# Arbitrary high order A-stable and B-convergent numerical methods for ODEs via deferred correction

## TL;DR

This paper develops a sequence of high-order, A-stable, B-convergent numerical methods for solving first-order ODEs using recursive deferred correction schemes based on the implicit midpoint method.

## Contribution

It introduces a new family of high-order deferred correction schemes that are A-stable and B-convergent, with proven order enhancement through correction steps.

## Key findings

- Numerical experiments confirm high accuracy of schemes DC2 to DC10.
- Theoretical order of accuracy is achieved in practice.
- Schemes demonstrate satisfactory stability on stiff and non-stiff ODEs.

## Abstract

This paper presents a sequence of deferred correction (DC) schemes built recursively from the implicit midpoint scheme for the numerical solution of general first order ordinary differential equations (ODEs). It is proven that each scheme is A-stable, satisfies a B-convergence property, and that the correction on a scheme DC(2j) of order 2j of accuracy leads to a scheme DC2j+2 of order 2j+2. The order of accuracy is guaranteed by a deferred correction condition. Numerical experiments with standard stiff and non-stiff ODEs are performed with the DC2, ..., DC10 schemes. The results show a high accuracy of the method. The theoretical orders of accuracy are achieved together with a satisfactory stability.