From h to p efficiently: Optimal implementation strategies for explicit time-dependent problems using the spectral/hp element method

We investigate the relative performance of a second-order Adams-Bashforth scheme and second- and fourth-order Runge-Kutta schemes when time-stepping a 2D linear advection problem discretised using a spectral/hp element technique for a range of different mesh sizes and polynomial orders. Numerical experiments explore the effects of short (2 wavelengths) and long (32 wavelengths) time integration for sets of uniform and non-uniform meshes. The choice of time-integration scheme and discretisation together fixes a CFL limit which imposes a restriction on the maximum time-step which can be taken to ensure numerical stability. The number of steps, together with the order of the scheme, affects not only the runtime but also the accuracy of the solution. Through numerical experiments we systematically highlight the relative effects of spatial resolution and choice of time integration on performance and provide general guidelines on how best to achieve the minimal execution time in order to obtain a prescribed solution accuracy. The significant role played by higher polynomial orders in reducing CPU-time while preserving accuracy becomes more evident, especially for uniform meshes, compared to what has been typically considered when studying this type of problem.

Article last modified on September 6, 2014 at 10:32 pm.