Explicit Runge-Kutta Schemes and Finite Elements with Symmetric Stabilization for First-Order Linear PDE Systems

Authors:
Erik Burman;Alexandre Ern;Miguel A. Fernández
Affiliations:
E.N.Burman@sussex.ac.uk;ern@cermics.enpc.fr;miguel.fernandez@inria.fr
Venue:
SIAM Journal on Numerical Analysis
Year:
2010

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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 space 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 smooth and rough solutions. The case of finite volumes is briefly discussed.