The Implementation of SIRK Methods for Differential Algebraic Equations

Abstract

The aim of this report is to survey some aspects of implict differential equations, differential algebraic equations, and their numerical solution. We will formulate an algorithm to solve implicitly defined differential systems by singly implicit Runge-Kutta methods, and see how estimates of the local error can be computed. Then we will give some convergence results for Runge-Kutta methods applied to differential algebraic systems. As a part of the project we have implemented an experimental version of the stiff ODE solver STRIDE by J.C. Butcher, K. Burrage and F.H. Chipman. The new code is capable of solving systems in implicit form and some differential algebraic systems. We have used the code to verify the convergence results and the local error estimates for SIRK methods. Furthermore we have tested the experimental code on some real applications.