A compact fourth-order implicit-explicit Runge-Kutta type scheme for numerical solution of the Kuramoto-Sivashinsky equation

TL;DR

This paper presents a new fourth-order implicit-explicit Runge-Kutta scheme combined with a compact finite difference spatial discretization for efficiently solving the one-dimensional Kuramoto-Sivashinsky equation, demonstrating superior accuracy and reliability.

Contribution

It introduces a novel fourth-order scheme that integrates MOL and partial fraction techniques, requiring only two linear solves per step, improving accuracy over existing methods.

Findings

The scheme achieves higher accuracy than existing methods.

Numerical tests confirm the scheme's reliability and efficiency.

The method is effective for both periodic and Dirichlet boundary conditions.

Abstract

This manuscript introduces a fourth-order Runge-Kutta based implicit-explicit scheme in time along with compact fourth-order finite difference scheme in space for the solution of one-dimensional Kuramoto-Sivashinsky equation with periodic and Dirichlet boundary conditions, respectively. The proposed scheme takes full advantage of method of line (MOL) and partial fraction decomposition techniques, therefore it just need to solve two backward Euler-type linear systems at each time step to get the solution. Performance of the scheme is investigated by testing it on some test examples and by comparing numerical results with relevant known schemes. It is found that the proposed scheme is more accurate and reliable than existing schemes to solve Kuramoto-Sivashinsky equation.