Collocation method

In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations. The idea is to choose a finitedimensional space of candidate solutions (usually, polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the given equation at the collocation points.
Ordinary differential equations
Suppose that the ordinary differential equation
is to be solved over the interval [t_{0}, t_{0} + h]. Choose 0 ≤ c_{1}< c_{2}< … < c_{n} ≤ 1.
The corresponding (polynomial) collocation method approximates the solution y by the polynomial p of degree n which satisfies the initial condition p(t_{0}) = y_{0}, and the differential equation p'(t) = f(t,p(t)) at all points, called the collocation points, t = t_{0} + c_{k}h where k = 1, …, n. This gives n + 1 conditions, which matches the n + 1 parameters needed to specify a polynomial of degree n.
All these collocation methods are in fact implicit Runge–Kutta methods. However, not all implicit Runge–Kutta methods are collocation methods. ^{[1]}
Example
Pick, as an example, the two collocation points c_{1} = 0 and c_{2} = 1 (so n = 2). The collocation conditions are
There are three conditions, so p should be a polynomial of degree 2. Write p in the form
to simplify the computations. Then the collocation conditions can be solved to give the coefficients
The collocation method is now given (implicitly) by
where y_{1} = p(t_{0} + h) is the approximate solution at t = t_{0} + h.
This method is known as the "trapezoidal rule." Indeed, this method can also be derived by rewriting the differential equation as
and approximating the integral on the righthand side by the trapezoidal rule for integrals.
References
 Ernst Hairer, Syvert Nørsett and Gerhard Wanner, Solving ordinary differential equations I: Nonstiff problems, second edition, Springer Verlag, Berlin, 1993. ISBN 3540566708.
 Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 1996. ISBN 0521553768 (hardback), ISBN 0521556554 (paperback).
Finite difference methods Heat Equation and related: FTCS scheme · Crank–Nicolson method Hyperbolic: Lax–Friedrichs method · Lax–Wendroff method · MacCormack method · Upwind scheme · Other: Alternating direction implicit method · Finitedifference timedomain methodFinite volume methods Finite element methods Other methods Spectral method · Pseudospectral method · Method of lines · Multigrid methods · Collocation method · Level set method · Boundary element method · Immersed boundary method · Analytic element method · Particleincell · Isogeometric analysisDomain decomposition methods Schur complement method · Fictitious domain method · Schwarz alternating method · Additive Schwarz method · Abstract additive Schwarz method · Neumann–Dirichlet method · Neumann–Neumann methods · Poincaré–Steklov operator · Balancing domain decomposition · BDDC · FETI · FETIDPCategories: Numerical differential equations
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