Multi-domain Fourier-continuation/WENO hybrid solver for conservation laws

Abstract

We introduce a multi-domain Fourier-Continuation/WENO hybrid method (FC-WENO) that enables high-order and non-oscillatory solution of systems of nonlinear conservation laws, and which enjoys essentially dispersionless, spectral character away from shocks, as well as mild CFL constraints. The hybrid scheme employs the expensive, shock-capturing WENO method in small regions containing discontinuities and 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 accuracy, stability and efficiency of the new FC-based method for conservation laws is investigated for both problems in which solutions are smooth and problems for which solutions contain shock discontinuities; in particular, in the latter case we compare the efficiency of the hybrid FC-WENO method to that of a purely WENO-based approach. In a variety of examples, including an Euler problem that governs the interaction of a strong shock with a very small entropy wave, we show that the hybrid strategy is several times faster than the pure WENO solver for a comparable level of accuracy.

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

“…As discussed in details in [39], the appearance of very small unstable eigenvalues through the Fourier continuation may render this approach weakly unstable. To remedy this, a very weak exponential filter is used to damp out high-frequency modes with minimal adverse impact on accuracy [20].…”

“…Thus the one-dimensional FC approximation described in [39] directly applies and we highlight the central elements of this approach below.…”

“…Their approach, also adopted here, uses the continuation of Gram polynomials to approximate the matching functionh. For a given approximation order the set of Gram polynomials are unique and thus their continuations can be carried out in a preprocessing stage and used to approximate the periodic extension of any given function f via its matching functionh as detailed in [6,39]. It is worth emphasizing that extensive experimentation with higher values of p does not show significant advantages for the class of problems considered in this work.…”

“…Finite volume version of the high-order WENO scheme have been used in shock wave bubble interactions in three dimensions [36]. It is notable that although the formal high-order accuracy of the WENO schemes is degraded near discontinuities and large solution gradients, the high-order scheme do indeed achieve higher level of accuracy/resolution in a more computational efficient fashion as demonstrated by recent investigations [24,25,39]. Recent research activities in extending and optimizing WENO schemes to orders up to 16 [37,38] may also be motivated by the computational efficiency of these higher order schemes over their lower order counterparts.…”

“…In this work, extending the one-dimensional version of the hybrid approach [39] to two-dimensional problems, we propose an attractive alternative in which we hybridize the WENO method with a recently proposed Fourier continuation (FC) method [1,6,7]. The FC approximation is based on a high-order periodic continuation of a (possibly non-periodic) function, yet being based on a Fourier method, the FC approximation has almost no dispersion (pollution) error and, utilizing an equispaced grid, provides a simple and efficient interface with WENO scheme and the multi-resolution analysis while allowing for an efficient temporal integration as compared to the Chebyshev-WENO scheme.…”