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://archive.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://archive.numdam.org/articles/10.1051/m2an/2011047/

[1] U.M. Ascher, S.J. Ruuth and R.J. Spiteri, 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. | MR | Zbl

[2] U.M. Ascher, S.J. Ruuth and B.T.R. Wetton, Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32 (1995) 797-823. | MR | Zbl

[3] M. Braack, E. Burman, V. John and G. Lube, Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Engrg. 196 (2007) 853-866. | MR | Zbl

[4] A.N. Brooks and T.J.R. Hughes, 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. | MR | Zbl

[5] E. Burman, A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal. 43 (2005) 2012-2033 (electronic). | MR | Zbl

[6] E. Burman, Consistent SUPG-method for transient transport problems : Stability and convergence. Comput. Methods Appl. Mech. Engrg. 199 (2010) 1114-1123. | MR | Zbl

[7] E. Burman and A. Ern, A continuous finite element method with face penalty to approximate Friedrichs' systems. ESAIM : M2AN41 (2007) 55-76. | Numdam | Zbl

[8] E. Burman, A. Ern and M.A. Fernández, Explicit Runge-Kutta schemes and finite elements with symmetric stabilization for first-order linear PDE systems. SIAM J. Numer. Anal. 48 (2010) 2019-2042. | MR | Zbl

[9] E. Burman and M.A. Fernández, Finite element methods with symmetric stabilization for the transient convection-diffusion-reaction equation. Comput. Methods Appl. Mech. Engrg. 198 (2009) 2508-2519. | MR | Zbl

[10] E. Burman and P. Hansbo, Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Engrg. 193 (2004) 1437-1453. | MR | Zbl

[11] E. Burman and G. Smith, Analysis of the space semi-discretized SUPG method for transient convection-diffusion equations. Technical report, University of Sussex (2010). | MR | Zbl

[12] E. Burman, J. Guzmán and D. Leykekhman, Weighted error estimates of the continuous interior penalty method for singularly perturbed problems. IMA J. Numer. Anal. 29 (2009) 284-314. | MR | Zbl

[13] B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comp. 52 (1989) 411-435. | MR | Zbl

[14] R. Codina, Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4295-4321. | MR | Zbl

[15] M. Crouzeix, Une méthode multipas implicite-explicite pour l'approximation des équations d'évolution paraboliques. Numer. Math. 35 (1980) 257-276. | MR | Zbl

[16] D.A. Di Pietro, A. Ern and J.-L. Guermond, Discontinuous Galerkin methods for anisotropic semidefinite diffusion with advection, SIAM J. Numer. Anal. 46 (2008) 805-831. | MR | Zbl

[17] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Appl. Math. Sci. 159 (2004). | MR | Zbl

[18] A. Ern and J.-L. Guermond, Discontinuous Galerkin methods for Friedrichs' systems. I. General theory. SIAM J. Numer. Anal. 44 (2006) 753-778. | MR | Zbl

[19] J.-L. Guermond, Stabilization of Galerkin approximations of transport equations by subgrid modeling. ESAIM : M2AN 33 (1999) 1293-1316. | Numdam | MR | Zbl

[20] J.-L. Guermond, Subgrid stabilization of Galerkin approximations of linear monotone operators. IMA J. Numer. Anal. 21 (2001) 165-197. | MR | Zbl

[21] J. Guzmán, Local analysis of discontinuous Galerkin methods applied to singularly perturbed problems. J. Numer. Math. 14 (2006) 41-56. | MR | Zbl

[22] F. Hecht, O. Pironneau, A. Le Hyaric and J. Morice, FreeFEM++, Version 3.14-0. http://www.freefem.org/ff++/.

[23] C. Johnson, U. Nävert and J. Pitkäranta, Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg. 45 (1984) 285-312. | Zbl

[24] C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp. 46 (1986) 1-26. | Zbl

[25] 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. | MR | Zbl

[26] D. Levy and E. Tadmor, From semidiscrete to fully discrete : stability of Runge-Kutta schemes by the energy method. SIAM Rev. 40 (1998) 40-73 (electronic). | MR | Zbl

[27] L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25 (2005) 129-155. | MR | Zbl

[28] 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). | MR | Zbl

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