Projected implicit Runge-Kutta methods for solving differential-algebraic equations

A differential-algebraic equation (DAE) is an equation of the form G(y;To solve Hessenberg DAE BVP's, of index two, Uri Ascher and Linda Petzold introduced Projected Implicit Runge-Kutta methods (PIRK methods). We restate some of their theorems. To extend their work, we describe a modified PIRK method, designed to solve more general semi-explicit DAE BVP's of index two. We prove convergence of the (linear and nonlinear) modified PIRK method. To solve the nonlinear equations associated with the nonlinear modified PIRK method we use a modified Newton method. We prove the modified Newton method converges two-step linearly to the modified PIRK solution. To confirm the theory, numerical results are reported. For the index two case, the modified PIRK method is similar to the method of selective projection, used by Ascher and Spiteri in the FORTRAN code COLDAE, however, there are key differences which we discuss.;Finally, we introduce a modified PIRK method to solve Hessenberg DAE BVP's of index three. We present theorems and numerical results to justify the method.