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In this work we present an efficient Galerkin least squares 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 its numerical support $\tilde{K}$. Gaining advantage from the well-known convective-diffusive effects of the Burgers' equation, computations start by choosing $\tilde{K}$ so it contains the support of the initial condition and, as solution diffuses out, $\tilde{K}$ is increased appropriately. By direct comparisons between numerical and analytic solutions and its asymptotic behavior, we conclude that the proposed scheme is accurate even for large times, and it can be applied to numerically investigate properties of this and similar equations on unbounded domains.

Numerical Simulations with the Galerkin Least Squares Finite Element Method for the Burgers' Equation on the Real Line