## 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 paper, 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.

**Authors**: Boisvert, Muir, and Spiteri

**Download:** BoisvertEtAl2013