Constraint Preserving Discontinuous Galerkin Method for Ideal Compressible MHD on 2-D Cartesian Grids
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Constraint Preserving Discontinuous Galerkin Method for Ideal Compressible MHD on 2-D Cartesian Grids Praveen Chandrashekar1
· Rakesh Kumar1
Received: 1 December 2019 / Revised: 27 April 2020 / Accepted: 26 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We propose a constraint preserving discontinuous Galerkin method for ideal compressible MHD in two dimensions and using Cartesian grids, which automatically maintains the global divergence-free property. The approximation of the magnetic field is achieved using Raviart– Thomas polynomials and the DG scheme is based on evolving certain moments of these polynomials which automatically guarantees divergence-free property. We also develop HLLtype multi-dimensional Riemann solvers to estimate the electric field at vertices which are consistent with the 1-D Riemann solvers. When limiters are used, the divergence-free property may be lost and it is recovered by a divergence-free reconstruction step. We show the performance of the method on a range of test cases up to fourth order of accuracy. Keywords Ideal compressible MHD · Divergence-free · Discontinuous Galerkin method · Multi-dimensional Riemann solvers
1 Introduction The equations governing ideal, compressible MHD are a mathematical model for plasma and form a system of non-linear hyperbolic conservation laws. While it is natural to try to use Godunov-type numerical methods which have been very successful for other non-linear hyperbolic conservation laws [44], the MHD equations have an additional feature in the form of a constraint on the magnetic field B, i.e., the divergence of B must be zero, which may not be satisfied by standard schemes. The non satisfaction of this constraint can yield wrong solutions and the methods can also be unstable [46]. Hence various strategies have been developed over the years to deal with this issue. Projection-based methods [11] use standard schemes to update the solution and á posteriori correct the magnetic field to make the divergence to be zero by solving an elliptic equation. Hyperbolic divergence cleaning methods have been developed in [20] by introducing an extra Lagrange multiplier or pressure
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Praveen Chandrashekar [email protected] Rakesh Kumar [email protected]
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TIFR Center for Applicable Mathematics, Bangalore, India 0123456789().: V,-vol
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Journal of Scientific Computing
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variable. Constrained transport methods [22,25] are designed to automatically keep some discrete measure of the divergence to be invariant. A key idea in most of these methods is the staggered storage of variables with the magnetic field components being located on the faces and the remaining hydrodynamic variables being located in cell centers. Divergencefree reconstruction of magnetic field have been developed in conjunction with approximate Riemann solvers [1–3,7] which also preserve a discrete divergence constraint. Another class of methods [10,15,31,39,48] aims to construct a stable scheme without expli
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