A Runge-Kutta Fehlberg method with phase-lag of order infinity for initial-value problems with oscillating solution

AbstractA Runge-Kutta method is developed for the numerical solution of initial-value problems with oscillating solution. Based on the Runge-Kutta Fehlberg 2(3) method, a Runge-Kutta method with phase-lag of order infinity is developed. Based on these methods we produce a new embedded Runge-Kutta Fehlberg 2(3) method with phase-lag of order infinity. This method is called as Runge-Kutta Fehlberg Phase Fitted method (RKFPF). The numerical results indicate that this new method is much more efficient, compared with other well-known Runge-Kutta methods, for the numerical solution of differential equations with oscillating solution, using variable step size

GPU Accelerated Discontinuous Galerkin Methods for Shallow Water Equations

We discuss the development, verification, and performance of a GPU accelerated discontinuous Galerkin method for the solutions of two dimensional nonlinear shallow water equations. The shallow water equations are hyperbolic partial differential equations and are widely used in the simulation of tsunami wave propagations. Our algorithms are tailored to take advantage of the single instruction multiple data (SIMD) architecture of graphic processing units. The time integration is accelerated by local time stepping based on a multi-rate Adams-Bashforth scheme. A total variational bounded limiter is adopted for nonlinear stability of the numerical scheme. This limiter is coupled with a mass and momentum conserving positivity preserving limiter for the special treatment of a dry or partially wet element in the triangulation. Accuracy, robustness and performance are demonstrated with the aid of test cases. We compare the performance of the kernels expressed in a portable threading language OCCA, when cross compiled with OpenCL, CUDA, and OpenMP at runtime.Comment: 26 pages, 51 figure