In direct collocation method, we are essentially performing variational calculus with the finite-dimensional subspace of piecewise linear cubic functions. In orthogonal collocation method, we instead use the finite-dimensional subspace spanned by the first N vectors in some [[Orthogonal polynomials|orthogonal polynomial]] basis, such as the [[Legendre polynomials]].
'''Orthogonal collocation''' is a method for the [[Numerical partial differential equations|numerical solution of partial differential equations]]. It uses [[Collocation method|collocation]] at the zeros of some [[orthogonal polynomials]] to transform the [[partial differential equation]] (PDE) to a set of [[Ordinary differential equation|ordinary differential equations]] (ODEs). The ODEs can then be solved by any method. It has been shown that it is usually advantageous to choose the collocation points as the zeros of the corresponding [[Jacobi polynomial]] (independent of the PDE system).
