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Error estimates for forward Euler shock capturing finite element approximations of the one-dimensional Burgers' equation

Erik Burman

Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK

https://doi.org/10.1142/S0218202515500517Cited by:2 (Source: Crossref)

Abstract

We propose an error analysis for a shock capturing finite element method for the Burgers' equation using the duality theory due to Tadmor. The estimates use a one-sided Lipschitz stability (Lip⁺-stability) estimate on the discrete solution and are obtained in a weak norm, but thanks to a total variation a priori bound on the discrete solution and an interpolation inequality, error estimates in L^p-norms (1 ≤ p < ∞) are deduced. Both first-order artificial viscosity and a nonlinear shock capturing term that formally is of second order are considered. For the discretization in time we use the forward Euler method. In the numerical section we verify the convergence order of the nonlinear scheme using the forward Euler method and a second-order strong stability preserving Runge–Kutta method. We also study the Lip⁺-stability property numerically and give some examples of when it holds strictly and when it is violated.

Keywords:

AMSC: 76M10, 65M12, 65M15, 65M60

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