the linear rational pseudospectral method for BVPs

Université - Mathématiques, Pérolles, CH-Fribourg,
Switzerland

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

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

former work 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.