A class of linear implicit methods for numerical solution of stiff ODE's is presented.These require only occasional calculation of the Jacobian matrix while maintaining stability. Especially, an effective second order stable algorithm with automatic stepsize control is designed and tested.
Abstract.Rational approximations of the form ~-,~=0 a~q/l'Ii=i (1 +7~q) to exp(-q), q E C, are studied with respect to order and error constant. It is shown that the maximum obtainable order is m + 1 and that the approximation of order m + 1 with bast absolute value of the error constant has ~'l =~ ..... Fn. As an application it is shown that the order of a v-stage semi-implicit Runge-Kutta method cannot exceed v + 1.
Summary. Based on the theory of Butcher series this paper developes the order conditions for Rosenbrock methods and its extensions to Runge-Kutta methods with exact Jacobian dependent coefficients. As an application a third order modified Rosenbrock method with local error estimate is constructed and tested on some examples.
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