# 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 $[t_{0},t_{0}+h]$. Choose $c_{k}$ from 0 ≤ c$_1$< c$_2$< ... < c$_n$ ≤ 1.
The corresponding (polynomial) collocation method approximates the solution y by the polynomial p of degree n which satisfies the initial condition $p(t_{0})=y_{0}$, and the differential equation $p'(t_{k})=f(t_{k},p(t_{k}))$ at all collocation points $t_{k}=t_{0}+c_{k}h$ for $k=1,\ldots ,n$. 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 c$_k$ in the Butcher tableau of a Runge–Kutta method are the collocation points. However, not all implicit Runge–Kutta methods are collocation methods.

### Example: The trapezoidal rule

Pick, as an example, the two collocation points c$_1$ = 0 and c$_2$ = 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
    ${\begin{aligned}\alpha &={\frac {1}{2h}}{\Bigf(t_{0}+h,p(t_{0}+h))-f(t_{0},p(t_{0})){\Big )},\\\beta &=f(t_{0},p(t_{0})),\\\gamma &=y_{0}.\end{aligned}}$
The collocation method is now given (implicitly) by
    $y_{1}=p(t_{0}+h)=y_{0}+{\frac {1}{2}}h{\Bigf(t_{0}+h,y_{1})+f(t_{0},y_{0}){\Big )},\,$
where y$_1$ = p(t$_0$ + h) is the approximate solution at t = t$_1$ = t$_0$ + 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 use the points of Gauss–Legendre quadrature as collocation points. The Gauss–Legendre method based on s points has order 2s. All Gauss–Legendre methods are A-stable.
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.

## Orthogonal collocation method

In direct collocation method, we are essentially performing variational calculus with the finite-dimensional subspace of piecewise linear functions (as in trapezoidal rule), or cubic functions, or other piecewise polynomial functions. In orthogonal collocation method, we instead use the finite-dimensional subspace spanned by the first N vectors in some orthogonal polynomial basis, such as the Legendre polynomials.