Generalized mono-implicit Runge-Kutta methods for stiff ordinary differential equations

Abstract

Ordinary differential equations (ODEs) arise in many applications. Typically these ODEs are sufficiently complicated that they must be solved using numerical methods. One of the well-known classes of numerical methods for ODEs is the class of Mono- Implicit Runge-Kutta (MIRK) methods. An important property of a MIRK method is its order; a method is of order p if its global error is O(hp). An issue with MIRK methods, when applied to certain ODEs, known as stiff ODEs, is that when they should be of order p, they perform as if their order is q, where q < p. This is called order reduction. This means that the MIRK methods will be inefficient when the ODE is stiff because the amount of computation that is performed is not consistent with the accuracy obtained. In this thesis, we derive generalizations of MIRK methods that can avoid order reduction when the ODE is stiff.