Reinterpretation of Multi-Stage Methods for Stiff Systems: A Comprehensive Review on Current Perspectives and Recommendations

Abstract

In this paper, we compare a multi-step method and a multi-stage method for stiff initial value problems. Traditionally, the multi-step method has been preferred than the multi-stage for a stiff problem, to avoid an enormous amount of computational costs required to solve a massive linear system provided by the linearization of a highly stiff system. We investigate the possibility of usage of multi-stage methods for stiff systems by discussing the difference between the two methods in several numerical experiments. Moreover, the advantages of multi-stage methods are heuristically presented even for nonlinear stiff systems through several numerical tests.

A Singularly P-Stable Multi-Derivative Predictor Method for the Numerical Solution of Second-Order Ordinary Differential Equations

“…Traces of ordinary differential equations can be found in various fields of mathematics, natural, or social sciences. Geometry, various engineering fields including analytical mechanics and electrical engineering, geology, physics, chemistry (in the analysis of nuclear chain reactions), biology (in the modeling of infectious diseases and genetic changes), ecology (in population modeling), and economy (in the modeling of dividend and stock price changes) are some of the scientific branches in which the ordinary differential equations play an essential role [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Since most of the differential equations that provide a relatively accurate model of the target phenomena have a complex and nonlinear form, finding an analytical solution for these problems is usually very difficult or even impossible.…”