#### 期刊菜单

The Numerical Analysis and Simulation Based on A Mixed Finite Element Method for the MRLW Equation
DOI: 10.12677/AAM.2019.812240, PDF, HTML, XML, 下载: 612  浏览: 750  科研立项经费支持

Abstract: A second-order backward-difference mixed finite element (MFE) method for modified regularized long wave (MRLW) equation is proposed and discussed in this paper. The spatial direction is approximated by the mixed Galerkin method using mixed linear space finite elements, and the time direction is considered by backward difference scheme with second-order convergence rate. The optimal error estimates for u in L2 and H1-norms and its flux q = ux and in L2-norm are derived. Some numerical results are given to test our theoretical analysis and illustrate the efficiency of the studied method.

1. 引言

$\left\{\begin{array}{l}{u}_{t}+{u}_{x}+6{u}^{2}{u}_{x}-\mu {u}_{xxt}=0,\text{ }\left(x,t\right)\in I×J\\ u\left(a,t\right)=u\left(b,t\right)=0,\text{ }t\in \stackrel{¯}{J},\\ u\left(x,0\right)={u}_{0}\left(x\right),\text{ }x\in \stackrel{¯}{I},\end{array}$ (1.1)

Mei和Chen [18] 针对RLW方程研究了基于Galerkin方法的显式两步方法。在本文中，我们的目的是对修正RLW方程研究一种显式两步混合元法，利用混合Galerkin方法逼近空间方向，向后二阶差分格式逼近时间方向。我们在全离散显式两步混合格式下，得到了未知函数u在 ${L}^{2}$ 模和 ${H}^{1}$ 模下最优误差估计和其辅助变量 $q={u}_{x}$ 基于 ${L}^{2}$ 模最优误差估计，并与其他的数值方法的准确性进行了比较。通过与文献 [18] [24] [25] 中的数值方法进行比较，我们得到了u与 $q={u}_{x}$ 的近似值。

2. 混合有限元数值方法

${u}_{x}=q$ (2.1)

${u}_{t}+q+6{u}^{2}q-\mu {q}_{xt}=0.$ (2.2)

$\left({u}_{x},w\right)=\left(q,w\right),\forall w\in {L}^{2}\left(\Omega \right)$ (2.3)

$\left({u}_{t},v\right)+\left(q,v\right)+6\left({u}^{2}q,v\right)+\mu \left({q}_{t},{v}_{x}\right)=0,\forall v\in {H}_{0}^{1}$ .(2.4)

${V}_{h}=\left\{{v}_{h}|{v}_{h}\in {C}^{0}\left(\stackrel{¯}{I}\right),{{v}_{h}|}_{{I}_{j}}\in {P}_{k}\left({I}_{j}\right),\forall {I}_{j}\in {T}_{h},{v}_{h}\left(a\right)={v}_{h}\left(b\right)=0\right\}\subset {H}_{0}^{1},$

${W}_{h}=\left\{{w}_{h}|{w}_{h}\in {L}^{2}\left(\Omega \right),{{w}_{h}|}_{{I}_{j}}\in {P}_{r}\left({I}_{j}\right),\forall {I}_{j}\in {T}_{h}\right\}\subset {L}^{2}\left(\Omega \right),$

$\left({u}_{hx},{w}_{h}\right)=\left({q}_{h},{w}_{h}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall {w}_{h}\in {W}_{h},$ (2.5)

$\left({u}_{ht},{v}_{h}\right)+\left({q}_{h},{v}_{h}\right)+6\left({\left({u}_{h}\right)}^{2}{q}_{h},{v}_{h}\right)+\mu \left({q}_{ht},{v}_{hx}\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall {v}_{h}\in {V}_{h}.$ (2.6)

3. 二阶向后差分混合方法和最优误差估计

3.1. 二阶向后差分方法和相关定理

$\left({u}_{x}^{n+1},w\right)=\left({q}^{n+1},w\right),\text{ }\forall w\in {L}_{2}\left(\Omega \right),$ (3.1)

$\left({u}_{t}^{n+1},v\right)+\left({q}^{n+1},v\right)+6\left({\left({u}^{n+1}\right)}^{2}{q}^{n+1},v\right)+\mu \left({q}_{t}^{n+1},{v}_{x}\right)=0,\text{ }\forall v\in {H}_{0}^{1}.$ (3.2)

$\left({u}_{x}^{n+1},w\right)=\left({q}^{n+1},w\right),\text{ }\forall w\in {L}_{2}\left(I\right),$ (3.3)

$\begin{array}{l}\left({\partial }_{t}^{2}{u}^{n+1},v\right)+\mu \left({\partial }_{t}^{2}{q}^{n+1},{v}_{x}\right)+\left({q}^{n+1},v\right)+6\left(2{\left({u}^{n}\right)}^{2}{q}^{n}-{\left({u}^{n-1}\right)}^{2}{q}^{n-1},v\right)\\ =\left({\tau }_{1}^{n+1},v\right)+\left({\tau }_{2}^{n+1},{v}_{x}\right)+6\left({\tau }_{3}^{n+1},v\right),\text{ }\forall v\in {H}_{0}^{1},\end{array}$ (3.4)

$\begin{array}{l}{\tau }_{1}^{n+1}={\partial }_{t}^{2}{u}^{n+1}-{u}_{t}^{n+1},\\ {\tau }_{2}^{n+1}=\mu \left({\partial }_{t}^{2}{q}_{x}^{n+1}-{q}_{xt}^{n+1}\right),\\ {\tau }_{3}^{n+1}=\left(2{\left({u}^{n}\right)}^{2}{q}^{n}-{\left({u}^{n-1}\right)}^{2}{q}^{n-1}\right)-{\left({u}^{n+1}\right)}^{2}{q}^{n+1}.\end{array}$ (3.5)

$‖{\tau }_{1}^{n+1}‖+‖{\tau }_{2}^{n+1}‖+‖{\tau }_{3}^{n+1}‖\le C\Delta {t}^{2}.$ (3.6)

$\left({\partial }_{t}^{2}{w}^{n+1},{w}^{n+1}\right)\ge \frac{1}{4\Delta t}\left[{‖{w}^{n+1}‖}^{2}+{‖2{w}^{n+1}-{w}^{n}‖}^{2}-{‖{w}^{n}‖}^{2}-{‖2{w}^{n}-{w}^{n-1}‖}^{2}\right].$ (3.7)

$\left({u}_{hx}^{n+1},{w}_{h}\right)=\left({q}_{h}^{n+1},{w}_{h}\right),\text{ }\forall {w}_{h}\in {W}_{h},$ (3.8)

$\left({\partial }_{t}^{2}{u}_{h}^{n+1},{v}_{h}\right)+\mu \left({\partial }_{t}^{2}{q}_{h}^{n+1},{v}_{hx}\right)+\left({q}_{h}^{n+1},{v}_{h}\right)+6\left(2{\left({u}_{h}^{n}\right)}^{2}{q}_{h}^{n}-{\left({u}_{h}^{n-1}\right)}^{2}{q}_{h}^{n-1},{v}_{h}\right)=0,\text{ }\forall {v}_{h}\in {V}_{h}.$ (3.9)

$\left(q-{\Pi }_{h}q,{v}_{hx}\right)=0,\text{ }{v}_{h}\in {V}_{h},$ (3.10)

$‖q-{\Pi }_{h}q‖\le C{h}^{k+1}{‖q‖}_{k+1},$ (3.11)

$\left({u}_{x}-{P}_{h}{u}_{x},{w}_{h}\right)=0,\text{ }{w}_{h}\in {W}_{h},$ (3.12)

$‖{u}_{x}-{P}_{h}{u}_{x}‖+h{‖u-{P}_{h}u‖}_{1}\le C{h}^{k+1}{‖u‖}_{k+1}$ (3.13)

${u}^{n}-{u}_{h}^{n}=\left({u}^{n}-{P}_{h}{u}^{n}\right)+\left({P}_{h}{u}^{n}-{u}_{h}^{n}\right)={\eta }^{n}+{\varsigma }^{n},$

${q}^{n}-{q}_{h}^{n}=\left({q}^{n}-{\Pi }_{h}{q}^{n}\right)+\left({\Pi }_{h}{q}^{n}-{q}_{h}^{n}\right)={\rho }^{n}+{\xi }^{n}.$

$\left({\xi }^{n+1}+{\rho }^{n+1},{w}_{h}\right)=\left({\varsigma }_{x}^{n+1},{w}_{h}\right),\text{ }\forall {w}_{h}\in {W}_{h},$ (3.14)

$\begin{array}{l}\left({\partial }_{t}^{2}{\varsigma }^{n+1},{v}_{h}\right)+\mu \left({\partial }_{t}^{2}{\xi }^{n+1},{v}_{hx}\right)+6\left(2\left({\left({u}^{n}\right)}^{2}{q}^{n}-{\left({u}_{h}^{n}\right)}^{2}{q}_{h}^{n}\right)-\left({\left({u}^{n-1}\right)}^{2}{q}^{n-1}-{\left({u}_{h}^{n-1}\right)}^{2}{q}_{h}^{n-1}\right),{v}_{h}\right)\\ +\left({\rho }^{n+1}+{\xi }^{n+1},{v}_{h}\right)=-\left({\partial }_{t}^{2}{\eta }^{n+1},{v}_{h}\right)+\left({\tau }_{1}^{n+1},{v}_{h}\right)+\left({\tau }_{2}^{n+1},{v}_{hx}\right)+6\left({\tau }_{3}^{n+1},{v}_{h}\right),\text{ }{v}_{h}\in {V}_{h}.\end{array}$ (3.15)

3.2. 最优误差估计

$\begin{array}{l}‖{u}^{n}-{u}_{h}^{n}‖+‖{q}^{n}-{q}_{h}^{n}‖\le C\left({h}^{k+1}+\Delta {t}^{2}\right),\\ {‖{u}^{n}-{u}_{h}^{n}‖}_{1}\le C\left({h}^{k}+\Delta {t}^{2}\right).\end{array}$

$\left({\partial }_{t}^{2}{\xi }^{n+1},{w}_{h}\right)=\left({\partial }_{t}^{2}{\varsigma }_{x}^{n+1},{w}_{h}\right)-\left({\partial }_{t}^{2}{\rho }^{n+1},{w}_{h}\right),\text{ }\forall {w}_{h}\in {W}_{h},$ (3.16)

$\begin{array}{l}\left({\partial }_{t}^{2}{\varsigma }^{n+1},{\varsigma }^{n+1}\right)+\mu \left({\partial }_{t}^{2}{\varsigma }_{x}^{n+1},{\varsigma }_{x}^{n+1}\right)\\ =\mu \left({\partial }_{t}^{2}{\rho }^{n+1},{\varsigma }_{x}^{n+1}\right)-6\left(2\left({\left({u}^{n}\right)}^{2}{q}^{n}-{\left({u}_{h}^{n}\right)}^{2}{q}_{h}^{n}\right)-\left({\left({u}^{n-1}\right)}^{2}{q}^{n-1}-{\left({u}_{h}^{n-1}\right)}^{2}{q}_{h}^{n-1}\right),{\varsigma }^{n+1}\right)\\ \text{ }+\left({\rho }^{n+1}+{\xi }^{n+1},{\varsigma }^{n+1}\right)+\left({\tau }_{1}^{n+1},{\varsigma }^{n+1}\right)+\left({\tau }_{2}^{n+1},{\varsigma }_{x}^{n+1}\right)+6\left({\tau }_{3}^{n+1},{\varsigma }^{n+1}\right)\\ \le C\left({‖2\left({\left({u}^{n}\right)}^{2}{q}^{n}-{\left({u}_{h}^{n}\right)}^{2}{q}_{h}^{n}\right)-\left({\left({u}^{n-1}\right)}^{2}{q}^{n-1}-{\left({u}_{h}^{n-1}\right)}^{2}{q}_{h}^{n-1}\right)‖}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\text{ }\\ \text{ }\end{array}+{‖{\rho }^{n+1}‖}^{2}+{‖{\xi }^{n+1}‖}^{2}+{‖{\tau }_{1}^{n+1}‖}^{2}+{‖{\tau }_{2}^{n+1}‖}^{2}+{‖{\tau }_{3}^{n+1}‖}^{2}+{‖{\varsigma }^{n+1}‖}^{2}+{‖{\varsigma }_{x}^{n+1}‖}^{2}\right).\end{array}$ (3.17)

$\begin{array}{l}{‖2\left({\left({u}^{n}\right)}^{2}{q}^{n}-{\left({u}_{h}^{n}\right)}^{2}{q}_{h}^{n}\right)-\left({\left({u}^{n-1}\right)}^{2}{q}^{n-1}-{\left({u}_{h}^{n-1}\right)}^{2}{q}_{h}^{n-1}\right)‖}^{2}\\ \le {‖2{\left({u}^{n}\right)}^{2}\left({q}^{n}-{q}_{h}^{n}\right)+2\left({\left({u}^{n}\right)}^{2}-{\left({u}_{h}^{n}\right)}^{2}\right){q}_{h}^{n}-\left({\left({u}^{n-1}\right)}^{2}\left({q}^{n-1}-{q}_{h}^{n-1}\right)+\left[{\left({u}^{n-1}\right)}^{2}-{\left({u}_{h}^{n-1}\right)}^{2}\right]{q}_{h}^{n-1}\right)‖}^{2}\\ \le C\underset{j=n-1}{\overset{n}{\sum }}\left[{‖{\left({u}^{j}\right)}^{2}‖}_{\infty }^{2}{\left(‖{\xi }^{j}‖+‖{\rho }^{j}‖\right)}^{2}+{\left({‖{u}^{j}‖}_{\infty }+{‖{u}_{h}^{j}‖}_{\infty }\right)}^{2}{‖{q}_{h}^{j}‖}_{\infty }^{2}{\left(‖{\eta }^{j}‖+‖{\varsigma }^{j}‖\right)}^{2}\right]\\ \le C\left({‖{\rho }^{n}‖}^{2}+{‖{\rho }^{n-1}‖}^{2}+{‖{\eta }^{n}‖}^{2}+{‖{\eta }^{n-1}‖}^{2}+{‖{\xi }^{n}‖}^{2}+{‖{\xi }^{n-1}‖}^{2}+{‖{\varsigma }^{n}‖}^{2}+{‖{\varsigma }^{n-1}‖}^{2}\right).\end{array}$ (3.18)

${‖{\xi }^{n+1}‖}^{2}\le C\left({‖{\rho }^{n+1}‖}^{2}+{‖{\varsigma }_{x}^{n+1}‖}^{2}\right).$ (3.19)

$\begin{array}{l}\left({\partial }_{t}^{2}{\varsigma }^{n+1},{\varsigma }^{n+1}\right)+\mu \left({\partial }_{t}^{2}{\varsigma }_{x}^{n+1},{\varsigma }_{x}^{n+1}\right)\\ =\mu \left({\partial }_{t}^{2}{\rho }^{n+1},{\varsigma }_{x}^{n+1}\right)-6\left(2\left({\left({u}^{n}\right)}^{2}{q}^{n}-{\left({u}_{h}^{n}\right)}^{2}{q}_{h}^{n}\right)-\left({\left({u}^{n-1}\right)}^{2}{q}^{n-1}-{\left({u}_{h}^{n-1}\right)}^{2}{q}_{h}^{n-1}\right),{\varsigma }^{n+1}\right)\\ \text{ }+\left({\rho }^{n+1}+{\xi }^{n+1},{\varsigma }^{n+1}\right)+\left({\tau }_{1}^{n+1},{\varsigma }^{n+1}\right)+\left({\tau }_{2}^{n+1},{\varsigma }_{x}^{n+1}\right)+6\left({\tau }_{3}^{n+1},{\varsigma }^{n+1}\right)\\ \le C\left({‖{\rho }^{n+1}‖}^{2}+{‖{\tau }_{1}^{n+1}‖}^{2}+{‖{\tau }_{2}^{n+1}‖}^{2}+{‖{\tau }_{3}^{n+1}‖}^{2}+\underset{i=n-1}{\overset{n+1}{\sum }}\left({‖{\varsigma }^{i}‖}^{2}+{‖{\varsigma }_{x}^{i}‖}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\text{ }\\ \text{ }\end{array}+{‖{\partial }_{t}^{2}{\rho }^{n+1}‖}^{2}+{‖{\rho }^{n}‖}^{2}+{‖{\rho }^{n-1}‖}^{2}+{‖{\eta }^{n}‖}^{2}+{‖{\eta }^{n-1}‖}^{2}\right).\end{array}$ (3.20)

$\begin{array}{l}{‖{\varsigma }^{n+1}‖}_{1}^{2}+{‖2{\varsigma }^{n+1}-{\varsigma }^{n}‖}_{1}^{2}-{‖{\varsigma }^{n}‖}_{1}^{2}-{‖2{\varsigma }^{n}-{\varsigma }^{n-1}‖}_{1}^{2}\\ \le C\left({‖{\rho }^{n+1}‖}^{2}+{‖{\tau }_{1}^{n+1}‖}^{2}+{‖{\tau }_{2}^{n+1}‖}^{2}+{‖{\tau }_{3}^{n+1}‖}^{2}+\underset{i=n-1}{\overset{n+1}{\sum }}\left({‖{\varsigma }^{i}‖}^{2}+{‖{\varsigma }_{x}^{i}‖}^{2}\right)\\ \begin{array}{c}\\ \end{array}+{‖{\partial }_{t}^{2}{\rho }^{n+1}‖}^{2}+{‖{\rho }^{n}‖}^{2}+{‖{\rho }^{n-1}‖}^{2}+{‖{\eta }^{n}‖}^{2}+{‖{\eta }^{n-1}‖}^{2}\right).\end{array}$ (3.21)

$\begin{array}{l}{‖{\varsigma }^{n+1}‖}_{1}^{2}+{‖2{\varsigma }^{n+1}-{\varsigma }^{n}‖}_{1}^{2}-{‖{\varsigma }^{n}‖}_{1}^{2}-{‖2{\varsigma }^{n}-{\varsigma }^{n-1}‖}_{1}^{2}\\ \le C\left({‖{\rho }^{n+1}‖}^{2}+{‖{\tau }_{1}^{n+1}‖}^{2}+{‖{\tau }_{2}^{n+1}‖}^{2}+{‖{\tau }_{3}^{n+1}‖}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{‖{\partial }_{t}^{2}{\rho }^{n+1}‖}^{2}+{‖{\rho }^{n}‖}^{2}+{‖{\rho }^{n-1}‖}^{2}+{‖{\eta }^{n}‖}^{2}+{‖{\eta }^{n-1}‖}^{2}\right).\end{array}$ (3.22)

$\begin{array}{c}{‖{\xi }^{n+1}‖}^{2}\le C\left({‖{\rho }^{n+1}‖}^{2}+{‖{\tau }_{1}^{n+1}‖}^{2}+{‖{\tau }_{2}^{n+1}‖}^{2}+{‖{\tau }_{3}^{n+1}‖}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{‖{\partial }_{t}^{2}{\rho }^{n+1}‖}^{2}+{‖{\rho }^{n}‖}^{2}+{‖{\rho }^{n-1}‖}^{2}+{‖{\eta }^{n}‖}^{2}+{‖{\eta }^{n-1}‖}^{2}\right).\end{array}$ (3.23)

4. 数值实验

$\begin{array}{l}{I}_{1}={\int }_{a}^{b}u\text{d}x\approx h\underset{j=1}{\overset{N}{\sum }}{u}_{j}^{n},\\ {I}_{2}={\int }_{a}^{b}\left({u}^{2}+\mu {\left({u}_{x}\right)}^{2}\right)\text{d}x\approx h\underset{j=1}{\overset{N}{\sum }}\left[{\left({u}_{j}^{n}\right)}^{2}+\mu {\left[{\left({u}_{x}\right)}_{j}^{n}\right]}^{2}\right]\text{\hspace{0.17em}},\\ {I}_{3}={\int }_{a}^{b}\left({u}^{4}-\mu {\left({u}_{x}\right)}^{2}\right)\text{d}x\approx h\underset{j=1}{\overset{N}{\sum }}\left[{\left({u}_{j}^{n}\right)}^{4}-\mu {\left[{\left({u}_{x}\right)}_{j}^{n}\right]}^{2}\right]\text{\hspace{0.17em}}.\end{array}$

$u\left(x,t\right)=\sqrt{c}\text{ }\text{sech}\left(p\left(x-\left(c+1\right)t-{x}_{0}\right)\right)$ (4.1)

${I}_{1}=\frac{\pi \sqrt{c}}{p},{I}_{2}=\frac{2c}{p}+\frac{2\mu pc}{3},{I}_{3}=\frac{4{c}^{2}}{3p}-\frac{2\mu pc}{3}.$

$u\left(x,0\right)=\sqrt{c}\text{ }\text{sech}\left(p\left(x-{x}_{0}\right)\right)$ (4.2)

Table 1. Invariants of single solitary wave. h = 0 , 125 , Δ t = 0.0125 , c 1 = 1 , c 2 = 0.5

Table 2. Convergence order and error in L2-norm for u of time. h = 0.125 , c = 1

Table 3. Convergence order and error in L2-norm for q of time. h = 0. 125 , c = 1

Table 4. Convergence order and error in L2-norm for u of space. Δ t = 0.01 , c = 1

Table 5. Convergence order and error in L2-norm for q of space. Δ t = 0.01 , c = 1

Figure 1. Surface for exact solution u

Figure 2. Surface for numerical solution uh

Figure 3. Surface for exact solution q

Figure 4. Surface for numerical solution qh

Figure 5. Comparison between u and uh

Figure 6. Comparison between q and qh

Figure 7. Numerical solution uh

Figure 8. Numerical solution qh

5. 结论

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