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 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

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

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.

All these collocation methods are in fact implicit Runge–Kutta methods. The coefficient ck in the Butcher tableau of a Runge–Kutta method are the collocation points. However, not all implicit Runge–Kutta methods are collocation methods. [1]

Example: The trapezoidal rule

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" for differential equations. 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.

Other examples

The Gauss–Legendre methods uses the points of Gauss–Legendre quadrature as collocation points. The Gauss–Legendre method based on s points has order 2s.[2] All Gauss–Legendre methods are A-stable.[3]

In fact, one can show that the order of a collocation method corresponds to the order of the quadrature rule that one would get using the collocation points as weights.
Notes

Ascher & Petzold 1998; Iserles 1996, pp. 43–44
Iserles 1996, pp. 47

Iserles 1996, pp. 63

References

Ascher, Uri M.; Petzold, Linda R. (1998), Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-412-8.
Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag, ISBN 978-3-540-56670-0.
Iserles, Arieh (1996), A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, ISBN 978-0-521-55655-2.

1 Examples of spectral methods
1.1 A concrete, linear example
1.1.1 Algorithm
1.2 A concrete, nonlinear example
2 A relationship with the spectral element method
3 See also

Mathematics Encyclopedia