Exact and nonstandard finite difference schemes for the generalized KdV–Burgers equation

Abstract

AbstractWe consider the generalized KdV–Burgers $\operatorname{KdVB}(p,m,q)$KdVB(p,m,q) equation. We have designed exact and consistent nonstandard finite difference schemes (NSFD) for the numerical solution of the $\operatorname{KdVB}(2,1,2)$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 $\operatorname{KdVB}(2,1,2)$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 $\operatorname{KdVB}(2,1,2)$KdVB(2,1,2) equation.