New interpolants for asymptotically correct defect control of BVODEs
The defect of a continuous approximate solution to an ODE is the amount by which that approximation fails to satisfy the ODE. A number of studies have explored the use of asymptotically correct defect estimates in the numerical solution of initial value ODEs (IVODEs). By employing an appropriately constructed interpolant to an approximate discrete solution to the ODE, various researchers have shown that it is possible to obtain estimates of the local error and/or the maximum defect that are asymptotically correct on each step, as the stepsize h →0. In this paper, we investigate the usefulness of asymptotically correct defect estimates for defect control in boundary value ODE (BVODE) codes. In the BVODE context, for a sequence of meshes which partition the problem interval, one computes a discrete numerical solution, constructs an interpolant, and estimates the maximum defect. The estimates (typically obtained by sampling the defect at a small number of points on each subinterval of the mesh) are used in a redistribution process to determine the next mesh and thus the availability of these more reliable maximum defect estimates can lead to improved meshes. As well, when such estimates are available, the code can terminate with more confidence that the defect is bounded throughout the problem domain by the user-prescribed tolerance. In this paper we employ a boot-strapping approach to derive interpolants that allow asymptotically correct defect estimates. Numerical results are included to demonstrate the validity of this approach.
KeywordsRunge-Kutta schemes Boundary value ODEs Defect control
Mathematics Subject Classifications (2000)65L05 65L10
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