A global divergence conforming DG method for hyperbolic conservation laws with divergence constraint

TL;DR

This paper introduces a globally divergence conforming discontinuous Galerkin method for curl-type hyperbolic conservation laws, ensuring divergence preservation and efficient implementation on Cartesian meshes.

Contribution

It develops a novel DG scheme that directly evolves face and cell moments, preserving divergence exactly and enabling local implementation without a global mass matrix.

Findings

The scheme preserves divergence exactly at the discrete level.

Numerical results demonstrate stability and accuracy for the induction equation.

The method is efficient for explicit time stepping schemes.

Abstract

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.