ICOSAHOM 2012 location:Gammarth, Tunisia date:25-29 June 2012
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. Stability and accuracy analysis techniques are used to assess the performance of the fully discrete scheme from the point of view of the user, who aims to choose the combination of mesh and numerical method that minimizes the computational time while fulfilling an accuracy requirement on a given frequency range.
In this framework, two scenarios are defined. In the first one, the mesh size can be freely chosen by the user. The efficiency then depends on a trade-off between accuracy and stability: increasing the accuracy of the scheme enables the use of fewer, larger elements, while favouring the stability results in larger time steps. Thus, we define a cost metric involving both stability and accuracy, following the considerations in Ref. [Bernardini and Pirozzoli, J. Comput. Phys. 228(11):4182-4199, 2009]. In the second scenario, the elements are assumed to be constrained to a very small size by geometrical features of the computational domain. In this case, the accuracy can be disregarded, and the computational efficiency is considered to depend only on the stability.
After reviewing relevant Runge-Kutta methods from the literature, we consider schemes of order q from 3 to 4, and number of stages up to q+4, for optimization. In the first scenario, a 8-stage, fourth-order Runge-Kutta scheme (named RKF84) is found to minimize the cost measure. In the second one, we derive one 7-stage, third-order scheme (RKC73) and one 8-stage, fourth-order scheme (RKC84) that maximize the stability limit. The performance of the three new schemes is thoroughly analyzed. Depending on the scenario, they outperfom the Runge-Kutta methods found in the literature by 16% to 27%. Their benefits are also illustrated with examples. For each of the new Runge-Kutta schemes, we provide the coefficients of a 2N-storage implementation.