A new robust compact difference scheme on graded meshes for the time-fractional nonlinear Kuramoto–Sivashinsky equation

Abstract

In this study, we explore a new robust compact difference method (CDM) on graded meshes for the time-fractional nonlinear Kuramoto–Sivashinsky (KS) equation. This equation exemplifies a fourth-order sub-diffusion equation marked by nonlinearity. Considering the weak singularity often exhibited by exact solutions of time-fractional partial differential equations (TFPDEs) near the initial time, we introduce the L2-1\(_{\sigma }\) scheme on graded meshes to discretize the Caputo derivatives. By employing a novel double reduction order approach, we obtain a triple-coupled nonlinear system of equations. To address the nonlinear term \(uu_{x}\), we use a fourth-order nonlinear CDM, while the second and fourth derivatives in space are treated using the fourth-order linear CDM. We prove the solvability through Browder theorem. Additionally, \(\alpha \)-robust stability and convergence are demonstrated by introducing a modified discrete Grönwall inequality. Finally, we present numerical examples to corroborate the findings of our theoretical analysis.