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Collocation method - Wikiwand

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}+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 = t1 = 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.