Modified Diagonally Implicit Runge–Kutta Methods

The experimental evidence indicates that the implementation of Newton's method in the numerical solution of systems of ordinary differential equations (ODE's) $y' = f(t,y)$, $y(a) = y_0 $, $t \in [a,b]$ by implicit computational schemes may cause difficulties. This is especially true if (i) $f(t,y)$ and/or $f'_y (t,y)$ are quickly varying in t and/or y and (ii) a low degree of accuracy is required. Such difficulties may also arise when diagonally implicit Runge–Kutta methods (DIRKM's) are used in the situation described by (i) and (ii). In this paper some modified DIRKM's (MDIRKM's) are derived. The use of MDIRKM's is an attempt to improve the performance of Newton's method in the case where f and $f'_y $ are quickly varying only in t. The stability properties of the MDIRKM's are studied. An error estimation technique for the new methods is proposed. Some numerical examples are presented.

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