Discontinuous Galerkin finite element methods for hyperbolic problems

Hyperbolic conservation laws model phenomena characterized by waves propagating at finite speeds; examples include the compressible Euler (sound waves), Maxwell (light waves), magnetohydrodynamic (magneto-acoustic and Alfven waves), and Einstein (gravitational waves) equations.  In recent years, the discontinuous Galerkin (DG) finite element method (FEM) has become one of the standard approaches for solving hyperbolic problems; however, there are still many open research directions to improve the stability, robustness, and computational efficiency of DG methods. Three key types of improvement that have guided our research are 

  1. Developing robust limiting strategies that prevent unphysical solutions;
  2. Developing new extensions to handle non-standard hyperbolic PDEs; and
  3. Developing novel time-stepping approaches for improved computational efficiency.