Time-dependent Hermite-Galerkin spectral method and its applications

TL;DR

This paper introduces a novel time-dependent Hermite-Galerkin spectral method for nonlinear convection-diffusion equations in unbounded domains, demonstrating its stability, spectral convergence, and superior accuracy through theoretical proofs and numerical examples.

Contribution

The paper develops a new THGSM with time-dependent scaling and translating factors, enhancing theoretical analysis and numerical performance for unbounded domain PDEs.

Findings

Proves stability and spectral convergence of the method.

Achieves higher accuracy than existing methods.

Successfully applies to KdVB, heat, and Burgers' equations.

Abstract

A time-dependent Hermite-Galerkin spectral method (THGSM) is investigated in this paper for the nonlinear convection-diffusion equations in the unbounded domains. The time-dependent scaling factor and translating factor are introduced in the definition of the generalized Hermite functions (GHF). As a consequence, the THGSM based on these GHF has many advantages, not only in theorethical proofs, but also in numerical implementations. The stability and spectral convergence of our proposed method have been established in this paper. The Korteweg-de Vries-Burgers (KdVB) equation and its special cases, including the heat equation and the Burgers' equation, as the examples, have been numerically solved by our method. The numerical results are presented, and it surpasses the existing methods in accuracy. Our theoretical proof of the spectral convergence has been supported by the numerical results.