On implicit Euler for high-order high-index DAEs (Q1612472)

The present paper deals with high order differential algebraic equations of the form \[ \begin{aligned} y^{(d)} &= f(t,y,y^{(1)},\dots, y^{(d-1)}, \lambda),\;y^{(j)}(t_0)= \eta_j,\;j= 0,1,\dots, d-1,\\ 0 &= g(t,y,y^{(1)},\dots, y^{(d-1)}),\;r\in [2,d+1].\end{aligned}\tag{1} \] It is well known that the application of the implicit Euler method to (1) in general fails to converge, since the accuracy of the \(y(t_n)\) estimate is to low to produce reasonable estimates of the derivatives of \(y\). As a remedy a slight modification of the implicit Euler method is suggested. Under appropriate assumptions uniform convergence is shown in the case \(d= r+1\). Moreover, generalizations of that approach to backward differentiation formulae are presented.