An efficient asymptotically correct error estimator for collocation solutions to singular index-1 DAEs
The authors consider linear index-1 differential-algebraic equations (DAEs) with properly stated leading term \[ A(t)(D(t)x(t))' + B(t)x(t)=g(t),\quad t\in (0,1], \tag{1} \] with appropriately smooth data. Further, it is assumed that the inherent ordinary differential equation (ODE) of (1) possesses a singularity of first kind at \(t=0\). The numerical solution of DAEs (1) by stiffly accurate collocation methods is analysed. A posteriori error estimation based on the defect correction is extended from singular ODEs, see \textit{W. Auzinger, O. Koch} and \textit{E. Weinmüller} [Numer. Algorithms 31, No. 1--4, 5--25 (2002; Zbl 1021.65038)], to singular index-1 DAEs (1). As the main result of the paper, using the convergence results for stiffly accurate collocation methods, it is pointed out that the error estimate is asymptotically correct. This means that while the global error of the collocation solution is of order \(h^s\), where \(h\) is the stepsize and \(s\) is the collocation degree, here \(s\) is assumed to be even, the deviation between the global error and the defect-based estimate is of order \(h^{s+1}\). Finally, two numerical examples are given for illustration. In the appendix, the interrelation between collocation methods and implicit Runge-Kutta methods as applied to DAEs is discussed.
scientific article; zbMATH DE number 5883653
scientific article