Efficient High-Order Methods for ODEs and DAEs

Abstract

We present methods for the high-order differentiation through ordinary differential equations (ODEs), and more importantly, differential algebraic equations (DAEs). First, methods are developed that assert that the requested derivatives are really those of the solution of the ODE, and not those of the algorithm used to solve the ODE. Next, high-order solvers for DAEs are developed that in a fully automatic way turn an n-th order solution step of the DAEs into a corresponding step for an ODE initial value problem. In particular, this requires the automatic high-order solution of implicit relations, which is achieved using an iterative algorithm that converges to the exact result in at most n + 1 steps. We give examples of the performance of the method.