Block multistep methods based on rational approximants

Abstract

Variable step direct block multistep method for general second order ODEs AIP Conf.A rational approximation for the parabolic equation method A parabolic wave equation based on a rational-cubic approximation J. Acoust. Soc. Am. 87, 619 (1990); 10.1121/1.398930 N−variable rational approximants and method of moments Abstract. In this study, the concept of block multistep methods based on rational approximants is introduced for the numerical solution of first order initial value problems. These numerical methods are also called rational block multistep methods. The main reason to consider block multistep methods in rational setting, is to improve the numerical accuracy and absolute stability property of existing block multistep methods that are based on polynomial approximants. For this pilot study, a 2-point explicit rational block multistep method is developed. Local truncation error and stability analysis for this new method are included as well. Numerical experimentations and results using some test problems are presented. Numerical results are satisfying in terms of numerical accuracy. Finally, future issues on the developments of rational block multistep methods are discussed.