A-Stable Method for the Solution of the Cauchy Problem for Stiff Systems of Ordinary Differential Equations

Abstract

For the solution of the Cauchy problem for the system of equations

$$\dot y = f\left( y \right)$$

((1))

there is constructed the Rosenbrock type method accurate to the fifth local order with a single computation of a Jacobian matrix per step of integration. Numerical experiments have shown high efficiency of the proposed method. The following approximation of the exponential function is taken as the basis of the method

$${e^x} \approx {\varphi _4}\left( x \right) \equiv 1 + \frac{x}{{1 - x}} - \frac{1}{2}\frac{{{x^2}}}{{{{\left( {1 - x} \right)}^2}}} + \frac{1}{6}\frac{{{x^3}}}{{{{\left( {1 - x} \right)}^3}}} + \frac{1}{{24}}\frac{{{x^4}}}{{{{\left( {1 - x} \right)}^4}}}$$

((2))

.