A Global Divergence Conforming DG Method for Hyperbolic Conservation Laws with Divergence Constraint

Abstract

We propose a globally divergence conforming discontinuous Galerkin (DG) method on Cartesian meshes for curl-type hyperbolic conservation laws based on directly evolving the face and cell moments of the Raviart-Thomas approximation polynomials. The face moments are evolved using a 1-D discontinuous Gakerkin method that uses 1-D and multi-dimensional Riemann solvers while the cell moments are evolved using a standard 2-D DG scheme that uses 1-D Riemann solvers. The scheme can be implemented in a local manner without the need to solve a global mass matrix which makes it a truly DG method and hence useful for explicit time stepping schemes for hyperbolic problems. The scheme is also shown to exactly preserve the divergence of the vector field at the discrete level. Numerical results using second and third order schemes for induction equation are presented to demonstrate the stability, accuracy and divergence preservation property of the scheme.

Constraint preserving discontinuous Galerkin method for ideal compressible MHD on 2-D Cartesian grids

“…For a proof of the above theorem, we refer the reader to [12], [14]. Note that this reconstruction is very local to each cell; it uses data in the cell and on its faces.…”

“…x , b ± y , α ij , β ij comes from a divergence-free vector field, then the reconstructed field is also divergence-free [14]. Hence, the vector field B will be globally divergence-free also.…”

“…The divergence-free reconstruction idea has been combined in a DG scheme in [6] for induction equation and for Maxwell's equations in [30]. DG schemes which automatically preserve the divergence condition have been developed in [36], [35] using central DG idea and in [24], [14] using Godunov approach. Maintaining the positivity of solutions is very important and recent work shows a close link between this property and a discrete divergence-free condition [50], see also [31].…”

“…In the present work, we develop a DG scheme based on tensor product polynomials and in particular using Raviart-Thomas polynomials for the approximation of the magnetic field. This builds on the initial work done for induction equation in [14] and is similar in spirit to [24], which developed up to third order schemes using Brezzi-Douglas Marin (BDM) polynomials (see [12], Section III.3.2) for the magnetic field.…”

Geometric Quasilinearization Framework for Analysis and Design of Bound-Preserving Schemes

“…Although a few globally divergence-free techniques (e.g. [30,16,6]) were developed and can enforce the condition (6.16), the local scaling limiter for (6.12) will destroy the globally divergence-free property. Notice that the locally divergence-free technique (e.g.…”