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.

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. 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 2 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. Pick, as an example, the two collocation points c 1 = 0 and c 2 = 1 (so n = 2). The collocation conditions are \begin{align} 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)). \end{align} 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 + \frac{1}{2} h \Big (f(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)) \, d\tau, \, and approximating the integral on the right-hand side by the trapezoidal rule for integrals. ==See also== Method of moments (electromagnetics) Boundary element method

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