A Runge-Kutta BVODE Solver with Global Error and Defect Control

Abstract

Boundary value ordinary differential equations (BVODEs) are systems of ODEs with boundary conditions imposed at two or more distinct points. The global error (GE) of a numerical solution to a BVODE is the amount by which the numerical solution differs from the exact solution. The defect is the amount by which the numerical solution fails to satisfy the ODEs and boundary conditions. Although GE control is often familiar to users, the defect controlled numerical solution can be interpreted as the exact solution to a perturbation of the original BVODE. Software packages based on GE control and on defect control are in wide use. The defect control solver, BVP_SOLVER, can provide an a posteriori estimate of the GE using Richardson extrapolation. In this article, we consider three more strategies for GE estimation based on (i) the direct use of a higher-order discretization formula (HO), (ii) the use of a higher-order discretization formula within a deferred correction (DC) framework, and (iii) the product of an estimate of the maximum defect and an estimate of the BVODE conditioning constant, and demonstrate that the HO and DC approaches have superior performance. We also modify BVP_SOLVER to introduce GE control .