The linear rational pseudospectral method with iteratively optimized poles for two-point boundary value problems

An algorithm is proposed which improves upon the polynomial pseudospectral method for solving linear two-point boundary value problems. In the latter, the collocation points are the vertical projection onto the interval of points equidistant or nearly equidistant on the circle, and they therefore accumulate in the vicinity of the extremities of the interval. Thus the method is well-suited for solving problems whose solutions have boundary layers but not as good at approximating solutions with large gradients (shocks) away from the extremities of their domain of definition. Our idea is to modify the polynomial ansatz by attaching a denominator so as to make it a rational interpolant. The denominator is then successively optimized in an iterative procedure with each step consisting of the solution of two problems: an optimization of the denominator for given values of the approximation to the solution u at the interpolation points and a collocation in the linear space of the rational interpolants with the just obtained xe denominator to obtain new approximate values of u. We show the efficiency of a Galerkin version of the method and discuss the power of the collocation version with several numerical examples.