Modified exponential-rational methods for the numerical olution of first order initial value problems

Abstract

Exponentially-fitted numerical methods are appealing because L-stability is guaranteed when solving initial value problems of the form Such numerical methods also yield the exact solution when solving the above-mentioned problem. Whilst rational methods have been well established in the past decades, most of them are not ‘completely’ exponentially-fitted. Recently, a class of one-step exponential-rational methods (ERMs) was discovered. Analyses showed that all ERMs are exponentially-fitted, hence implying L-stability. Several numerical experiments showed that ERMs are more accurate than existing rational methods in solving general initial value problem. However, ERMs have two weaknesses: every ERM is non-uniquely defined and may return complex values. Therefore, the purpose of this study was to modify the original ERMs so that these weaknesses will be overcome. This study discusses the generalizations of the modified ERMs and the theoretical analyses involved such as consistency, stability and convergence. Numerical experiments showed that the modified ERMs and the original ERMs are found to have comparable accuracy; hence modified ERMs are preferable to original ERMs.