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Implicit-explicit Runge–Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations

Published online by Cambridge University Press:  03 February 2012

Erik Burman  and
Alexandre Ern
Erik Burman
Affiliation:
Department of Mathematics, University of Sussex, Brighton, BN1 9QH, UK. [email protected]
Alexandre Ern
Affiliation:
Université Paris-Est, CERMICS, École des Ponts, 77455 Marne la Vallée Cedex 2, France; [email protected]
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Abstract

We analyze a two-stage implicit-explicit Runge–Kutta scheme for time discretization of advection-diffusion equations. Space discretization uses continuous, piecewise affine finite elements with interelement gradient jump penalty; discontinuous Galerkin methods can be considered as well. The advective and stabilization operators are treated explicitly, whereas the diffusion operator is treated implicitly. Our analysis hinges on L2-energy estimates on discrete functions in physical space. Our main results are stability and quasi-optimal error estimates for smooth solutions under a standard hyperbolic CFL restriction on the time step, both in the advection-dominated and in the diffusion-dominated regimes. The theory is illustrated by numerical examples.

Keywords

Stabilized finite elementsstabilityerror boundsimplicit-explicit Runge–Kutta schemesunsteady convection-diffusion
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 4 , July 2012 , pp. 681 - 707
Copyright
© EDP Sciences, SMAI, 2012

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