Runge - Kutta Methods , Trees , and Mathematica
نویسنده
چکیده
This paper presents a simple and elementary proof of Butcher's theorem on the order conditions of Runge-Kutta methods. It is based on a recursive definition of rooted trees and avoids combinatorial tools such as labelings andFà a di Bruno's formula. This strictly recursive approach can easily and elegantly be implemented using modern computer algebra systems like Mathematica for automatically generating the order conditions. The full, but short source code is presented and applied to some instructive examples.
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Another Approach to Runge-Kutta Methods
The condition equations are derived by the introduction of a system of equivalent differential equations, avoiding the usual formalism with trees and elementary differentials. Solutions to the condition equations are found by direct numerical optimization, during which simplifying assumptions upon the Runge-Kutta coefficients may or may not be used. Depending on the optimization criterion, diff...
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A Runge–Kutta method takes small time steps, to approximate the solution to an initial value problem. How accurate is this approximation? If the error is asymptotically proportional to hp, where h is the stepsize, the Runge–Kutta method is said to have “order” p. To find p, write the exact solution, after a single time-step, as a Taylor series, and compare with the Taylor series for the approxi...
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