david/extrapolation_vs_IDC

Most initial value ODEs are efficiently solved by modern integration methods of moderate order. For some problems, such as chaotic systems, methods of very high order may be required. The construction of very high order methods is challenging: very high order Runge–Kutta methods are subject to vast numbers of order conditions while very high order linear multistep methods tend to have poor stability properties.

Two well-known approaches exist for the construction of high-order methods: extrapolation and deferred correction. Both are usually viewed as iterative methods, since they build up a high order solution based on repeated computation of low order approximations. However, when the base method on which they are built is a one-step method (i.e., a Runge–Kutta method), both extrapolation and deferred correction result in what is in fact a Runge–Kutta method – with very many stages.

In the first part of this project, this connection with Runge–Kutta methods will be used to study and compare the stability and accuracy of extrapolation and deferred correction methods based on the forward Euler method. The Python package NodePy will be used and extended to compare stability regions, leading truncation errors, and computational cost of each type of method. We will also consider the effect of varying method parameters such as the extrapolation step number sequence or the location of the nodes in deferred correction.

In the second part of the project, the methods will be compared in practice by implementing them in an adaptive error control framework and applying them to a small set of test problems, including some classical problems and a three-body chaotic problem. We will also compare these results with those obtained from the very high order multistep methods of Glaser and Rokhlin. If time permits, we will construct and analyze extrapolation and deferred correction methods based on higher-order Runge-Kutta and linear multistep methods.

It is expected that the results of this project will be suitable for the preparation of an article in a leading internation journal on numerical analysis, such as SINUM.