Arbitrary Order Spectral Deferred CorrectionMethods for Differential Algebraic Equation


Abstract: In this paper, we discuss the fundamentals of a new class of numerical methods for differential algebraic equation (DAE) initial value problems. For the nonlinear
collocation formulation, deferred (or defect) correction techniques are introduced as
preconditioners, and the resulting preconditioned nonlinear system is solved using
Newton-Krylov schemes such as the Newton-GMRES method. The least squares
based orthogonal polynomial approximations are computed using the Gaussian type
quadratures, and spectral integration is used to avoid the numerically unstable
differentiation operator. The resulting numerical methods are of arbitrary order and
very stable. Preliminary results show that these new methods are very competitive
with existing DAE solvers.