A Non-Linear Absolutely-Stable Explicit Numerical Integration Algorithm for Stiff Initial-Value Problems

Abstract

The time-step in integration process has two restrictions. The first one is the time step restriction due to accuracy requirement τ_ac and the second one is the time-step restriction due to stability requirement τ_st. The most of explicit methods have small stability regions and consequently small τ_st. It obliges us to solve stiff problems with small step size τ_st << τ_ac. The implicit methods work well with stiff problems but these methods require more work per step than the explicit methods. In this study, a non-linear absolutly stable explicit one step numerical integration algorithm is proposed for solving non linear stiff initial-value problems in ordinary differential equations. The algorithm is based on deriving a non-linear relation between the dependent variable and its derivatives from the well known Taylor expansion. The accuracy of the method depends on some unknown parameter inserted in Taylor expansion and determined from the error analysis. The accuracy and stability properties of the method are investigated and shown to yield at least third-order and A-stable. The results obtained in the numerical experiments show the efficiency of the present method in solving stiff initial value problems.