Spectral Collocation Methods for Differential-Algebraic Equations with Arbitrary Index

Authors:
Can Huang;Zhimin Zhang
Affiliations:
Department of Mathematics, Michigan State University, East Lansing, USA 48824;Department of Mathematics, Wayne State University, Detroit, USA 48202 and Beijing Computational Science Research Center, Beijing, China 100084
Venue:
Journal of Scientific Computing
Year:
2014

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Abstract

In this paper, a symmetric Jacobi---Gauss collocation scheme is explored for both linear and nonlinear differential-algebraic equations (DAEs) of arbitrary index. After standard index reduction techniques, a type of Jacobi---Gauss collocation scheme with $$N$$N knots is applied to differential part whereas another type of Jacobi---Gauss collocation scheme with $$N+1$$N+1 knots is applied to algebraic part of the equation. Convergence analysis for linear DAEs is performed based upon Lebesgue constant of Lagrange interpolation and orthogonal approximation. In particular, the scheme for nonlinear DAEs can be applied to Hamiltonian systems. Numerical results are performed to demonstrate the effectiveness of the proposed method.