Optimal RungeKutta schemes for discontinuous Galerkin space discretizations applied to wave propagation problems
We study the performance of methods of lines combining discontinuous Galerkin spatial discretizations and explicit RungeKutta time integrators, with the aim of deriving optimal RungeKutta schemes for wave propagation applications. We review relevant RungeKutta methods from literature, and consider schemes of order q from 3 to 4, and number of stages up to q+4, for optimization. From a user point of view, the problem of the computational efficiency involves the choice of the best combination of mesh and numerical method; two scenarios are defined. In the first one, the element size is totally free, and a 8-stage, fourth-order RungeKutta scheme is found to minimize a cost measure depending on both accuracy and stability. In the second one, the elements are assumed to be constrained to such a small size by geometrical features of the computational domain, that accuracy is disregarded. We then derive one 7-stage, third-order scheme and one 8-stage, fourth-order scheme that maximize the stability limit. The performance of the three new schemes is thoroughly analyzed, and the benefits are illustrated with two examples. For each of these RungeKutta methods, we provide the coefficients for a 2N-storage implementation, along with the information needed by the user to employ them optimally.
rungekutta discontinuous galerkin wave propagation computational efficiency