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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. E.N.Burman@sussex.ac.uk
Alexandre Ern
Affiliation:
Université Paris-Est, CERMICS, École des Ponts, 77455 Marne la Vallée Cedex 2, France; ern@cermics.enpc.fr
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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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References

Références

Ascher, U.M., Ruuth, S.J. and Spiteri, R.J., Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Special issue on time integration (Amsterdam, 1996). Appl. Numer. Math. 25 (1997) 151–167. Google Scholar
Ascher, U.M., Ruuth, S.J. and Wetton, B.T.R., Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32 (1995) 797–823. Google Scholar
Braack, M., Burman, E., John, V. and Lube, G., Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Engrg. 196 (2007) 853–866. Google Scholar
Brooks, A.N. and Hughes, T.J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. FENOMECH’81, Part I, Stuttgart (1981). Comput. Methods Appl. Mech. Engrg. 32 (1982) 199–259. Google Scholar
Burman, E., A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal. 43 (2005) 2012–2033 (electronic). Google Scholar
Burman, E., Consistent SUPG-method for transient transport problems : Stability and convergence. Comput. Methods Appl. Mech. Engrg. 199 (2010) 1114–1123. Google Scholar
E. Burman and A. Ern, A continuous finite element method with face penalty to approximate Friedrichs’ systems. ESAIM : M2AN41 (2007) 55–76.
Burman, E., Ern, A. and Fernández, M.A., Explicit Runge-Kutta schemes and finite elements with symmetric stabilization for first-order linear PDE systems. SIAM J. Numer. Anal. 48 (2010) 2019–2042. Google Scholar
Burman, E. and Fernández, M.A., Finite element methods with symmetric stabilization for the transient convection-diffusion-reaction equation. Comput. Methods Appl. Mech. Engrg. 198 (2009) 2508–2519. Google Scholar
Burman, E. and Hansbo, P., Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Engrg. 193 (2004) 1437–1453. Google Scholar
E. Burman and G. Smith, Analysis of the space semi-discretized SUPG method for transient convection-diffusion equations. Technical report, University of Sussex (2010).
Burman, E., Guzmán, J. and Leykekhman, D., Weighted error estimates of the continuous interior penalty method for singularly perturbed problems. IMA J. Numer. Anal. 29 (2009) 284–314. Google Scholar
Cockburn, B. and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comp. 52 (1989) 411–435. Google Scholar
Codina, R., Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4295–4321. Google Scholar
Crouzeix, M., Une méthode multipas implicite-explicite pour l’approximation des équations d’évolution paraboliques. Numer. Math. 35 (1980) 257–276. Google Scholar
Di Pietro, D.A., Ern, A. and Guermond, J.-L., Discontinuous Galerkin methods for anisotropic semidefinite diffusion with advection, SIAM J. Numer. Anal. 46 (2008) 805–831. Google Scholar
A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Appl. Math. Sci. 159 (2004).
Ern, A. and Guermond, J.-L., Discontinuous Galerkin methods for Friedrichs’ systems. I. General theory. SIAM J. Numer. Anal. 44 (2006) 753–778. Google Scholar
Guermond, J.-L., Stabilization of Galerkin approximations of transport equations by subgrid modeling. ESAIM : M2AN 33 (1999) 1293–1316. Google Scholar
Guermond, J.-L., Subgrid stabilization of Galerkin approximations of linear monotone operators. IMA J. Numer. Anal. 21 (2001) 165–197. Google Scholar
Guzmán, J., Local analysis of discontinuous Galerkin methods applied to singularly perturbed problems. J. Numer. Math. 14 (2006) 41–56. Google Scholar
F. Hecht, O. Pironneau, A. Le Hyaric and J. Morice, FreeFEM++, Version 3.14-0. http://www.freefem.org/ff++/.
Johnson, C., Nävert, U. and Pitkäranta, J., Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg. 45 (1984) 285–312. Google Scholar
Johnson, C. and Pitkäranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp. 46 (1986) 1–26. Google Scholar
P. Lesaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, in Mathematical aspects of Finite Elements in Partial Differential Equations, edited by C. de Boors. Academic Press (1974) 89–123.
Levy, D. and Tadmor, E., From semidiscrete to fully discrete : stability of Runge–Kutta schemes by the energy method. SIAM Rev. 40 (1998) 40–73 (electronic). Google Scholar
Pareschi, L. and Russo, G., Implicit-explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25 (2005) 129–155. Google Scholar
H.-G. Roos, M. Stynes and L. Tobiska, Robust numerical methods for singularly perturbed differential equations, Convection-diffusion-reaction and flow problems. Springer Series in Computational Mathematics, 2nd edition. Springer-Verlag, Berlin 24 (2008).