SIAM Journal on Scientific and Statistical Computing
Journal of Computational Physics
Runge-Kutta Software with Defect Control four Boundary Value ODEs
SIAM Journal on Scientific Computing
A new mesh selection strategy for ODEs
Applied Numerical Mathematics
A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra
SIAM Journal on Matrix Analysis and Applications
Numerical approximation of nonlinear BVPs by means of BVMs
Applied Numerical Mathematics
Estimating conditioning of BVPs for ODEs
Mathematical and Computer Modelling: An International Journal
International Journal of Computing Science and Mathematics
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Codes for the numerical solution of two-point boundary value problems can now handle quite general problems in a fairly routine and reliable manner. When faced with particularly challenging equations, such as singular perturbation problems, the most efficient codes use a highly non-uniform grid in order to resolve the non-smooth parts of the solution trajectory. This grid is usually constructed using either a pointwise local error estimate defined at the grid points or else by using a local residual control. Similar error estimates are used to decide whether or not to accept a solution. Such an approach is very effective in general providing that the problem to be solved is well conditioned. However, if the problem is ill conditioned then such grid refinement algorithms may be inefficient because many iterations may be required to reach a suitable mesh on which to compute the solution. Even worse, for ill conditioned problems an inaccurate solution may be accepted even though the local error estimates may be perfectly satisfactory in that they are less than a prescribed tolerance. The primary reason for this is, of course, that for ill conditioned problems a small local error at each grid point may not produce a correspondingly small global error in the solution. In view of this it could be argued that, when solving a two-point boundary value problem in cases where we have no idea of its conditioning, we should provide an estimate of the condition number of the problem as well as the numerical solution. In this paper we consider some algorithms for estimating the condition number of boundary value problems and show how this estimate can be used in the grid refinement algorithm.