Volume 46, Number 4, July-August 2012
|Page(s)||681 - 707|
|Published online||03 February 2012|
Implicit-explicit Runge–Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations
1 Department of Mathematics, University of Sussex, Brighton, BN1 9QH, UK
2 Université Paris-Est, CERMICS, École des Ponts, 77455 Marne la Vallée Cedex 2, France
Received: 28 October 2010
Revised: 31 May 2011
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.
Mathematics Subject Classification: 5M12 / 65M15 / 65M60
Key words: Stabilized finite elements / stability / error bounds / implicit-explicit Runge–Kutta schemes / unsteady convection-diffusion
© EDP Sciences, SMAI, 2012
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