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With James Rossmanith, I am developping a numerical method for nonlinear hyperbolic conservation laws which is able to take a greatly enhanced timestep by mixing ideas of implicit and explicit time-step schemes. So far, we have seen that this method allows a CFL restriction that is independent of of method order of accuracy. The CFL restriction depends only on the problem dimension and the choice of basis :
| Problem | CFL |
|---|---|
| 1D | 1.0 |
| 2D | 0.75 |
| 3D | 0.6 |
Since this allows the method to take almost two orders of magnitudes fewer timesteps to obtain the same solution as an Runge-Kutta DG method, parallel computing via domain decomposition is incredibly efficient for the RIDG method. Here is a strong scaling study of RIDG vs RKDG vs LIDG for a 2D periodic problem:
More details can be found in these papers
The Regionally-Implicit Discontinuous Galerkin Method: Improving the Stability of DG-FEM