High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements

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

“…As an alternative method for eliminating the ill-conditioning without incurring the high O(N 3 ) computational cost that arises from the evaluation of the singular value decomposition [6] mentioned above (which, in application to solution of PDEs is prohibitively expensive), the contribution [4] proposes the following strategy: at first the restriction of the function f to certain very small subintervals of [0, 1] containing the endpoints 0 and 1 (boundary intervals) are projected onto a space of discrete orthogonal polynomials (Gram polynomials ( [25])) of small degree, and Fourier continuation of orthogonal bases of these spaces are precomputed using fine submeshes. The continuation of the original function then proceeds, at FFT speeds, using the projection coefficients of the function at the boundary segments.…”

“…(As noted in [4], this calculation should be performed in high-precision arithmetic, and the small number of associated coefficients should be stored for use as part of the any general domain PDE solver.) To include the odd modes in the continuation of the basis function we consider functionsζ (x) such that they approximate, in the least squares sense, φ (x) and −φ (x + ∆ + d) in ∆ right and 1 + d + ∆ left , respectively:…”

“…Hybridization of that solver with a WENO method, which is left for future work, is expected to give rise to higher efficiencies than the CD6-WENO hybrid. But, in any case, owing to its superior control of pollution error (see Figure 1) and general ability to handle complex, non-periodic domains at high order (as discussed in detail in the introduction sections of references [4,5]), the FC5-WENO9 method is itself expected to be significantly more efficient and flexible than CD6-WENO in applications involving general geometries Table 2 Error in entropy amplification, Mach = 6.0, T = 2.5 N D = 160, N P = 33, ∆t = 2.1 × 10 −4…”

“…(Here we restrict our considerations to one dimensional problems. In view of the contributions [1,4,5], which concern FC methods for PDEs in two and three spatial dimensions, however, we expect our FC-WENO approach should be applicable and, indeed, highly competitive for systems of conservation laws in both two-and three-dimensional space.) The FC approximation is based on a high-order periodic continuation of a (possibly nonperiodic) function, yet being a Fourier method, the FC approximation retains the equi-spaced grid points, has no dispersion (pollution) error, and, owing to its reliance on the Fast Fourier Transform, it is highly efficient.…”

“…In the present work we propose an alternative approach in which we hybridize the WENO method with a recently proposed Fourier continuation (FC) method [1,4,5]. (Here we restrict our considerations to one dimensional problems.…”