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.
@article{M2AN_2012__46_4_681_0,
author = {Burman, Erik and Ern, Alexandre},
title = {Implicit-explicit Runge-Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {46},
year = {2012},
pages = {681-707},
doi = {10.1051/m2an/2011047},
mrnumber = {2891466},
zbl = {1281.65123},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2012__46_4_681_0}
}
Burman, Erik; Ern, Alexandre. Implicit-explicit Runge-Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) pp. 681-707. doi : 10.1051/m2an/2011047. http://gdmltest.u-ga.fr/item/M2AN_2012__46_4_681_0/
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