Computational and Applied Math Proseminar

Department of Mathematics and Statistics, Arizona State University

Thursday, April 18, 2002, 12:15 p.m. in GWC Room 604

J.-P. Berrut

Department of Mathematics, Universite de Fribourg

The Linear Rational Collocation Method for Boundary Value Problems

Abstract The polynomial pseudospectral (collocation) method for the solution of differential equations replaces the derivatives in the differential operator with differentiation matrices of Lagrange basis polynomials at the collocation points. Since the latter are projections onto the interval of equidistant or nearly equidistant points on the circle, they accumulate in the vicinity of the domain boundary. As a consequence, the application of the discretized differential operator by means of the corresponding matrix is ill-conditioned.

A recurrent application of such operators occurs in the time marching solution of the systems of ordinary differential equations arising from discretizing time evolution problems by the method of lines. Kosloff and Tal-Ezer [1] have suggested reducing the ill-conditioning by a change of variable that shifts the collocation points toward the interior of the domain. Many such shifts have since been studied: when conformal, they maintain spectral convergence in space.

In the present work we propose to use the same idea to fight the ill-conditioning in another application of such discretized operators, namely in the iterative solution of the systems of equations arising from solving boundary value problems. For that purpose we use the linear rational collocation method suggested in [2] as an alternate way of using shifted points with time evolution problems.

We will restrict ourselves here to one-dimensional problems, although the generalization to more variables is straightforward. After presenting the method, we prove its convergence in the constant coefficient case, demonstrate the influence of the shift of points on the pseudospectrum of the (preconditioned) differential operators and describe with several examples the gain in iteration number as compared with the classical polynomial pseudospectral method.

[1] Kosloff D., Tal-Ezer H., A modified Chebyshev pseudospectral method with an N^(-1) time step restriction, J. Comput. Phys. 104(1993)457--469.
[2] Baltensperger R., Berrut J.-P., The linear rational collocation method, J. Comp. Appl. Math. 134(2001)243-258.

For further information please contact: mittelmann@asu.edu