Collocation method

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.

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.