A global divergence conforming DG method for hyperbolic conservation laws with divergence constraint
This work addresses the challenge of maintaining divergence constraints in numerical simulations of curl-type hyperbolic conservation laws, which is important for applications like magnetohydrodynamics.
The paper proposes a globally divergence-conforming discontinuous Galerkin method for hyperbolic conservation laws with divergence constraints, achieving exact divergence preservation at the discrete level. Numerical results for the induction equation demonstrate stability and accuracy up to third order.
We propose a globally divergence conforming discontinuous Galerkin (DG) method on Cartesian meshes for {\em 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.