Performance Evaluation of an Extrapolation Method for Ordinary Differential Equations with Error-free Transformation

TL;DR

This paper evaluates an extrapolation method enhanced with error-free transformation (EFT) for solving ordinary differential equations, demonstrating improved efficiency especially for large linear and small nonlinear problems.

Contribution

It introduces the application of EFT to explicit extrapolation methods for ODEs, enabling more efficient computations with existing BLAS functions.

Findings

Effective for large linear ODEs

Efficient for small nonlinear ODEs

Particularly beneficial with harmonic sequence

Abstract

The application of error-free transformation (EFT) is recently being developed to solve ill-conditioned problems. It can reduce the number of arithmetic operations required, compared with multiple precision arithmetic, and also be applied by using functions supported by a well-tuned BLAS library. In this paper, we propose the application of EFT to explicit extrapolation methods to solve initial value problems of ordinary differential equations. Consequently, our implemented routines can be effective for large-sized linear ODE and small-sized nonlinear ODE, especially in the case when harmonic sequence is used.