Implicit-explicit Runge-Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 4, pp. 681-707.

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.

DOI : 10.1051/m2an/2011047
Classification : 5M12, 65M15, 65M60
Mots-clés : stabilized finite elements, stability, error bounds, implicit-explicit Runge-Kutta schemes, unsteady convection-diffusion 
@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 },
     pages = {681--707},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {4},
     year = {2012},
     doi = {10.1051/m2an/2011047},
     mrnumber = {2891466},
     zbl = {1281.65123},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2011047/}
}
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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 , Tome 46 (2012) no. 4, pp. 681-707. doi : 10.1051/m2an/2011047. http://www.numdam.org/articles/10.1051/m2an/2011047/

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