A note on conditioning, stability and collocation matrices

摘要

The concept of a well-conditioned BVP, as considered in recent literature, essentially corresponds to the concept of a uniformly stable IVP. But this terminology suggests a shift from absolute to relative quantities, which makes it more difficult to understand the conditioning of numerical discretizations, especially for higher order differential operators.In this note we wish to make it clear that the notion of well-conditioning of BVPs is not always a direct analogue of the u The condition number of a discretization of a well-conditioned BVP generally grows unboundedly when higher accuracy is desired, because it contains a discretization of a differential operator. This condition number gets large particularly rapidly if a direct discretization of a higher order differential operator is attempted.For collocation, an implementation which uses a local (Runge-Kutta) solution representation produces a much better conditioning, because the repeated numerical differentiation is avoided. Instead, the highest derivative is approximated locally by a (discontinuous) piece-wise polynomial, and local integrations yield lower solution derivatives. Upon imposing the collocation equations the highest derivative approximation can be locally eliminated, resulting in a sophisticated discretization of the corresponding first order system.