A semi-discrete large-time behavior preserving scheme for the augmented Burgers equation

TL;DR

This paper investigates the large-time behavior of solutions to the augmented Burgers equation, establishing decay rates, analyzing asymptotics, and proposing a semi-discrete scheme that preserves these properties with numerical validation.

Contribution

The paper introduces a semi-discrete numerical scheme that preserves the large-time asymptotic behavior of the augmented Burgers equation, including the influence of the convolution term.

Findings

Established $L^1$-$L^p$ decay rates for solutions.

Proposed a scheme with correcting factors for non-local terms.

Numerical experiments confirm the scheme's accuracy.

Abstract

In this paper we analyze the large-time behavior of the augmented Burgers equation. We first study the well-posedness of the Cauchy problem and obtain $L^1$-$L^p$ decay rates. The asymptotic behavior of the solution is obtained by showing that the influence of the convolution term $K*u_{xx}$ is the same as $u_{xx}$ for large times. Then, we propose a semi-discrete numerical scheme that preserves this asymptotic behavior, by introducing two correcting factors in the discretization of the non-local term. Numerical experiments illustrating the accuracy of the results of the paper are also presented.