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

Abstract

We consider the generalized KdV-Burgers KdVB(p, m, q) equation. We have designed exact and consistent nonstandard finite difference schemes (NSFD) for the numerical solution of the 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 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 KdVB(2, 1, 2) equation.

Exact and nonstandard finite difference schemes for the Burgers equationB(2,2)

“…Koroglu et.al [9] have presented a NSFD scheme with theta method which includes the implicit Euler and a Crank-Nicolson type discretization for the numerical solution of the modified Korteweg-de Vries (MKdV) equation. In [10], the author have designed exact and consistent nonstandard finite difference schemes for the numerical solution of the KdVB( 2, 1, 2 ) equation. In these studies exact finite difference schemes and NSFD schemes are obtained by means of travelling wave solution of the PDE under consideration.…”