Collocation Method - Ordinary Differential Equations

Ordinary Differential Equations

Suppose that the ordinary differential equation

is to be solved over the interval . Choose 0 ≤ c₁< c₂< … < 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(t₀) = y₀, and the differential equation p'(t) = f(t,p(t)) at all points, called the collocation points, t = t₀ + c_kh where k = 1, …, 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 coefficient 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.