Explicit Runge-Kutta schemes and finite elements with symmetric stabilization for first-order linear PDE systems

Burman, Erik, Ern, Alexandre and Fernández, Miguel A (2010) Explicit Runge-Kutta schemes and finite elements with symmetric stabilization for first-order linear PDE systems. SIAM Journal on Numerical Analysis, 48 (6). pp. 2019-2042. ISSN 0036-1429

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Abstract

We analyze explicit Runge-Kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a first-order linear differential operator in space of Friedrichs-type. For the time discretization, we consider explicit second- and third-order Runge¿Kutta schemes. We identify a general set of properties on the spatial stabilization, encompassing continuous and discontinuous finite elements, under which we prove stability estimates using energy arguments. Then, we establish L^2-norm error estimates with (quasi-)optimal convergence rates for smooth solutions in space and time. These results hold under the usual CFL condition for third-order Runge¿Kutta schemes and any polynomial degree in space and for second-order Runge¿Kutta schemes and first-order polynomials in space. For second-order Runge¿Kutta schemes and higher polynomial degrees in space, a tightened 4/3-CFL condition is required. Numerical results are presented for the advection and wave equations

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Depositing User: Erik Burman
Date Deposited: 06 Feb 2012 19:25
Last Modified: 21 Jun 2012 11:14
URI: http://sro.sussex.ac.uk/id/eprint/20476
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