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

Besse, Nicolas; Kröner, Dietmar

ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 6, pp. 1177-1202.

We present the convergence analysis of locally divergence-free discontinuous Galerkin methods for the induction equations which appear in the ideal magnetohydrodynamic system. When we use a second order Runge Kutta time discretization, under the CFL condition $\Delta t\sim h^{4/3}$, we obtain error estimates in $L^2$ of order $𝒪(\Delta t^2+h^{m+1/2})$ where $m$ is the degree of the local polynomials.

Classification : 76W05, 65J10
Mots clés : magnetohydrodynamics, discontinuous-Galerkin methods, convergence analysis

[1] S. Alinhac and P. Gérard, Opérateurs pseudo-différentiels et théorème de Nash-Moser, CNRS Éditions (1991). | MR | Zbl

[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. | MR | Zbl

[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 $\nabla ·𝐁$ on the numerical solution of the magnetohydrodynamic equations. J. Comput. Phys. 35 (1980) 426. | MR | Zbl

[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. | Zbl

[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. | Zbl

[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. | Zbl

[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. | Zbl

[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). | Zbl

[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. | Zbl

[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. | Numdam | Zbl

[14] K.O. Friedrichs, Symmetric positive linear differential equations. Comm. Pure Appl. Math. XI (1958) 333-418. | Zbl

[15] V. Gilrault and P.-A. Raviart, Finite element methods for the Navier-Stokes equatons, Theory and algorithms. Springer Ser. Comput. Math. 5 (1986). | Zbl

[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. | Zbl

[17] F. Li, C.-W. Shu, Locally divergence-free discontinuous Galerkin methods for MHD equations. SIAM J. Sci. Comput. 27 (2005) 413-442. | Zbl

[18] J.-L. Lions and J. Petree, Sur une classe d'espaces d'interpolation. Publ. I.H.E.S. 19 (1964) 5-68. | Numdam | Zbl

[19] J.C. Nédélec, Mixed finite element in ${ℝ}^3$. Numer. Math. 35 (1980) 315-341. | Zbl

[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). | Zbl

[22] G. Tóth, The $\nabla ·𝐁=0$ constraint in shock-capturing magnetohydrodynamics codes. J. Comput. Phys. 161 (2000) 605. | MR | Zbl

[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. | Zbl

[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. | Zbl