Solve DAEs with Orthogonal Collocation on Finite Elements

Discretization of a continuous time representation allow large-scale nonlinear programming (NLP) solvers to find solutions at specified intervals in a time horizon. There are many names and related techniques for obtaining mathematical relationships between derivatives and non-derivative values. Some of the terms that are relevant to this discussion include orthogonal collocation on finite elements, direct transcription, Gauss pseudospectral method, Gaussian quadrature, Lobatto quadrature, Radau collocation, Legendre polynomials, Chebyshev polynomials, Jacobi polynomials, Laguerre polynomials, any many more. There are many papers that discuss the details of the derivation and theory behind these methods. The purpose of this section is to give a practical introduction to orthogonal collocation on finite elements with Lobatto quadrature for the numerical solution of differential algebraic equations. See http://apmonitor.com/do/index.php/Main/OrthogonalCollocation

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