A numerical study of third-order equation with time-dependent coefficients: KdVB equation

TL;DR

This paper develops a numerical method for the linear Korteweg-de Vries Burgers equation with time-dependent coefficients, analyzing stability, convergence, and dispersion-dissipation properties through Legendre-Petrov-Galerkin techniques.

Contribution

It extends existing numerical methods to a new class of dispersive equations with time-varying parameters, providing stability, convergence proofs, and practical insights.

Findings

Proves stability and convergence of the numerical scheme.

Analyzes dispersion-dissipation relations for various profiles.

Demonstrates accuracy and efficiency of the proposed algorithms.

Abstract

In this article we present a numerical analysis for a third-order differential equation with non-periodic boundary conditions and time-dependent coefficients, namely, the linear Korteweg-de Vries Burgers equation. This numerical analysis is motived due to the dispersive and dissipative phenomena that government this kind of equations. This work builds on previous methods for dispersive equations with constant coefficients, expanding the field to include a new class of equations which until now have eluded the time-evolving parameters. More precisely, throughout the Legendre-Petrov-Galerkin method we prove stability and convergence results of the approximation in appropriate weighted Sobolev spaces. These results allow to show the role and trade off of these temporal parameters into the model. Afterwards, we numerically investigate the dispersion-dissipation relation for several profiles, further provide insights into the implementation method, which allow to exhibit the accuracy and efficiency of our numerical algorithms.