Additive methods for the numerical solution of ordinary differential equations

Abstract

Consider a system of differential equations x = f(x). Most methods for the numerical solution of such a system may be characterized by a pair of matrices (A, B) and make no special use of any structure inherent in the system. In this article, methods which are characterized by a triple of matrices (A; Bx, B2) are considered. These methods are applied in an additive fashion to a decomposition / = fx + f2 and some methods have pronounced advantages when one term of the decomposition is linear. This article obtains algebraic conditions which give the order of convergence of such methods. Some simple examples are displayed. 1. Introduction. Consider an initial value problem for a system of« differential equations, x'=/(x), x(f0) = x0. Butcher [1] showed that many methods for the numerical solution of the initial value problem may be characterized by a pair of matrices {A,B). Such methods make no special allowance for any structure in the differential system, although in many cases the system occurs naturally in a form where f=fx+f2, and frequently one term in this decomposition is linear. To take account of such structure, this article examines certain methods characterized by a triple of matrices {A;BX, B2). These methods are used in an additive fashion with a decomposition f=fx + f2, which may be time dependent. Since the results extend to methods characterized by r + 1 matrices {A\BX,B2,. .. , Br), used with a decomposition f=fi + /2 + •••+/., it is possible to approximate each equation in the differential system in a different way. For example, special methods for certain high order differential equations may be interpreted as additive methods used with a particular decomposition. In this article, a general decomposition is treated. An alternative approach was adopted by Lawson [3]. Lawson considered a decomposition f=fx + f2 with fx linear and integrated the linear term before applying a numerical method to the differential system. To indicate possible advantages in the use of additive methods, consider the trapezoidal rule used with a step length h. This gives jKm) .^«-D + l/Cy-*-») + |/Ö^m)), m = 1,2,3,..., where >>(m) is an approximation to x{tm), with tm = f0 + mh, m = 0, 1, 2,. .. . In

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“…Additive Runge-Kutta. One method of splitting an ODE is to partition the right hand side of an IVP into Λ parts [10,1,21,24]: …”

“…In the literature, the definition of ARK for the case of Λ = 2 usually is presented in the form of (2.2) [1,21,10,24]. We reformulate the definition of ARK (Λ = 2) below to suit our analysis.…”

A note on B-stability of splitting methods

“…To take into account the special structure of the right hand side of the equation, let us consider the class of Additive Runge-Kutta methods defined in the following way [7].…”

Additive Runge-Kutta Methods for Stiff Ordinary Differential Equations

“…Introduction. In a recent article [2] the authors showed that certain pairs of methods may be used in an additive fashion to solve an initial value problem for a system of n differential equations x'=f(t,x),…”

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