Third order explicit method for the stiff ordinary differential equations

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 stability property of the Runge-Kutta method depend on stability region of the method. The stability function of the explicit methods is the polynomial. The stability regions of the polynomials are relatively small. The most of explicit methods have small stability regions and consequently small τ st . It obliges us to solve the ODE with the small step size τ st ≪τac. The goal of our article is to construct the third order explicit methods with enlarged stability region (with the big τ st : τ st ≥τ ac ). To achieve this aim we construct the third order polynomials: 1−z+z2/2−z3/6+∑ i=4 n d i z i with the enlarge stability regions. Then we derive the formula for the embedded Runge-Kutta third order accuracy methods with the stability functions equal to above polynomials. The methods can use only three arrays of the storage. It gives us opportunity to solve large systems of differential equations.