Collocation Schemes for Nonlinear Index 1 DAEs with a Singular Point

Abstract

We analyze the convergence behavior of collocation schemes applied to approximate solutions of BVPs in nonlinear index 1 DAEs, which exhibit a critical point at the left boundary. Such a critical point of the DAE causes a singularity in the inherent nonlinear ODE system. In particular, we focus on the case when the inherent ODE system is singular with a singularity of the first kind and apply polynomial collocation to the original DAE system. We show that for a well-posed boundary value problem in DAEs having a sufficiently smooth solution, the global error of the collocation scheme converges with the so-called stage order. Due to the singularity, superconvergence does not hold in general. The theoretical results are supported by numerical experiments.

An efficient asymptotically correct error estimator for collocation solutions to singular index-1 DAEs

“…In [4] it has been shown to be successfully applicable to nonlinear singular ODEs. The convergence properties of collocation methods for nonlinear singular DAEs are currently under investigation [12], at least for the semilinear case…”