Convergence of locally divergence-free discontinuous-Galerkin methods for the induction equations of the 2D-MHD system

Mathematical Modelling and Numerical Analysis

Volumn 39, Issue 6, 2005, Pages 1177-1202

  Besse, Nicolas ^a   Kröner, Dietmar ^b  

^a UNIVERSITÉ DE STRASBOURG   (France)

^b UNIVERSITY OF FREIBURG   (Germany)

Author keywords

Convergence analysis; Discontinuous Galerkin methods; Magnetohydrodynamics

Indexed keywords

EID: 30344483198     PISSN: 0764583X     EISSN: None     Source Type: Journal    
DOI: 10.1051/m2an:2005051     Document Type: Article

References (24)

1. S. Alinhac and P. Gérard, Opérateurs pseudo- différentiels et théorème de Nash-Moser, CNRS Éditions (1991).
2. G.A. Baker, W.N. Jureidini and O.A. Karakashian, A piecewise solenoidal vector fields and the stokes problem. SIAM J. Numer. Anal. 27 (1990) 1466-1485.
3. D.S. Balsara and D.S. Spicer, A piecewise solenoidal vector fields and the stokes problem. J. Comput. Phys. 149 (1999) 270-292.
4. J.U. Brackbill and D.C. Barnes. The effect of nonzero ∇ · B on the numerical solution of the magnetohydrodynamic equations. J. Comput. Phys. 35 (1980) 426.
5. P.G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of numerical analysis, P.G. Ciarlet and J.-L. Lions, Eds., North-Holland (1991) 17-351.
6. B. Cockburn, Discontinuous Galerkin methods for convection dominated problems, in High-order methods for computational physics, Springer, Berlin. Lect. Notes Comput. Sci. Eng. 9 (1999) 69-224.
7. B. Cockburn, F. Li and C.-W. Shu, Locally divergence-free discontinuous Galerkin-methods for the Maxwell equations. J. Comput. Phys. 194 (2004) 588-610.
8. M. Costabel, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains. Math. Method Appl. Sci. 12 (1990) 365-368.
9. M. Costabel and M. Dauge, Un résultat de densité pour les équations de Maxwell régularisées dans un domaine lipschitzien. C. R. Acad. Sci. Paris Sér. I 327 849-854 (1998).
10. W. Dai and P.R. Woodward, On the divergence-free condition and conservation laws in numerical simulations for supersonic magnetohydrodynamic flow. Astrophys. J. 494 (1998) 317.
11. A. Dedner, F. Kemm, D. Kröner, C.-D. Munz, T. Schnitzer and M. Wesenberg, Hyperbolic Divergence cleaning for the MHD equations. J. Comput. Phys. 175 (2002) 645-673.
12. C.R. Evans and J.F. Hawley, Simulation of magnetohydrodynamic flows, a constrained transport method. Astrophys. J. 332 (1988) 659.
13. C. Foias and R. Temam, Remarques sur les équations de Navier-Stokes et les phénoménes successifs de bifurcation. Ann. Sci. Norm. Sup. Pisa Sér. IV 5 (1978) 29-63.
14. K.O. Friedrichs, Symmetric positive linear differential equations. Comm. Pure Appl. Math. XI (1958) 333-418.
15. V. Gilrault and P.-A. Raviart, Finite element methods for the Navier-Stokes equatons, Theory and algorithms. Springer Ser. Comput. Math. 5 (1986).
16. O.A. Karakashian and W.N. Jureidini, A nonconforming finite element method for the stationary Navier-Stokes equations. SIAM J. Numer. Anal. 35 (1998) 93-120.
17. F. Li, C.-W. Shu, Locally divergence-free discontinuous Galerkin methods for MHD equations. SIAM J. Sci. Comput. 27 (2005) 413-442.
18. J.-L. Lions and J. Petree, Sur une classe d'espaces d'interpolation. Publ. I.H.E.S. 19 (1964) 5-68.
19. 3. Numer. Math. 35 (1980) 315-341.
20. K.G. Powell, An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension), ICASE-Report 94-24 (NASA CR-194902), NASA Langley Research Center, Hampton, VA 23681-0001 (1994).
21. P.-A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite elements methods, in Proc. of the conference held in Rome, 10-12 Dec. 1975, A. Dold, B. Eckmann, Eds., Springer, Berlin, Heidelberg, New York. Lect. Notes Math. 606 (1977).
22. G. Tóth, The ∇ · B = 0 constraint in shock-capturing magnetohydrodynamics codes. J. Comput. Phys. 161 (2000) 605.
23. L. Ying, A second order explicit finite element scheme to multidimensional conservation laws and its convergence. Sci. China Ser. A 43 (2000) 945-957.
24. Q. Zhang and C.-W. Shu, Error Estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Numer. Anal. 42 (2004) 641-666.