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A quasi-randomized Runge-Kutta method

Authors: Ibrahim Coulibaly and Christian Lécot
Journal: Math. Comp. 68 (1999), 651-659
MSC (1991): Primary 65L06; Secondary 65C05
MathSciNet review: 1627781
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Abstract | References | Similar Articles | Additional Information

Abstract: We analyze a quasi-Monte Carlo method to solve the initial-value problem for a system of differential equations $y^{\prime}(t) = f (t,y(t))$. The function $f$ is smooth in $y$ and we suppose that $f$ and $D_y^1f$ are of bounded variation in $t$ and that $D_{y}^2 f$ is bounded in a neighborhood of the graph of the solution. The method is akin to the second order Heun method of the Runge-Kutta family. It uses a quasi-Monte Carlo estimate of integrals. The error bound involves the square of the step size as well as the discrepancy of the point set used for quasi-Monte Carlo approximation. Numerical experiments show that the quasi-randomized method outperforms a recently proposed randomized numerical method.

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Additional Information

Ibrahim Coulibaly
Affiliation: Laboratoire de Mathématiques, Université de Savoie, Campus scientifique, 73376 Le Bourget-du-Lac cedex, France

Christian Lécot
Affiliation: Laboratoire de Mathématiques, Université de Savoie, Campus scientifique, 73376 Le Bourget-du-Lac cedex, France

Keywords: Runge-Kutta method, quasi-Monte Carlo method, discrepancy
Received by editor(s): July 18, 1997
Article copyright: © Copyright 1999 American Mathematical Society