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

Erik Burman; Alexandre Ern

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 4, page 681-707
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Burman, Erik, and Ern, Alexandre. "Implicit-explicit Runge–Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations." ESAIM: Mathematical Modelling and Numerical Analysis 46.4 (2012): 681-707. <http://eudml.org/doc/277842>.

@article{Burman2012,
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.},
author = {Burman, Erik, Ern, Alexandre},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Stabilized finite elements; stability; error bounds; implicit-explicit Runge–Kutta schemes; unsteady convection-diffusion; stabilized finite elements; implicit-explicit Runge-Kutta schemes; semidiscretization; advection-diffusion equations; smooth solution; convergence; numerical experiment},
language = {eng},
month = {2},
number = {4},
pages = {681-707},
publisher = {EDP Sciences},
title = {Implicit-explicit Runge–Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations},
url = {http://eudml.org/doc/277842},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Burman, Erik
AU - Ern, Alexandre
TI - Implicit-explicit Runge–Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/2//
PB - EDP Sciences
VL - 46
IS - 4
SP - 681
EP - 707
AB - 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.
LA - eng
KW - Stabilized finite elements; stability; error bounds; implicit-explicit Runge–Kutta schemes; unsteady convection-diffusion; stabilized finite elements; implicit-explicit Runge-Kutta schemes; semidiscretization; advection-diffusion equations; smooth solution; convergence; numerical experiment
UR - http://eudml.org/doc/277842
ER -

References

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