Numerical simulations with the finite element method for the Burgers' equation on the real line
This work provides a practical numerical method for solving Burgers' equation on infinite domains, which is useful for researchers studying convective-diffusive systems.
The paper presents a second-order finite element scheme for simulating Burgers' equation on the real line with compactly supported initial conditions, achieving high accuracy by adaptively increasing the domain size. Numerical comparisons with analytic solutions confirm the scheme's accuracy.
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.