A Priori Error Analysis of a Discontinuous Galerkin Scheme for the Magnetic Induction Equation

References

[1] N. Besse and D. Kröner, Convergence of locally divergence-free discontinuous-Galerkin methods for the induction equations of the 2D-MHD system, ESAIM Math. Model. Numer. Anal. 39 (2005), no. 6, 1177–1202. 10.1051/m2an:2005051Search in Google Scholar

[2] 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), no. 3, 426–430. 10.1016/0021-9991(80)90079-0Search in Google Scholar

[3] F. Brezzi, B. Cockburn, L. D. Marini and E. Süli, Stabilization mechanisms in discontinuous Galerkin finite element methods, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 25–28, 3293–3310. 10.1016/j.cma.2005.06.015Search in Google Scholar

[4] 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), no. 6, 2019–2042. 10.1137/090757940Search in Google Scholar

[5] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Stud. Math. Appl. 4, North-Holland, Amsterdam, 1978. 10.1115/1.3424474Search in Google Scholar

[6] B. Cockburn and C. W. Shu, The Runge–Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems, J. Comput. Phys. 141 (1998), no. 2, 199–224. 10.1006/jcph.1998.5892Search in Google Scholar

[7] D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, Math. Appl. 69, Springer, Berlin, 2012. 10.1007/978-3-642-22980-0Search in Google Scholar

[8] L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, American Mathematical Society, Providence, 2007. Search in Google Scholar

[9] F. G. Fuchs, K. H. Karlsen, S. Mishra and N. H. Risebro, Stable upwind schemes for the magnetic induction equation, ESAIM Math. Model. Numer. Anal. 43 (2009), no. 5, 825–852. 10.1051/m2an/2009006Search in Google Scholar

[10] S. Gottlieb and C. W. Shu, Total variation diminishing Runge–Kutta schemes, Math. Comp. 67 (1998), no. 221, 73–85. 10.1090/S0025-5718-98-00913-2Search in Google Scholar

[11] B. Gustafsson, H.-O. Kreiss and J. Oliger, Time Dependent Problems and Difference Methods, Pure Appl. Math. (Hoboken) 24, John Wiley & Sons, Hoboken, 1995. Search in Google Scholar

[12] E. Hairer, S. P. Nrsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer Ser. Comput. Math. 8, Springer, Heidelberg, 2010. Search in Google Scholar

[13] E. Hairer, S. P. Nrsett and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer Ser. Comput. Math. 14, Springer, Heidelberg, 2010. Search in Google Scholar

[14] P. Houston, I. Perugia, A. Schneebeli and D. Schötzau, Interior penalty method for the indefinite time-harmonic Maxwell equations, Numer. Math. 100 (2005), no. 3, 485–518. 10.1007/s00211-005-0604-7Search in Google Scholar

[15] P. Houston, I. Perugia and D. Schötzau, Mixed discontinuous Galerkin approximation of the Maxwell operator, SIAM J. Numer. Anal. 42 (2004), no. 1, 434–459. 10.1137/S003614290241790XSearch in Google Scholar

[16] U. Koley, S. Mishra, N. H. Risebro and M. Svärd, Higher order finite difference schemes for the magnetic induction equations, BIT 49 (2009), no. 2, 375–395. 10.1007/s10543-009-0219-ySearch in Google Scholar

[17] U. Koley, S. Mishra, N. H. Risebro and M. Svärd, Higher-order finite difference schemes for the magnetic induction equations with resistivity, IMA J. Numer. Anal. 32 (2012), no. 3, 1173–1193. 10.1093/imanum/drq030Search in Google Scholar

[18] H.-O. Kreiss and J. Lorenz, Initial-Boundary Value Problems and Navier–Stokes Equations, Pure Appl. Math. 136, Academic Press, New York, 1989. Search in Google Scholar

[19] B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, Front. Appl. Math., Society for Industrial and Applied Mathematics, Philadelphia, 2008. 10.1137/1.9780898717440Search in Google Scholar

[20] T. Sarkar, Interior penalty discontinuous Galerkin method for the magnetic induction equation with resistivity, Appl. Math. Comput. 314 (2017), 212–227. 10.1016/j.amc.2017.07.018Search in Google Scholar

[21] T. Sarkar and C. Praveen, Stabilized discontinuous Galerkin method for the magnetic induction equation, submitted. Search in Google Scholar

[22] E. Tadmor, From semidiscrete to fully discrete: Stability of Runge–Kutta Schemes by the energy method. II, SIAM Proc. Appl. Math. 109 (2002), 25–49. Search in Google Scholar

[23] G. Tóth, The ∇ ⋅ B = 0 constraint in shock-capturing magnetohydrodynamics codes, J. Comput. Phys., 161 (2000), no. 2, 605–652. 10.1006/jcph.2000.6519Search in Google Scholar

[24] H. Yang and F. Li, Stability analysis and error estimates of an exactly divergence-free method for the magnetic induction equations, ESAIM Math. Model. Numer. Anal. 50 (2016), no. 4, 965–993. 10.1051/m2an/2015061Search in Google Scholar

[25] Q. Zhang and C. W. Shu, Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws, SIAM J. Numer. Anal. 44 (2006), no. 4, 1703–1720. 10.1137/040620382Search in Google Scholar

[26] Q. Zhang and C. W. Shu, Stability analysis and a priori error estimates of the third order explicit Runge–Kutta discontinuous Galerkin method for scalar conservation laws, SIAM J. Numer. Anal. 48 (2010), no. 3, 1038–1063. 10.1137/090771363Search in Google Scholar