Analysis of parallel diagonally implicit iteration of Runge-Kutta methods

Abstract

16lJVan der Houwcn, P.J. and B.P. Sommcijcr. Analysis of parallel diagonally implicit iteration of Runge-Kutta methods, Applied Numerical Mathematics 11 (1993) 169-188.In this paper, we analyze parallel, diagonally implicit iteration of Rungc-Kutta methods (PDIRK methods) for solving large systems of stiff equations on parallel computers. Like Newton-iterated backward differentiation formulas (BDFs), these PD IRK methods are such that in each step the (sequential) costs consist of solving a number of linear systems with the same matrix of coefficients and with the same dimension as the system of differential equations. Although for PDIRK methods the number of linear systems is usually higher than for Newton iteration of BDFs, the more computationally intensive work of computing the matrix of coefficients and its LU-decomposition arc identical. The advantage of PDIRK methods over Newton-iterated BDFs is their unconditional stability (A-stability for Gauss-based methods and L-stability for Radau-based methods) for any order of accuracy.Special characteristics of the PD IRK methods will be studied, such as the rate of convergence, the influence of particular predictors on the resulting stability properties, and the stiff error constants in the global error.Keywords. Diagonally implicit Runge-Kutta methods: parallelism. l. IntroductionConsider the initial value problem for systems of ordinary differential equations (ODEs) m dimension d y'(t) =f(t, y(t)), ( 1.1) In this paper, we analyze integration methods based on iteration of implicit Runge-Kutta lRK) methods of collocation type. Such RK methods possess both a large step-point order and a large stage order. Furthermore, by a suitable choice of the collocation parameters, these RK methods are unconditionally stable for any order of accuracy.We shall employ the diagonally implicit iteration-type methods proposed in [11,12]. TheseCorrespondence to: B.P.

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