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 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

Given the ordinary differential equation

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

which should be solved over the interval [t0,t0+h]. Denote the collocation points by c1, …, cn. For simplicity, it is assumed that the collocation points are all different.

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

If we take n = 2, c1 = 0 and c2 = 1, then p is the quadratic polynomial which satisfies

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)). \,

Hence p is of the form

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

and its coefficients satisfy

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

It follows that

y_1 = 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}t, \,

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).
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