Author:

Keywords:

Discontinuous Galerkin, Runge-Kutta, Wave Propagation, Computational Efficiency, Science & Technology, Technology, Physical Sciences, Computer Science, Interdisciplinary Applications, Physics, Mathematical, Computer Science, Physics, Wave propagation, Computational efficiency, COMPUTATIONAL ACOUSTICS, LOW-DISSIPATION, TIME INTEGRATION, OPTIMIZATION, DISPERSION, AEROACOUSTICS, 01 Mathematical Sciences, 02 Physical Sciences, 09 Engineering, Applied Mathematics, 40 Engineering, 49 Mathematical sciences, 51 Physical sciences

Abstract:

We study the performance of methods of lines combining discontinuous Galerkin spatial discretizations and explicit Runge-Kutta time integrators, with the aim of deriving optimal Runge-Kutta schemes for wave propagation applications. We review relevant Runge-Kutta 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 Runge-Kutta 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 Runge-Kutta methods, we provide the coefficients for a 2N-storage implementation, along with the information needed by the user to employ them optimally.