An Assessment of the Efficiency of Nodal Discontinuous Galerkin Spectral Element Methods

Although high order discontinuous Galerkin spectral element methods (DGSEMs) formally require more work than high order finite difference methods, they are not necessarily more expensive in practice. The matrix-vector multiplication used to compute the spatial derivatives compares favorably to the work needed to solve the tri-diagonal systems required by compact finite difference schemes. The stiffness of the two approximations is not significantly different because of the stretching used in finite difference methods to reduce the accuracy loss near boundaries. Implicit methods can give speedups close to two orders of magnitude when memory and time accuracy are not issues. However for time accurate problems much of the advantage is lost because the increase in the time step need not be enough to counterbalance the increased work per time step. Parallelism complicates the situation because the explicit time integration of the DGSEM is effectively parallelized. Overall, the evidence suggests that implicit methods are not the panacea for reducing the stiffness barriers for high order DGSEMs.