High order stiffly stable composite multistep methods for numerical integration of stiff differential equations

Abstract

Abstract.Let y =f (y,t) with y(to) =Y0 possess a solution y(t) for t~to. Set t~ =to +nh, n = 1, 2 ...... Let Yo denote the approximate solution of y(tn) defined by the composite multistep methodwith .//n--f(yn,tn) and _N = 1,2 ...... It is conjectured that the method is stiffly stable with order p=l for all l:>l and shown to be so for l= 1 ..... 25. The method is intrinsically efficient in that l future approximate solution values are established simultaneously in an iterative solution process with only one function evaluation per iteration for each of the I future time points.Step and order control are easily implemented, in that the approximate solution at only one past point appears in each component multistep formula of the method and in that the local truncation error for the first component multistep formula of the method is easily evaluated as h kU~l --YlVlf , PREn denotes the value at tiv~ of tile Lagrange interpolating polynomial where u~v~ passing through the points Y(N-1)z+j at t(~-l)~+j with j = -1, 0,..., l-1. Introduction.Many dynamical systems are characterized by stiff differential equations. For example, a linear electrical network with a large ratio of maximum to minimum moduli of natural frequencies is stiff. To achieve efficient numerical integration of stiff differential equations, it is necessary to choose a numerical integration method with care. One property sought in selecting a method is stiff-stability.Gear [1] having originated the definition of stiff stability went on to

Two-parameter, arbitrary order, exponential approximations for stiff equations

“…Finally, consider the 2-block one step method as proposed in Watts and Shampine [14] In the special cases when Mi = 3/4 and p2 = 0, we have the second-order method described in [1], while for pl = 0, p2 = 1, (4.4) is the fourth-order Newton-Cotes method studied in [14]. Since 0 < p., p2 < 1 and Mi + M2 ^ 1 m both cases, the corresponding methods are A -stable.…”

A review of recent developments in solving ODEs

“…Several attempts have been made to reduce the linear algebra costs associated with implicit RKFs. The earliest successful approach was by Bickart and Picel [1973]. (Although their approach was applied to composite multistep formulas, it is equally applicable to implicit RKFs.)…”

Arithmetic tests forA-stability,A[α]-stability, and stiff-stability

“…)r/+ do (7) • t See also the references cited in [1][2][3][4][5][6][7][8][9][10][11] and the sources citing [11]. t* As a matter of convenience, the polynomial Dss.s.y(r/) is derived from the polynomial -P(-2, cx~), not P(-2,~,).…”