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. Suppose that the ordinary differential equation is to be solved over the interval [ t 0 , t 0 + c k h ] {displaystyle } . Choose c k {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 p ( t 0 ) = y 0 {displaystyle p(t_{0})=y_{0}} , and the differential equation p ′ ( t k ) = f ( t k , p ( t k ) ) {displaystyle p'(t_{k})=f(t_{k},p(t_{k}))} at all collocation points t k = t 0 + c k h {displaystyle t_{k}=t_{0}+c_{k}h} for k = 1 , … , n {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. Pick, as an example, the two collocation points c1 = 0 and c2 = 1 (so n = 2). The collocation conditions are There are three conditions, so p should be a polynomial of degree 2. Write p in the form