Abstract

We consider the generalized KdV–Burgers \(\operatorname{KdVB}(p,m,q)\) equation. We have designed exact and consistent nonstandard finite difference schemes (NSFD) for the numerical solution of the \(\operatorname{KdVB}(2,1,2)\) equation. In particular, we have proposed three explicit and three fully implicit exact finite difference schemes. The proposed NSFD scheme is linearly implicit. The chosen numerical experiment consists of tanh function. The NSFD scheme is compared with a standard finite difference(SFD) scheme. Numerical results show that the NSFD scheme is accurate and efficient in the numerical simulation of the kink-wave solution of the \(\operatorname{KdVB}(2,1,2)\) equation. We see that while the SFD scheme yields numerical instability for large step sizes, the NSFD scheme provides reliable results for long time integration. Local truncation error reveals that the NSFD scheme is consistent with the \(\operatorname{KdVB}(2,1,2)\) equation.

中文翻译：

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广义KdV-Burgers方程的精确和非标准有限差分格式

摘要

我们考虑广义的KdV-Burgers \（\ operatorname {KdVB}（p，m，q）\）方程。我们已经为\（\ operatorname {KdVB}（2,1,2）\）方程的数值解设计了精确一致的非标准有限差分方案（NSFD）。特别是，我们提出了三种显式和三种完全隐式的精确有限差分方案。提出的NSFD方案是线性隐式的。所选的数值实验由tanh函数组成。将NSFD方案与标准有限差分（SFD）方案进行比较。数值结果表明，NSFD方案在\（\ operatorname {KdVB}（2,1,2）\）的扭折波解的数值模拟中是准确有效的方程。我们看到，虽然SFD方案对于大步长产生数值不稳定，但NSFD方案为长时间积分提供了可靠的结果。局部截断错误表明，NSFD方案与\（\ operatorname {KdVB}（2,1,2）\）公式一致。