# 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

${\displaystyle y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0},}$

is to be solved over the interval ${\displaystyle [t_{0},t_{0}+c_{k}h]}$. Choose ${\displaystyle c_{k}}$ from 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 ${\displaystyle p(t_{0})=y_{0}}$, and the differential equation ${\displaystyle p'(t_{k})=f(t_{k},p(t_{k}))}$

at all collocation points ${\displaystyle t_{k}=t_{0}+c_{k}h}$ for ${\displaystyle k=1,\ldots ,n}$. 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 coefficients 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

${\displaystyle p(t_{0})=y_{0},\,}$
${\displaystyle p'(t_{0})=f(t_{0},p(t_{0})),\,}$
${\displaystyle p'(t_{0}+h)=f(t_{0}+h,p(t_{0}+h)).\,}$

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

${\displaystyle p(t)=\alpha (t-t_{0})^{2}+\beta (t-t_{0})+\gamma \,}$

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

{\displaystyle {\begin{aligned}\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{aligned}}}

The collocation method is now given (implicitly) by

${\displaystyle y_{1}=p(t_{0}+h)=y_{0}+{\frac {1}{2}}h{\Big (}f(t_{0}+h,y_{1})+f(t_{0},y_{0}){\Big )},\,}$

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

${\displaystyle y(t)=y(t_{0})+\int _{t_{0}}^{t}f(\tau ,y(\tau ))\,{\textrm {d}}\tau ,\,}$

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

### Other examples

The Gauss–Legendre methods use 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.

## 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.