| This study investigates the application of local extrapolation algorithms to the numerical solution of stiff ordinary initial-value problems expressed in the partitioned form(UNFORMATTED TABLE OR EQUATION FOLLOWS){dollar}{dollar}eqalign{lcub}mu{lcub}dzover dt{rcub}&= F(z,y,t),qquad zin Rsp{lcub}nsb1{rcub},qquad yin Rsp{lcub}nsb2{rcub},cr{lcub}dyover dt{rcub}&= f(z,y,t),cr{rcub}{dollar}{dollar}(TABLE/EQUATION ENDS)where {dollar}mu{dollar} is a small positive parameter. The main emphasis is on those sufficient conditions which guarantee that the error estimates of the basic discretizations (and hence the rates of convergence that may be expected) are uniformly valid with respect to {dollar}mu{dollar}.; Within the class of linear multistep methods, it is shown that, under mild smoothness and stability assumptions, the discretization errors of the {dollar}mu{dollar}-stable one-step schemes admit uniformly valid general asymptotic expansions in terms of the integration stepwise {dollar}h{dollar}. However, the stepwise restrictions inherent in these schemes render them unsuitable for integration in stiff regions. In the special case of the Boundary-layer equation{dollar}{dollar}mu{lcub}dzover dt{rcub} = F(z,t),qquad zin Rsp{lcub}nsb1{rcub},{dollar}{dollar}we present sufficient conditions under which certain explicit one-step discretizations are not only A-stable but admit uniformly valid error estimates in both stiff and nonstiff regions. These error appraisals furnish rough estimates of the convergence rates that may be expected in stiff regions when the stepwise {dollar}h{dollar} is chosen independently of {dollar}mu{dollar}. By considering compound discretizations based on these explicit A-stable schemes and on a conventional explicit one-step method, we also derive analogous results for more general partitioned systems.; In illustration of the theory, a number of representative test problems were solved by extrapolation algorithms based on these methods. The results of these calculations, performed on a Honeywell DPS-8/49 computer, provide practical confirmation of this uniform property, even in cases where the coupling between the subsystems considered was relatively strong. In all cases, reasonably accurate results were obtained for stepsizes of practical interest. |
