Sequential regularization methods for higher index DAEs with constraint singularities: linear index-$2$ case

Standard stabilization techniques for higher index differential-algebraic equations (DAEs) often involve elimination of the algebraic solution components. This may not work well if there are singularity points where the constraint Jacobian matrix becomes rank deficient. This paper proposes instead a sequential regularization method (SRM)—a functional iteration procedure for solving problems with isolated singularities which have smooth differential solution components.For linear index-2 DAEs we consider both initial value problems (IVPs) and boundary value problems (BVPs). The convergence of the SRM is described and proved in detail. We believe that this is the first convergence proof for any method for DAEs with this type of constraint singularities. Moreover, the regularization parameter in our method is not necessarily very small, so the SRM is an important improvement over usual regularization methods. Various aspects of the subsequent numerical discretization of the regularized problems are discussed as well and some numerical verifications are carried out.