Multi-dimensional hybrid Fourier continuation-WENO solvers for conservation laws

We introduce a multi-dimensional point-wise multi-domain hybrid Fourier-Continuation/WENO technique (FC-WENO) that enables high-order and non-oscillatory solution of systems of nonlinear conservation laws, and essentially dispersionless, spectral, solution away from discontinuities, as well as mild CFL constraints for explicit time stepping schemes. The hybrid scheme conjugates the expensive, shock-capturing WENO method in small regions containing discontinuities with the efficient FC method in the rest of the computational domain, yielding a highly effective overall scheme for applications with a mix of discontinuities and complex smooth structures. The smooth and discontinuous solution regions are distinguished using the multi-resolution procedure of Harten [J. Comput. Phys. 115 (1994) 319-338]. We consider a WENO scheme of formal order nine and a FC method of order five. The accuracy, stability and efficiency of the new hybrid method for conservation laws are investigated for problems with both smooth and non-smooth solutions. The Euler equations for gas dynamics are solved for the Mach 3 and Mach 1.25 shock wave interaction with a small, plain, oblique entropy wave using the hybrid FC-WENO, the pure WENO and the hybrid central difference-WENO (CD-WENO) schemes. We demonstrate considerable computational advantages of the new FC-based method over the two alternatives. Moreover, in solving a challenging two-dimensional Richtmyer-Meshkov instability (RMI), the hybrid solver results in seven-fold speedup over the pure WENO scheme. Thanks to the multi-domain formulation of the solver, the scheme is straightforwardly implemented on parallel processors using message passing interface as well as on Graphics Processing Units (GPUs) using CUDA programming language. The performance of the solver on parallel CPUs yields almost perfect scaling, illustrating the minimal communication requirements of the multi-domain strategy. For the same RMI test, the hybrid computations on a single GPU, in double precision arithmetics, displays fiveto six-fold speedup over the hybrid computations on a single CPU. The relative speedup of the hybrid computation over the WENO computations on GPUs is similar to that on CPUs, demonstrating the advantage of hybrid schemes technique on both CPUs and GPUs.