Iterative solution of the systems arising from
the linear rational pseudospectral method for BVPs
 

Jean-Paul BERRUT

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.