Optimized point shifts and poles in the linear rational pseudospectral method for boundary value problems

Authors:
Jean-Paul Berrut;Hans D. Mittelmann
Affiliations:
Département de Mathématiques, Université de Fribourg, CH-1700 Fribourg/Pérolles, Switzerland;Department of Mathematics, Arizona State University, Tempe, Arizona 85287-1804, USA
Venue:
Journal of Computational Physics
Year:
2005

Citing 7
Cited 3

Minimizing multimodal functions of continuous variables with the “simulated annealing” algorithm: Corrigenda for this article is available here

ACM Transactions on Mathematical Software (TOMS)
A modified Chebyshev pseudospectral method with an O(N–1) time step restriction

Journal of Computational Physics
A Fast Adaptive Numerical Method for Stiff Two-Point Boundary Value Problems

SIAM Journal on Scientific Computing
The barycentric weights of rational interpolation with prescribed poles

Journal of Computational and Applied Mathematics - Special issue: dedicated to William B. Gragg on the occasion of his 60th Birthday
Spectral methods in MatLab

Spectral methods in MatLab
Improving the accuracy of the matrix differentiation method for arbitrary collocation points

Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
Adaptive point shifts in rational approximation with optimized denominator

Journal of Computational and Applied Mathematics - Special Issue: Proceedings of the 10th international congress on computational and applied mathematics (ICCAM-2002)

Algorithm 882: Near-Best Fixed Pole Rational Interpolation with Applications in Spectral Methods

ACM Transactions on Mathematical Software (TOMS)
Adaptive rational spectral methods for the linear stability analysis of nonlinear fourth-order problems

Journal of Computational Physics
Barycentric rational interpolation with asymptotically monitored poles

Numerical Algorithms

Quantified Score

Hi-index	31.45

Visualization

Abstract

Due to their rapid - often exponential - convergence as the number N of interpolation/collocation points is increased, polynomial pseudospectral methods are very efficient in solving smooth boundary value problems. However, when the solution displays boundary layers and/or interior fronts, this fast convergence will merely occur with very large N. To address this difficulty, we present a method which replaces the polynomial ansatz with a rational function r and considers the physical domain as the conformal map g of a computational domain. g shifts the interpolation points from their classical position in the computational domain to a problem-dependent position in the physical domain. Starting from a map by Bayliss and Turkel we have constructed a shift that can in principle accomodate an arbitrary number of fronts. Its parameters as well as the poles of r are optimized. Numerical results demonstrate how g best accomodates interior fronts while the poles also handle boundary layers.