An Explicit Discontinuous Galerkin Scheme with Divergence Cleaning for Magnetohydrodynamics
The explicit space-time expansion discontinuous Galerkin scheme (Gassner et al., J. Sci. Comp. 34(3):260–286, 2008) is applied for solving ideal and viscous magnetohydrodynamic equations. Based on a Taylor expansion in space and time about the barycenter of each cell at the old time level, this predictor-corrector strategy enables each cell to have its own time step whereas the high order of accuracy in time is retained. Thus, it may significantly speed up computations. The discontinuous Galerkin method together with the local time-stepping algorithm allows for an efficient local sub-cycling for a divergence cleaning using a hyperbolic transport correction (Dedner et al., J. Comput. Phys. 175(2):645–673, 2002). Convergence tests and test problems are performed to challenge the capabilities of the space-time expansion scheme.
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- 1.Dedner, A., Kemm, F., Kröner, D., Munz, C.-D., Schnitzer, T. and Wesenberg, M. Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys. 175, 2 (2002), 645–673Google Scholar
- 2.Gassner, G., Lörcher, F. and Munz, C.-D. A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes. J. Comput. Phys. 224, 2 (2007), 1049–1063Google Scholar
- 3.Gassner, G., Lörcher, F. and Munz, C.-D. A discontinuous Galerkin scheme based on a space-time expansion II. Viscous flow equations in multi dimensions. J. Sci. Comp. 34, 3 (2008), 260–286Google Scholar
- 4.Gassner, G. Discontinuous Galerkin Methods for the Unsteady Compressible Navier–Stokes Equations. Dissertation, Universität Stuttgart, 2009Google Scholar
- 5.Käser, M. and Dumbser, M. An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes I. The two-dimensional isotropic case with external source terms. Geo. J. Int. 166, 2 (2006), 855–877Google Scholar
- 6.Li, F. and Shu, C.-W. Locally divergence-free discontinuous Galerkin methods for MHD equations. J. Sci. Comp. 22–23, 1 (2005), 413–442Google Scholar
- 7.Li, S. An HLLC Riemann solver for magneto-hydrodynamics. J. Comput. Phys. 203, 1 (2005), 344–357Google Scholar
- 8.Orszag, S. A. and Tang, C. M. Small-scale structure of two-dimensional magnetohydrodynamic turbulence. J. Fluid Mech., (1979), 90–129Google Scholar
- 9.Persson, P.-O. and Peraire, J. Sub-cell shock capturing for discontinuous Galerkin methods. Proc. of the 44th AIAA Aerospace Sciences Meeting and Exhibit, (January 2006)Google Scholar
- 11.Warburton, T. C. and Karniadakis, G. E. A discontinuous Galerkin method for the viscous MHD equations. J. Comput. Phys. 152, 2 (1999), 608–641Google Scholar