Modified Diagonally Implicit Runge-Kutta Methods

Experimental evidence indicates that the implementation of Newton's method in the numerical solution of ordinary differential equations (y'=f(t,y), y(a)=y_circle, t in [a,b]) by implicit computational schemes may cause difficulties. This is especially true in the situation where (i) f(t,y) and/or f'(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 (DlRKM's) are used. In this paper some modifications in the DlRKM's are suggested so that the modified DlRKM's (MDlRKM's) will perform better than the corresponding DlRKM's when the functions f and f' are quickly varying only in t and (ii) is satisfied. An error estimation technique for the 2-stage MDlRKM's is proposed. Finally, it is shown that the MDlRKM's are more efficient than the corresponding DlRKM's when linear systems of ordinary differential equations are solved in the situation described by (i) and (ii).