Doubling the Degree of Precision Without Doubling the Grid When Solving a Differential Equation with a Pseudo-Spectral Collocation Method

Abstract

In the conventional pseudo-spectral collocation method to solve an ordinary first order differential equation, the derivative is obtained from Lagrange interpolation and has degree of precision N for a grid of (N+1) points. In the present, novel method Hermite interpolation is used as point of departure. From this the second order derivative is obtained with degree of precision (2N+1) for the same grid as above. The associated theorem constitutes the main result of this paper. Based on that theorem a method in put forward in which the differential equation and the differentiated differential equation are simultaneously collocated. In this method every grid point counts for two. The double collocation leads to a solution accuracy which is superior to the precision obtained with the conventional method for the same grid. This superiority is demonstrated by 3 examples, 2 linear problems and a non-linear one. In the examples it is shown that the accuracy obtained with the present method is comparable to the solution accuracy of the standard method with twice the number of grid points. However, the condition number of the present method grows like N ³ as compared to N ² in the standard method.