Exact and nonstandard finite difference schemes for the generalized KdV\u2013Burgers equation

C Koroglu

doi:10.1186/s13662-020-02584-2

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.

Highlights

1 Introduction Nonlinear ordinary differential equations (ODEs) and nonlinear partial differential equations (PDEs) play a very important role in describing some complex physical phenomena arising in various fields of science and engineering such as condensed matter, plasma physics, nonlinear quantum, nonlinear optics, biophysics, fluid mechanics, theory of turbulence and phase transitions
Many scientists pay attention to the research into the exact solution of these PDEs but, in general, it is difficult to obtain the exact solution of some partial differential equations
Numerical solutions obtained by the nonstandard finite difference schemes (NSFD) scheme are compared with the standard finite difference (SFD) scheme

Summary

Introduction

Nonlinear ordinary differential equations (ODEs) and nonlinear partial differential equations (PDEs) play a very important role in describing some complex physical phenomena arising in various fields of science and engineering such as condensed matter, plasma physics, nonlinear quantum, nonlinear optics, biophysics, fluid mechanics, theory of turbulence and phase transitions. The exact finite difference scheme is an important issue for the construction of new numerical algorithms in ODE and PDE, and it plays a key role in determining the appropriate denominator function It is a special NSFD scheme which is available if the solution of the ODE exists [24]. 5. Numerical solutions obtained by the NSFD scheme are compared with the standard finite difference (SFD) scheme.

Conclusion

Methods