Doubling the degree of precision without doubling the grid when solving a differential equation with a pseudo-spectral collocation method (Q1610564)

A pseudospectral collocation method for initial value problems of systems of first-order ordinary differential equations is presented. The new idea is to collocate the differential equation \(y'= f(x,y)\) and its differentiated version \(y''= (f_x+ f_y\cdot f)(x,y)\) simultaneously, using Hermite rather than Lagrange interpolation. The accuracy obtained by this approach with \(N\) gridpoints is shown in numerical examples to be comparable with the accuracy obtained by the conventional pseudospectral collocation method with \(2N\) gridpoints. However, the condition number is approximately \(N\) times larger than for the conventional method.