A stiffly stable integration process using cyclic composite methods

Abstract

A set of stiffly stable, cychc composite multlstep methods, expressed as l t a,~ymt+j-h ~ fl,jYmt+j=O with ~= 1,. ,l, j=--k+~ ]--I of orders 3 through 7 is established Each method exhibits better stability properties than those of the backward dffferentmtlon formula of the same order. A new varmble-step-slze, variable-order integration algorithm incorporating these new cyclic methods is described This algorithm, realized as a Fortran program, is ideally suited for the numemcal mtegratmn of stiff systems of first order ordinary differential equations y = f(y, t) wherein the Jacobian matrix fy(y, t) has complex eigenvalues near the Imaginary axis Numermal results to support th~s clmm are presented, mcludmg a comparison of the new program with GEAR, Hmdmarsh's tuned versmn of Gear's widely accepted computer program DIFSUB, and with EPISODE, Byrne and Hmdmarsh's counterpart of DIFSUB and GEAR using a varmble-step form of the backward dffferentmuon formulas.Key Words and Phrases stiff dffferentml equatmns, stiffly stable methods, composite multlstep methods, cyclic methods, numerical mtegratlon, ordinary differential equatmns, initial value problems, multlstep formulas, numermal mtegratmn program, Fortran code STINT CR Categorms 5 16, 5 17 The Algorithm. STINT. STiff {differential equatmns) INTegrator. ACM Trans Math. Software 4, 4 {Dec. 1978), 399-403.

A Performance Evaluation of Some FORTRAN Subroutines for the Solution of Stiff Oscillatory Ordinary Differential Equations

“…(1.1) when some of the inequalities in (1.6) hold. In a recent study [8] four FORTRAN subroutines, namely STRIDE, BLEND, STINT, and DIRK [4,21,22,1], which should be suitable for problems with properties (1.4)-(1.6), were evaluated on the basis of their performance in solving a sequence of model problems. These problems were derived from the linear system of equations [9], for which the real components satisfy ~j _> -2 × 104, 1 _< j _< 488, and the imaginary components are greater than or equal to zero.…”

Implementation of defect correction methods for stiff differential equations

“…This linear problem was also used by Tendler, Bickart and Picel [27] to demonstrate the performance of their code STINT (cf. [27], p. 355). Note, however, that they stop the integration at x1= 15 (in contrast to x~=25 of Ehle [3]).…”

“…The eigenvalues of the Jacobian are _+i and 10_+90 i. The stability regions of the cyclic composite multistep methods of Tendler, Bickart and Picel [27] lead to a 45~ reduction of function evaluations when compared with GEAR, but COLODE and IDEC are still more efficient. An improvement of the performance of the BDF codes for problems of this type is to be expected from modified control mechanisms (compare Skelboe [24] and Shampine [23] The last example is case 3 of problem 1 of Byrne, Hindmarsh, Jackson and Brown [2].…”

A review of recent developments in solving ODEs

“…Mihelcic [1977] derives A(a)-stable cyclic LMFs that have order greater than their step number. Tendler et al [1978] derive A(a)-stable cyclic LMFs that are kth order, k step for k = 3, 4, . .…”