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

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 $K\ast u_{xx}$ 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.

Accepted for publication in ESAIM: Mathematical Modelling and Numerical Analysis

References in corpus (2)

• Large-time asymptotics, vanishing viscosity and numerics for 1-D scalar conservation laws
• Spikes and diffusion waves in one-dimensional model of chemotaxis

Cited by in corpus (1)

• Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection