Formal series and numerical integrators, Part II: Application to index-2 differential-algebraic systems Original Research Article

This paper is in two parts. In the first part, we presented a unified approach for deriving and dealing with the formal series that arise when studying the convergence of numerical integrators for systems of ordinary differential equations (ODEs) and differential algebraic equations (DAEs). This general approach was applied for systems of ODEs. The particular case of Hamiltonian systems was studied in some detail. In this the second part, we study in a unified way convergence of a general class of Runge-Kutta type methods (the partitioned Runge-Kutta methods) for systems of index 2 DAEs in Hessenberg form, using tools developed in the first part to handle the relevant formal expansions, the DA2-series. Our approach allows us to obtain sharp estimates for the global errors, even in cases where the method does not satisfy the algebraic constraints, the errors for the algebraic variables affect the approximations to the differential variables, and inconsistent initial values are used.