Third order explicit method for the stiff ordinary differential equations
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.
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- 1.Hairer E., Wanner G. (1991) Solving ordinary differential equations 2. Stiff problems. Springer-Verlag Berlin Heidelbers New York London Paris Tokyo.Google Scholar
- 2.Lebedev V.I., Medovikov A.A. (1995) The methods of the second order accuracy with variable time steps. Izvestiya Vuzov. Matematika. N10. Russia.Google Scholar
- 3.P.J. van der Houwen, B.P. Sommeijer (1980) On the internal stability of explicit m-stage Runge-Kutta methods for large m-values. Z.Angew. Math. Mech. 60.Google Scholar
- 4.Lebedev V.I. (1987) Explicit difference scheams with time-variable steps for solution of stiff system of equations. Preprint DNM AS USSR N177.Google Scholar