t}{\eta }_{x}^{n}‖}^{2}.\end{array}$ (47)

$\begin{array}{l}{‖{\partial }_{t}{\eta }^{n}‖}_{1}^{2}\le C\left(‖{\xi }^{n}‖+{‖{\partial }_{t}{\xi }^{n}‖}^{2}+{‖{\tau }^{n}‖}^{2}+{‖{\lambda }^{n}‖}^{2}+{h}^{2}{‖{u}_{t}^{n}‖}_{1}^{2}+\Delta t{\int }_{{t}^{n-1}}^{{t}^{n}}{‖{u}_{t}‖}^{2}\text{d}t+{‖{\xi }^{n-1}‖}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+C\Delta t\sum _{k=1}^{n}\left(‖{\xi }^{k}‖+{‖{\partial }_{t}{\xi }^{k}‖}^{2}+{‖{\tau }^{k}‖}^{2}+{‖{\lambda }^{k}‖}^{2}+{h}^{2}{‖{u}_{t}^{k}‖}_{1}^{2}+\Delta t{\int }_{{t}^{k-1}}^{{t}^{k}}{‖{u}_{t}‖}^{2}\text{d}t\right)\text{ }\text{ }.\end{array}$ (48)

${‖{\epsilon }^{n}‖}^{2}\le \frac{1}{2}{‖{\epsilon }^{n}‖}^{2}+C\left({‖{\eta }^{n}‖}_{1}{}^{2}+{‖{\partial }_{t}{\eta }_{x}^{n}‖}^{2}+‖{\xi }^{n}‖+{‖{\lambda }^{n}‖}^{2}+\Delta t{\int }_{{t}^{n-1}}^{{t}^{n}}{‖{u}_{t}‖}^{2}\text{d}t+{‖{\eta }^{n-1}‖}^{2}+{‖{\xi }^{n-1}‖}^{2}\right).$ (49)

$\begin{array}{l}{‖{\epsilon }^{n}‖}^{2}\le C\left({‖{\xi }^{n}‖}^{2}+{‖{\partial }_{t}{\xi }^{n}‖}^{2}+{‖{\tau }^{n}‖}^{2}+{‖{\lambda }^{n}‖}^{2}+{h}^{2}{‖{u}_{t}^{n}‖}_{1}^{2}+\Delta t{\int }_{{t}^{n-1}}^{{t}^{n}}{‖{u}_{t}‖}^{2}\text{d}t+{‖{\xi }^{n-1}‖}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+C\Delta t\sum _{k=1}^{n}\left({‖{\xi }^{k}‖}^{2}+{‖{\partial }_{t}{\xi }^{k}‖}^{2}+{‖{\tau }^{k}‖}^{2}+{‖{\lambda }^{k}‖}^{2}+{h}^{2}{‖{u}_{t}^{k}‖}_{1}^{2}+\Delta t{\int }_{{t}^{k-1}}^{{t}^{k}}{‖{u}_{t}‖}^{2}\text{d}t\right)\text{ }\text{ }.\end{array}$ (50)

${‖{\partial }_{t}{\xi }^{n}‖}^{2}\le \frac{C}{\Delta t}{\int }_{{t}^{n-1}}^{{t}^{n}}{‖{\xi }_{t}‖}^{2}\text{d}s,\text{\hspace{0.17em}}n\ge 1,$

${‖{\tau }^{n}‖}^{2}\le C\Delta t{\int }_{{t}^{n-1}}^{{t}^{n}}{‖{u}_{xtt}‖}^{2}\text{d}s,\text{\hspace{0.17em}}{‖{\lambda }^{n}‖}^{2}\le C\Delta t{\int }_{{t}^{n-1}}^{{t}^{n}}{‖{u}_{xtt}‖}^{2}\text{d}s,\text{\hspace{0.17em}}n\ge 1.$

5. 数值算例

$\left\{\begin{array}{l}\left(a\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{t}+\sigma {u}_{x}+\delta u{u}_{x}-\alpha {u}_{xx}-\beta {u}_{xxt}=g\left(x,t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(x,t\right)\in \Omega ×\Gamma ,\\ \left(b\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(a,t\right)=u\left(b,t\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \overline{\Gamma },\\ \left(c\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(x,0\right)={u}_{0}\left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in \Omega ,\end{array}$ (51)

$u\left(x,t\right)={\text{e}}^{-t}\mathrm{sin}\left(\text{π}x\right),\text{\hspace{0.17em}}p\left(x,t\right)={\text{e}}^{-t}\mathrm{sin}\left(\text{π}x\right)+\frac{1}{2}{\text{e}}^{-2t}{\left(\mathrm{sin}\left(\text{π}x\right)\right)}^{2}.$

Figure 1. The numerical solution of uh(x,t)

Figure 2. The numerical solution of ph(x,t)

Table 1. Error estimate and order of convergence

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