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Numerical methods for systems with measurable coefficients. (English) Zbl 0725.65071
The initial value problem for the n-dimensional system \(dy/dt=f(t,y)\), \(y(t_ 0)=y_ 0\), where f is smooth in y and bounded and measurable in t, is considered. A family of numerical algorithms to solve this problem, which is loosely akin to the Runge-Kutta method, is given. Firstly a discretization with respect to y is done and the dependence on t is retained only through mean properties. Mean values of m-fold self- substitution of f are allowed at m different values of t. By estimating the resulting means with Monte Carlo simulation one obtains actual numerical procedures, named Runge-Kutta Monte Carlo (RKMC) methods, which simulate estimates in the statistical sense for the solution.
One parameter families of second and third order RKMC methods are emphasized and it is stated that these algorithms give stable, qualitative correct answers with a small number of steps, which can be insufficient for a Runge-Kutta method of the same order to give meaningful results.

MSC:
65L05 Numerical methods for initial value problems
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65C05 Monte Carlo methods
34A34 Nonlinear ordinary differential equations and systems, general theory
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References:
[1] Butcher, J.C., The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods, (1987), Wiley Chichester · Zbl 0616.65072
[2] Dekker, K.; Verwer, J.G., Stability of Runge-Kutta methods for stiff nonlinear differential equations, (1984), North Holland New York · Zbl 0571.65057
[3] Stengle, G., A stochastic algorithm for numerical solution of ordinary differential equations with measurable coefficients. Preprint, Bethlehem, PA · Zbl 0725.65071
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