An automatic continuation strategy for the solution of singularly perturbed nonlinear boundary value problems

Abstract

In a recent paper, the present authors derived an automatic continuation algorithm for the solution of linear singular perturbation problems. The algorithm was incorporated into two general-purpose codes for solving boundary value problems, and it was shown to deal effectively with a large test set of linear problems. The present paper describes how the conintuation algorithm for linear problems can be extended to deal with the nonlinear case. The results of exstensive numerical testing on a set of nonlinear singular perturbation problems are given, and these clearly demonstrate the efficacy of continuation for solving such problems.

Transient Behavior and Gelation of Free Radical Polymerizations in Continuous Stirred Tank Reactors

“…There is only a compound tolerance parameter for code ACDC, [44] which has been varied between 10 À6 and 10 À8 , and numerical results agree to 8 decimal places.…”

Kinetic Modeling of Non‐Linear Polymerization

“…This leads to extreme numerical sensitivity [36] of the 2PBVP needed for solution by the method of characteristics. Only rather recently a suitable method [37] could be found to solve them.…”

An improved general kinetic analysis of non-linear irreversible polymerisations

“…Only recently (Costa and Dias, 2003) could a solution be found, thanks to the use of the integrator in automatic continuation code ACDC (Cash et al, 1995(Cash et al, , 2001Bashir-Ali et al, 1998).…”

“…Code ACDC (Cash et al, 2001) can deal with the "stiff" systems (Hairer and Wanner, 2002) found in free-radical polymerisation (discussed in next section) and this is a relatively easy problem for that method. Prediction of weight-average molecular weight takes about 30 s CPU before gelation, and each prediction of sol fraction and M w after gelation takes about 59 s. Computations were performed with a PIV 1.5 GHz running Linux (computing times are about the same with a dual G4 Power PC running Apple MacOSX, also with a program compiled with GNU C compiler gcc-2.95 and associated Fortran compiler).…”