Efficient extrapolation methods for ODEs
Technical Report
·
OSTI ID:6358818
Gragg proposed a way of solving the initial value problem for a system of ordinary differential equations (ODEs) which is based on repeated extrapolation of the explicit midpoint rule. To advance the integration of y' = f(x, y) from an approximation y..omega.. to y(x/sub 0/) to an approximation of y(x/sub 0/ + H), one must repeatedly select an integer n/sub i/ and integrate from x/sub 0/ to x/sub 0/ + H using the midpoint rule and a step size h/sub i/ = H/n/sub i/. The approximations to y(x/sub 0/ + H), which are of order two, can be combined in a linear (polynomial extrapolation) or a nonlinear (rational extrapolation) way to form high order approximations. If polynomial extrapolation is done, this process amounts to a convenient way of constructing explicit Runge-Kutta formulas of increasing order. Bulirsch and Stoer worked out the many practical details involved in turning the basic idea into a code. The resulting code and its successors are so effective that extrapolation of the explicit midpoint rule is one of the most popular ways to solve the initial value problem for ODEs. Each of the Runge-Kutta formulas generated by the extrapolation process depends on the sequence (n/sub i/) selected. There are several criteria for selecting a sequence, such as the number of evaluations of f made (the number of stages of the Runge-Kutta formula), the absolute stability, and the efficiency of the resulting formula. In this paper we consider the efficiency as it is affected by the choice of (n/sub i).
- Research Organization:
- Sandia National Labs., Albuquerque, NM (USA)
- DOE Contract Number:
- AC04-76DP00789
- OSTI ID:
- 6358818
- Report Number(s):
- SAND-83-0041; ON: DE83009390
- Country of Publication:
- United States
- Language:
- English
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