Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional KdV-Burgers-Kuramoto equation

TL;DR

This paper introduces a fully discrete local discontinuous Galerkin method for solving the time-fractional KdV-Burgers-Kuramoto equation, demonstrating stability and convergence through theoretical analysis and numerical tests.

Contribution

It develops a novel fully discrete LDG scheme combining finite differences in time with spatial DG methods for the fractional KBK equation, with proven stability and error estimates.

Findings

Unconditionally stable scheme

Error convergence rate of O(h^{k+1}+(\Delta t)^2+(\Delta t)^{rac{ ext{ extalpha}}{2}}h^{k+1/2})

Numerical results confirm efficiency and accuracy

Abstract

In this paper, a fully discrete local discontinuous Galerkin (LDG) finite element method is considered for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation. The scheme is based on a finite difference method in time and local discontinuous Galerkin methods in space. We prove that our scheme is unconditional stable and $L^2$ error estimate for the linear case with the convergence rate $O(h^{k+1}+(\Delta t)^2+(\Delta t)^\frac{\alpha}{2}h^{k+1/2})$. Numerical examples are presented to show the efficiency and accuracy of our scheme.