Accuracy, Resolution and Stability Properties of a Modified Chebyshev Method

While the Chebyshev pseudospectral method provides a spectrally accurate method, integration of partial differential equations with spatial derivatives of order M requires time steps of approximately O(N−2M) for stable explicit solvers. Theoretically, time steps may be increased to O(N−M) with the use of a parameter, α-dependent mapped method introduced by Kosloff and Tal-Ezer [ J. Comput. Phys., 104 (1993), pp. 457–469]. Our analysis focuses on the utilization of this method for reasonable practical choices for N, namely N ≲ 30, as may be needed for two- or three dimensional modeling. Results presented confirm that spectral accuracy with increasing N is possible both for constant α (Hesthaven, Dinesen, and Lynov [J. Comput. Phys., 155 (1999), pp. 287–306]) and for α scaled with N, α sufficiently different from 1 (Don and Solomonoff [SIAM J. Sci. Comput., 18 (1997), pp. 1040–1055]). Theoretical bounds, however, show that any realistic choice for α, in which both resolution and accuracy considerations are imposed, permits no more than a doubling of the time step fora stable explicit integrator in time, much less than the O(N) improvement claimed by Kosloff and Tal-Ezer. On the other hand, by choosing α carefully, it is possible to improve on the resolution of the Chebyshev method; in particular, one may achieve satisfactory resolution with fewer than π points per wavelength. Moreover, this improvement is noted not only for waves with the minimal resolution but also for waves sampled up to about 8 points per wavelength. Our conclusions are verified by calculation of phase and amplitude errors for numerical solutions of first and second order one-dimensional wave equations. Specifically, while α can be chosen such that the mapped method improves the accuracy and resolution of the Chebyshev method, for practical choices of N, it is not possible to achieve both single precision accuracy and gain the advantage of an O(N−M) time step

Stable high-order finite-difference methods based on non-uniform grid point distributions

It is well known that high-order finite-difference methods may become unstable due to the presence of boundaries and the imposition of boundary conditions. For uniform grids, Gustafsson, Kreiss, and Sundstr¨om theory and the summation-by-parts method provide sufficient conditions for stability. For non-uniform grids, clustering of nodes close to the boundaries improves the stability of the resulting finite-difference operator. Several heuristic explanations exist for the goodness of the clustering, and attempts have been made to link it to the Runge phenomenon present in polynomial interpolations of high degree. By following the philosophy behind the Chebyshev polynomials, a non-uniform grid for piecewise polynomial interpolations of degree q_N is introduced in this paper, where N + 1 is the total number of grid nodes. It is shown that when q = N, this polynomial interpolation coincides with the Chebyshev interpolation, and the resulting finite-difference schemes are equivalent to Chebyshev collocation methods. Finally, test cases are run showing how stability and correct transient behaviours are achieved for any degree q<N through the use of the proposed non-uniform grids. Discussions are complemented by spectra and pseudospectra of the finite-difference operators

Spectral methods based on prolate spheroidal wave functions for hyperbolic PDEs

We examine the merits of using prolate spheroidal wave functions (PSWFs) as basis functions when solving hyperbolic PDEs using pseudospectral methods. The relevant approximation theory is reviewed and some new approximation results in Sobolev spaces are established. An optimal choice of the band-limit parameter for PSWFs is derived for single-mode functions. Our conclusion is that one might gain from using the PSWFs over the traditional Chebyshev or Legendre methods in terms of accuracy and efficiency for marginally resolved broadband solutions

Chebyshev pseudospectral methods for conservation laws with source terms and applications to multiphase flow

Pseudospectral methods are well known to produce superior results for the solution of partial differential equations whose solutions have a certain amount of regularity. Recent advances have made possible the use of spectral methods for the solution of conservation laws whose solutions may contain shocks. We use a recently described Super Spectral Viscosity method to obtain stable approximations of Systems of Nonlinear Hyperbolic Conservation Laws. A recently developed postprocessing method, which is theoretically capable of completely removing the Gibbs phenomenon from the Super Spectral Viscosity approximation, is examined. The postprocessing method has shown great promise when applied in some simple cases. We discuss its application to more complicated problems and examine the possibility of the method being used as a black box postprocessing method. Applications to multiphase fluid flow are made