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 finite-dimensional 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 [t0, t0 + h]. Choose 0 ≤ c1< c2< … < cn ≤ 1.
The corresponding (polynomial) collocation method approximates the solution y by the polynomial p of degree n which satisfies the initial condition p(t0) = y0, and the differential equation p'(t) = f(t,p(t)) at all points, called the collocation points, t = t0 + ckh where k = 1, …, n. This gives n + 1 conditions, which matches the n + 1 parameters needed to specify a polynomial of degree n.
Pick, as an example, the two collocation points c1 = 0 and c2 = 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 y1 = p(t0 + h) is the approximate solution at t = t0 + 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 right-hand side by the trapezoidal rule for integrals.
- Ernst Hairer, Syvert Nørsett and Gerhard Wanner, Solving ordinary differential equations I: Nonstiff problems, second edition, Springer Verlag, Berlin, 1993. ISBN 3-540-56670-8.
- Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 1996. ISBN 0-521-55376-8 (hardback), ISBN 0-521-55655-4 (paperback).
Finite difference methods Finite volume methods Finite element methods Other methods Domain decomposition methods
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