Summary
The extrapolated midpoint rule is a popular way to solve the initial value problem for a system of ordinary differential equations. As originally formulated by Gragg, the results are smoothed to remove the weak instability of the midpoint rule. It is shown that this smoothing is not necessary. A cheaper smoothing scheme is proposed. A way to exploit smoothing to increase the robustness of extrapolation codes is formulated.
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References
Bader, G., Deuflhard, P.: A semi-implicit mid-point rule for stiff systems of ordinary differential equations. Preprint Nr. 114. University of Heidelberg, 1981
Bulirsch, R., Stoer, J.: Numerical treatment of ordinary differential equations by extrapolation methods. Numer. Math.8, 1–13 (1966(
Deuflhard, P.: Order and stepsize control in extrapolation methods. Numer. Math. (to appear)
Gragg, W.B.: Repeated extrapolation to the limit in the numerical solution of ordinary differential equations. Thesis, University of California at Los Angeles, 1964
Gragg, W.B.: On extrapolation algorithms for ordinary initial value problems. SIAM J. Numer. Anal.2, 384–403 (1965)
Hussels, H.G.: Schrittweitensteuerung bei der Integration gewöhnlicher Differentialgleichungen mit Extrapolationsverfahren. Diplomarbeit Universität Köln, Math. Inst. 1973
Milne, W.E., Reynolds, R.R.: Stability of a numerical solution of differential equations. J. ACM6, 196–203 (1959) and7, 45–56 (1960)
Shampine, L.F.: Limiting precision in differential equation solvers. Math. Comput.28, 141–144 (1974)
Shampine, L.F.: Implementation of Rosenbrock codes. ACM Trans. Math. Software8, 93–113 (1982)
Shampine, L.F.: Type-insensitive ODE codes based on extrapolation methods. SIAM J. Sci. Stat. Comput. (to appear)
Shampine, L.F., Baca, L.S.: Should the extrapolated midpoint rule be smoothed? Report SAND 82-0317. Sandia National Laboratories, Albuquerque, NM, 1982
Shampine, L.F., Watts, H.A., Davenport, S.M.: Solving nonstiff ordinary differential equations — the state of the art. SIAM Rev.18, 376–411 (1976)
Stoer, J.: Extrapolation methods for the solution of initial value problems and their practical realization, pp. 1–21. In: Proc. Conf. on Numer. Soln of Ordinary Differential Equations, University of Texas at Austin. Nr. 362 in Lecture Notes in Math. Berlin, Heidelberg, New York: Springer, 1974
Stoer, J., Bulirsch, R.: Introduction to numerical analysis. Berlin, Heidelberg, New York: Springer, 1980
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Shampine, L.F., Baca, L.S. Smoothing the extrapolated midpoint rule. Numer. Math. 41, 165–175 (1983). https://doi.org/10.1007/BF01390211
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DOI: https://doi.org/10.1007/BF01390211