A new decoupling technique for the Hermite cubic collocation equations arising from boundary value problems

Abstract

We present a new decoupling technique for solving the linear systems arising from Hermite cubic collocation solutions to boundary value problems with both Dirichlet and Neumann boundary conditions. While the traditional approach yields a linear system of order 2 N×2 N with bandwidth 2, our technique decouples this system into two systems, one with a tridiagonal system of order N−1× N−1 and the other with the identity matrix of order N× N. Besides cutting the work in half, our new approach results in a new tridiagonal system that exhibits the same desirable properties (e.g. symmetric, positive definite) as in the case of finite difference approximations. We validate our theoretical work with a number of experimental results, demonstrating both accuracy and stability.