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Mathematics of Computation

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Embedded diagonally implicit Runge-Kutta algorithms on parallel computers

Authors: P. J. van der Houwen, B. P. Sommeijer and W. Couzy
Journal: Math. Comp. 58 (1992), 135-159
MSC: Primary 65L06; Secondary 65Y05
MathSciNet review: 1106986
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Abstract: This paper investigates diagonally implicit Runge-Kutta methods in which the implicit relations can be solved in parallel and are singly diagonal-implicit on each processor. The algorithms are based on diagonally implicit iteration of fully implicit Runge-Kutta methods of high order. The iteration scheme is chosen in such a way that the resulting algorithm is $ A(\alpha )$-stable or $ L(\alpha )$-stable with $ \alpha $ equal or very close to $ \pi /2$. In this way, highly stable, singly diagonal-implicit Runge-Kutta methods of orders up to 10 can be constructed. Because of the iterative nature of the methods, embedded formulas of lower orders are automatically available, allowing a strategy for step and order variation.

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Keywords: Runge-Kutta methods, parallelism
Article copyright: © Copyright 1992 American Mathematical Society