# Kortewego-de Vries方程的Hermite函数谱配置方法Generalized Hermite Function Spectral-Collocation Method for KdV Equations

DOI: 10.12677/AAM.2019.84070, PDF, HTML, XML, 下载: 402  浏览: 518  国家自然科学基金支持

Abstract: The paper deals with the numerical solutions of initial value problem of KdV equation on un-bounded interval that approximates the solution of the Kortewego-de Vries equation, in which Hermite-Gauss nodes are used to collocate nodes of spectral collocation method. Selecting appro-priately the relaxation factor   involved in the generalized Hermite functions approximation enables us to fit the asymptotic behaviors of exact solutions at infinity closely. Numerical results demonstrate its efficiency and high accuracy of this approach. Especially, it is much easier to deal with nonlinear equation.

1. 引言

$R=\left\{x|-\infty $t>0,u\left(x,t\right)$ 是实值函数。为方便，记 $\frac{\partial }{\partial x}$${\partial }_{x}$ 。考虑Kortewego-de Vries方程初值问题 [1] [2] [3]

$\left\{\begin{array}{ccc}{\partial }_{t}u+\gamma u{\partial }_{x}u+\nu {\partial }_{x}^{3}u-\mu {\partial }_{x}^{5}u=0,& x\in R,& t>0,\\ u\left(x,t\right)\to 0,& |x|\to \infty ,& t>0,\\ u\left(x,0\right)=\phi \left(x\right)& x\in R,& \end{array}$ (1)

2. Lagrange插值函数及其微分矩阵

Hermite函数 ${\stackrel{^}{H}}_{n}\left(y\right)$ 定义为

${\stackrel{^}{H}}_{n}\left(y\right)=\frac{1}{\sqrt{{2}^{n}n!}}{e}^{-{y}^{2}/2}{H}_{n}\left(y\right),n\ge 0,y\in R.$

${\varphi }_{j}\left(y\right)=\frac{{e}^{-{y}^{2}/2}}{{e}^{-{y}_{j}^{2}/2}}\frac{{H}_{N+1}\left(y\right)}{\left(y-{y}_{j}\right){H}_{{}_{N+1}}^{\text{'}}\left({y}_{j}\right)},j=0,1,\cdots ,N,$

${p}_{N}^{\left(m\right)}\left(y\right)=\underset{j=0}{\overset{N}{\sum }}u\left({y}_{j}\right){\varphi }_{j}^{\left(m\right)}\left(y\right).$

${D}^{\left(m\right)}={\left({\varphi }_{j}^{\left(m\right)}\left({y}_{k}\right)\right)}_{0\le k,j\le N},D:={D}^{\left(1\right)},$

${\varphi }_{j}^{\left(1\right)}\left({y}_{k}\right)=\left\{\begin{array}{cc}\frac{{\stackrel{^}{H}}_{N}\left({y}_{k}\right)}{\left({y}_{k}-{y}_{j}\right){\stackrel{^}{H}}_{N}\left({y}_{j}\right)},& k\ne j,\\ 0,& k=j.\end{array}$ (2)

${D}^{\left(m\right)}={D}^{m}$ (3)

${H}_{n}^{\sigma }\left(x\right)={\stackrel{^}{H}}_{n}\left(\sigma x\right).$

${\varphi }_{j}^{\sigma }\left(x\right)=\frac{{\text{e}}^{-{\left(\sigma x\right)}^{2}/2}}{{\text{e}}^{-{\left(\sigma {x}_{j}\right)}^{2}/2}}\frac{{H}_{N+1}\left(\sigma x\right)}{\left(x-{x}_{j}\right){H}_{{}_{N+1}}^{\text{'}}\left(\sigma {x}_{j}\right)},j=0,1,\cdots ,N.$

${D}_{\sigma }^{\left(m\right)}={\left({\left({\varphi }_{j}^{\sigma }\left({y}_{k}\right)\right)}^{\left(m\right)}\right)}_{0\le k,j\le N},{D}_{\sigma }:={D}_{\sigma }^{\left(1\right)},$

${\left({\varphi }_{j}^{\sigma }\left({x}_{k}\right)\right)}^{\left(1\right)}=\left\{\begin{array}{cc}\frac{{H}_{N}^{\sigma }\left({x}_{k}\right)}{\left({x}_{k}-{x}_{j}\right){H}_{N}^{\sigma }\left({x}_{j}\right)},& k\ne j,\\ 0,& k=j.\end{array}$

${\varphi }_{j}^{\sigma }\left(x\right)={\varphi }_{j}\left(y\right)|{}_{y=\sigma x}$ 和(3)式知 $m$ 阶微分矩阵：

${D}_{\sigma }^{\left(m\right)}={D}_{\sigma }^{m}={\sigma }^{m}{D}^{m}$

3. Kortewego-de Vries方程的Hermite函数谱配置方法

3.1. 三阶Kortewego-de Vries方程配置方法

$\left\{\begin{array}{ccc}{\partial }_{t}u+u{\partial }_{x}u+{\partial }_{x}^{3}u=0,& x\in R,& t>0,\\ u\left(x,t\right)\to 0,& |x|\to \infty ,& t>0,\\ u\left(x,0\right)=\phi \left(x\right)& x\in R,& \end{array}$ (4)

${p}_{N}^{\sigma }\left(x,t\right)=\underset{j=0}{\overset{N}{\sum }}u\left({x}_{j},t\right){\varphi }_{j}^{\sigma }\left(x\right),x\in R$ ，逼近式(4)的解 $u\left(x,t\right)$ ，将其代入式(4)，得

$\left\{\begin{array}{cc}{\partial }_{t}{p}_{N}^{\sigma }\left({x}_{k},t\right)+{p}_{N}^{\sigma }\left({x}_{k},t\right){\partial }_{x}{p}_{N}^{\sigma }\left({x}_{k},t\right)+{\partial }_{x}^{3}{p}_{N}^{\sigma }\left({x}_{k},t\right)=0,& k=0,1,\cdots ,N,t>0,\\ {p}_{N}^{\sigma }\left({x}_{k},0\right)=\phi \left({x}_{k}\right),& k=0,1,\cdots ,N.\end{array}$ (5)

$\left\{\begin{array}{l}{u}^{\prime }\left({x}_{0},t\right)=-\underset{j=0}{\overset{N}{\sum }}u\left({x}_{j},t\right){\left({\varphi }_{j}^{\sigma }\left({x}_{0}\right)\right)}^{\left(3\right)}-u\left({x}_{0},t\right)\underset{j=0}{\overset{N}{\sum }}u\left({x}_{j},t\right){\left({\varphi }_{j}^{\sigma }\left({x}_{0}\right)\right)}^{\left(1\right)},\\ \begin{array}{cccc}& & & \begin{array}{cccc}\begin{array}{ccc}\begin{array}{cccc}\cdots & & \cdots & \end{array}& \cdots & \end{array}& & & \end{array}\end{array}\\ {u}^{\prime }\left({x}_{N},t\right)=-\underset{j=0}{\overset{N}{\sum }}u\left({x}_{j},t\right){\left({\varphi }_{j}^{\sigma }\left({x}_{N}\right)\right)}^{\left(3\right)}-u\left({x}_{N},t\right)\underset{j=0}{\overset{N}{\sum }}u\left({x}_{j},t\right){\left({\varphi }_{j}^{\sigma }\left({x}_{N}\right)\right)}^{\left(1\right)},\\ u\left({x}_{j},0\right)=\phi \left({x}_{j}\right)\begin{array}{cc},& j=0,1,\cdots N.\end{array}\end{array}$ (6)

$X\left(t\right)={\left(u\left({x}_{0},t\right),u\left({x}_{1},t\right),\cdots ,u\left({x}_{N},t\right)\right)}^{T},{X}_{0}={\left(\phi \left({x}_{0}\right),\phi \left({x}_{1}\right),\cdots ,\phi \left({x}_{N}\right)\right)}^{T}$ ，式(6)的矩阵形式为

$\left\{\begin{array}{cc}\frac{\text{d}X\left(t\right)}{\text{d}t}=-{\sigma }^{3}{D}^{3}X\left(t\right)-\sigma A\left(t\right)DX\left(t\right)& t>0\\ X\left(0\right)={X}_{0}.& \end{array}$ (7)

3.2. 数值结果

$\left\{\begin{array}{c}\left(I+\frac{1}{2}\tau {\sigma }^{3}D\right)X\left(t+\tau \right)=\left(I-\frac{1}{2}\tau {\sigma }^{3}{D}^{3}\right)X\left(t\right)-\frac{1}{2}\tau \sigma \left(A\left(t+\tau \right)DX\left(t+\tau \right)+A\left(t\right)DX\left(t\right)\right),\\ t=0,\tau ,2\tau ,\cdots T-\tau ,\\ X\left(0\right)={X}_{0}.\end{array}$ (8)

$\left\{\begin{array}{c}\left(I+\frac{1}{2}\tau {\sigma }^{3}D\right){X}_{n+1}\left(t+\tau \right)=\left(I-\frac{1}{2}\tau {\sigma }^{3}{D}^{3}\right)X\left(t\right)-\frac{1}{2}\tau \sigma \left(A\left(t+\tau \right)D{X}_{n}\left(t+\tau \right)+A\left(t\right)DX\left(t\right)\right),\\ t=0,\tau ,2\tau ,\cdots T-\tau ,\\ X\left(0\right)={X}_{0}.\end{array}$ (9)

$u=12\kappa {\text{sech}}^{2}\left(\kappa \left(x-4{\kappa }^{2}t-{x}_{0}\right)\right),$ (10)

${L}^{\infty }$ -范数

${E}_{N}=\underset{0\le j\le N}{\mathrm{max}}|u\left({x}_{j},t\right)-{P}_{N}^{\sigma }\left({x}_{j},t\right)|$

Figure 1. $\kappa =0.3,\sigma =0.3$

Figure 2. $\kappa =0.3,\sigma =0.3,\tau =0.001$

3.3. 五阶Kortewego-de Vries方程配置方法

${p}_{N}^{\sigma }\left(x,t\right)=\underset{m=0}{\overset{N}{\sum }}u\left({x}_{j},t\right){\varphi }_{j}^{\sigma }\left(x\right),x\in R$ ，逼近式(1)的解，将其代入式(1)，得

$\left\{\begin{array}{cc}{\partial }_{t}{p}_{N}^{\sigma }\left({x}_{k},t\right)+\gamma {p}_{N}^{\sigma }\left({x}_{k},t\right){\partial }_{x}{p}_{N}^{\sigma }\left({x}_{k},t\right)+\nu {\partial }_{x}^{3}{p}_{N}^{\sigma }\left({x}_{k},t\right)-\mu {\partial }_{x}^{5}{p}_{N}^{\sigma }\left({x}_{k},t\right)=0,& k=0,1,\cdots ,N,t>0,\\ {p}_{N}^{\sigma }\left({x}_{k},0\right)=\phi \left({x}_{k}\right)& k=0,1,\cdots ,N.\end{array}$ (11)

$\left\{\begin{array}{cc}\frac{\text{d}X\left(t\right)}{\text{d}t}=\left(\mu {\sigma }^{5}{D}^{5}-\nu {\sigma }^{3}{D}^{3}\right)X\left(t\right)-\gamma \sigma A\left(t\right)DX\left(t\right),& t>0\\ X\left(0\right)={X}_{0}.& \end{array}$ (12)

4. 数值结果

$\left\{\begin{array}{c}\left(I-\frac{1}{2}\tau \left(\mu {\sigma }^{5}{D}^{5}-\nu {\sigma }^{3}{D}^{3}\right)\right)X\left(t+\tau \right)=\left(I+\frac{1}{2}\tau \left(\mu {\sigma }^{5}{D}^{5}-\nu {\sigma }^{3}{D}^{3}\right)\right)X\left(t\right)\\ -\frac{1}{2}\tau \gamma \sigma \left(A\left(t+\tau \right)DX\left(t+\tau \right)+A\left(t\right)DX\left(t\right)\right),\\ t=0,\tau ,2\tau ,\cdots T-\tau ,\\ X\left(0\right)={X}_{0}.\end{array}$ (13)

$\left\{\begin{array}{c}\left(I-\frac{1}{2}\tau \left(\mu {\sigma }^{5}{D}^{5}-\nu {\sigma }^{3}{D}^{3}\right)\right){X}_{n+1}\left(t+\tau \right)=\left(I+\frac{1}{2}\tau \left(\mu {\sigma }^{5}{D}^{5}-\nu {\sigma }^{3}{D}^{3}\right)\right)X\left(t\right)\\ -\frac{1}{2}\tau \gamma \sigma \left(A\left(t+\tau \right)D{X}_{n}\left(t+\tau \right)+A\left(t\right)DX\left(t\right)\right)\\ t=0,\tau ,2\tau ,\cdots T-\tau ,\\ X\left(0\right)={X}_{0}.\end{array}$ (14)

$u\left(x,t\right)={\eta }_{0}+A{\text{sech}}^{4}\left(\kappa \left(x-ct-{x}_{0}\right)\right),$

$A=\frac{105{\nu }^{2}}{169\mu \gamma },\kappa =\sqrt{\frac{\nu }{52\mu }},c=\gamma {\eta }_{0}+\frac{36{\nu }^{2}}{169\mu }.$

$u\left(x,t\right)$ 作为测试函数，用 ${E}_{N}$ 度量数值误差。图3给出 $t=1$ 和不同的 $\tau$${\mathrm{log}}_{10}{E}_{N}$${\mathrm{log}}_{10}N$ 的变化关系。可以看出，数值误差随 $N$ 的增加及数值误差 $\tau$ 的减小而快速衰减，空间方向达到谱精度；图4给出 $t=1$ 和不同的伸缩因子 $\sigma$ 数值误差 ${\mathrm{log}}_{10}{E}_{N}$${\mathrm{log}}_{10}N$ 的变化关系，可以发现对适当小的 $\sigma$ 数值误差更小。但如何选取最佳伸缩因子是一个未解决的问题。

Figure 3. $\sigma =0.365,\mu =\gamma =1,\nu =1.1,{\eta }_{0}=0$

Figure 4. $\tau =0.001,\mu =\gamma =1,\nu =1.1,{\eta }_{0}=0$

5. 结论

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