Numerical simulations with the finite element method for the Burgers' equation on the real line

TL;DR

This paper introduces a second-order finite element scheme for accurately simulating Burgers' equation on the entire real line, effectively handling initial conditions with compact support and demonstrating reliable numerical results.

Contribution

The paper presents a simple, accurate finite element method for Burgers' equation on the real line, with a practical approach for domain extension during simulations.

Findings

The scheme achieves high accuracy in numerical simulations.

It effectively captures the asymptotic behavior of solutions.

The method is suitable for studying properties of equations on infinite domains.

Abstract

In this paper we present a simple and accurate second order finite element scheme to simulate the Burgers' equation on the whole real line and subjected to initial conditions with compact support. The numerical simulations are performed by considering a sequence of auxiliary spatially dimensionless Dirichlet's problems parameterized by the domain's semidiameter L. Gaining advantage from the well-known convective-diffusive effects of the Burgers' equation, computations start by choosing L larger than the semidiameter of the support of the initial condition and, as solution diffuses out, L is increased appropriately. By direct comparisons between numerical and analytic solutions and its asymptotic behavior, we conclude this simple scheme is very accurate and can be applied to numerically investigate properties of this and similar equations on infinite domains.