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Comparison

A comparison between one-step and multistep methods must take several factors into consideration. The fourth order Runge-Kutta method requires four evaluations of f at each step, while the fourth order Adams-Bashforth method (once past the starting values) requires only one and the predictor-corrector method only two. Thus, for a given step size h, the latter two methods may well be considered faster than Runge-Kutta. However, if Runge-Kutta is more accurate and therefore can use fewer steps, them the difference in speed will be reduced and perhaps eliminated. The Adams-Moulton and backward differentiation formulas also require that the difficulty in solving the implicit equation at each step be taken into account. All multistep methods have the possible disadvantage that error in earlier steps can feed back into later calculations with unfavorable consequences. On the other hand, the underlying polynomial approximations in multistep methods make it easy to approximate the solution at points between the mesh points, should this be desirable. Multistep methods have become popular largely because its is relatively easy to estimate the error at each step and to adjust the order or the step size to control it.



Dinesh Manocha
Sun Mar 29 02:59:14 EST 1998