An exponential integrator/WENO discretization for sonic-boom simulation on modern computer hardware

Abstract

Recently a splitting approach has been presented for the simulation of sonicboom propagation. Splitting methods allow one to divide complicated partial differential equations into simpler parts that are solved by specifically tailored numerical schemes. The present work proposes a second order exponential integrator for the numerical solution of sonic-boom propagation modelled through a dispersive equation with Burgers' nonlinearity. The linear terms are efficiently solved in frequency space through FFT, while the nonlinear terms are efficiently solved by a WENO scheme. The numerical method is designed to be highly parallelisable and therefore takes full advantage of modern computer hardware. The new approach also improves the accuracy compared to the splitting method and it reduces oscillations. The enclosed numerical results illustrate that parallelisation on a CPU results in a speedup of 22 times faster than the straightforward sequential version. The GPU implementation further accelerates the runtime by a factor 3, which improves to 5 when single precision is used instead of double precision.

Exponential Integrators for Resistive Magnetohydrodynamics: Matrix-free Leja Interpolation and Efficient Adaptive Time Stepping

“…In some problems, for an appropriate choice of A, the action of the matrix function can be computed directly (using FFT techniques). This approach has been used, among others, for kinetic equations (Crouseilles et al 2018;Caliari et al 2021), water waves (Klein & Roidot 2011;Caliari et al 2021), sonic boom propagation (Einkemmer et al 2021), and the simulation of Bose-Einstein condensates (Besse et al 2017). However, due to the fully nonlinear nature and the fundamentally complex structure of the MHD equations, such an approach is not feasible.…”