Projected Implicit Runge–Kutta Methods for Differential-Algebraic Equations

In this paper a new class of numerical methods, Projected Implicit Runge–Kutta methods, is introduced for the solution of index-2 Hessenberg systems of initial and boundary value differential-algebraic equations (DAEs). These types of systems arise in a variety of applications, including the modeling of singular optimal control problems and parameter estimation for differential-algebraic equations such as multibody systems. The new methods appear to be particularly promising for the solution of DAE boundary value problems, where the need to maintain stability in the differential part of the system often necessitates the use of methods based on symmetric discretizations. Previously defined symmetric methods have severe limitations when applied to these problems, including instability, oscillation, and loss of accuracy; the new methods overcome these difficulties. For linear problems we define an essential underlying boundary value ODE and prove well-conditioning of the differential (or state-space) solution components. This is then used to prove stability and superconvergence for the corresponding numerical approximations for linear and nonlinear problems.