Continuous numerical methods for ODEs with defect control

Over the last decade several general-purpose numerical methods for ordinary differential equations (ODEs) have been developed which generate a continuous piecewise polynomial approximation that is defined for all values of the independent variable in the range of interest. For such methods it is possible to introduce measures of the quality of the approximate solution based on how well the piecewise polynomial satisfies the ODE. This leads naturally to the notion of “defect-control”. Numerical methods that adopt error estimation and stepsize selection strategies in order to control the magnitude of the associated defect can be very effective and such methods are now being widely used. In this paper we will review the advantages of this class of numerical methods and present examples of how they can be effectively applied. We will focus on numerical methods for initial value problems (IVPs) and boundary value problems (BVPs) where most of the developments have been introduced but we will also discuss the implications and related developments for other classes of ODEs such as delay differential equations (DDEs) and differential algebraic equations (DAEs).