Collocation method

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In mathematics, a collocation method is a method for the numerical solution of ordinary differential equation and 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.

[edit] Ordinary differential equations

Suppose that the ordinary differential equation

y'(t) = f(t,y(t)), \quad y(t_0)=y_0,

is to be solved over the interval [t0, t0+h]. Denote the collocation points by 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 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.

All these collocation methods are in fact implicit Runge–Kutta methods. However, not all Runge–Kutta methods are collocation methods.

[edit] Example

Pick, as an example, the two collocation points c1 = 0 and c2 = 1 (so n = 2). The collocation conditions are

p(t_0) = y_0, \,
p'(t_0) = f(t_0, p(t_0)), \,
p'(t_0+h) = f(t_0+h, p(t_0+h)). \,

There are three conditions, so p should be a polynomial of degree 2. Write p in the form

p(t) = \alpha (t-t_0)^2 + \beta (t-t_0) + \gamma \,

to simplify the computations. Then the collocation conditions can be solved to give the coefficients

\begin{align} \alpha &= \frac{1}{2h} \Big( f(t_0+h, p(t_0+h)) - f(t_0, p(t_0)) \Big), \\ \beta &= f(t_0, p(t_0)), \\ \gamma &= y_0. \end{align}

The collocation method is now given (implicitly) by

y_1 = p(t_0 + h) = y_0 + \frac12h \Big (f(t_0+h, y_1) + f(t_0,y_0) \Big), \,

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

y(t) = y(t_0) + \int_{t_0}^t f(\tau, y(\tau)) \,\textrm{d}\tau, \,

and approximating the integral on the right-hand side by the trapezoidal rule for integrals.

[edit] References

  • 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).