Fully discrete scheme for the fifth-order KdV-Burgers-Fisher equation using Strang splitting and Fourier collocation methods

TL;DR

This paper introduces a fully discrete numerical scheme combining Strang splitting and Fourier collocation methods to solve the fifth-order KdV-Burgers-Fisher equation with spectral accuracy and proven convergence.

Contribution

The study develops and analyzes a novel fully discrete scheme for the KBF equation, achieving second-order temporal and spectral spatial convergence.

Findings

Numerical results confirm the theoretical error estimates.

The scheme demonstrates spectral accuracy in space under regularity assumptions.

Second-order convergence in time is achieved and verified.

Abstract

Operator splitting is an effective technique for the numerical solution of nonlinear partial differential equations by decomposing a complex problem into simpler subproblems. In this study, we present and analyze a fully discrete scheme for the fifth-order Korteweg-de Vries-Burgers-Fisher equation (KBF) by combining Strang splitting for time discretization with the Fourier collocation method for spatial discretization. In particular, the Fourier collocation method is an essential component of the proposed fully discrete scheme and yields spectral accuracy in space under suitable regularity assumptions. The KBF equation describes the interaction of reaction, dissipative, and dispersive mechanisms by incorporating the Fisher reaction term together with Burgers-type diffusion and higher-order KdV dispersion. The equation is split into a linear operator and a nonlinear operator, and the resulting subproblems are solved within the Strang splitting framework. Convergence is analyzed in the Sobolev space $H^s$. The local error is derived using operator-theoretic arguments in Banach spaces together with Lie commutator estimates, while the global error is obtained using the Lady Windermere's fan argument. The analysis yields second-order convergence in time and spectral convergence in space. Numerical results confirm the theoretical error estimates and demonstrate the accuracy of the proposed fully discrete scheme.