An implementation variant of the polynomial finite difference method with orthogonal collocation and adjustable element length

The polynomial finite difference method, an easy-to-use variant of the finite difference method for the numerical solution of differential and differential–algebraic equations, has been recently presented [Wu, B., & White, R.E. (2004). Computers & Chemical Engineering, 28, 303–309]. In this work, it is shown that the polynomial finite difference method can be seen as a collocation method with finite elements of equal size with uniform distribution of collocation points within each element. We show that the same type of implementation can be improved if one uses orthogonal distribution of collocation points, without significantly affecting the computational effort. The suggested method is further improved with the use of Michelsen's technique for step-size adjustment to solve stiff differential equations with a semi-implicit third order method. Several examples that show improvements of one or two orders of magnitude of the proposed approach over the implementation by Wu and White are presented.