Symmetric collocation methods for linear differential-algebraic boundary value problems

We present symmetric collocation methods for linear differential-algebraic boundary value problems without restrictions on the index or the structure of the differential-algebraic equation. In particular, we do not require a separation into differential and algebraic solution components. Instead, we use the splitting into differential and algebraic equations (which arises naturally by index reduction techniques) and apply Gauß-type (for the differential part) and Lobatto-type (for the algebraic part) collocation schemes to obtain a symmetric method which guarantees consistent approximations at the mesh points. Under standard assumptions, we show solvability and stability of the discrete problem and determine its order of convergence. Moreover, we show superconvergence when using the combination of Gauß and Lobatto schemes and discuss the application of interpolation to reduce the number of function evaluations. Finally, we present some numerical comparisons to show the reliability and efficiency of the new methods.