In mathematics, a collocation method is a method for the numerical solution of ordinary differential equation and partial differential equations and integral equations. The idea 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. Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... Pope Pius XI, depicted in this window at Cathedral of Our Lady of Peace, Honolulu, was ordinary of the universal Roman Catholic Church and local ordinary of Rome. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ...

Ordinary differential equations

Given the ordinary differential equation In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...

which should be solved over the interval [t₀,t₀+h]. Denote the collocation points by c₁, …, c_n. For simplicity, it is assumed that the collocation points are all different.

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 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. However, not all Runge–Kutta methods are collocation methods.

Example

If we take n = 2, c₁ = 0 and c₂ = 1, then p is the quadratic polynomial which satisfies

Hence p is of the form

and its coefficients satisfy

It follows that

where y₁ = p(t₀ + h) is the approximate solution at t = t₀ + h.

This method is known as the trapezoidal rule. Indeed, this method can also be derived by rewriting the differential equation as

and approximating the integral on the right-hand side by the trapezoidal rule for integrals. The function f(x) (in blue) is approximated by a linear function (in red). ...

References

• Ernst Hairer, Syvert Nørsett and Gerhard Wanner, Solving ordinary differential equations I: Nonstiff problems, second edition, Springer Verlag, Berlin, 1993. ISBN 3-540-56670-8.
• Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 1996. ISBN 0-521-55376-8 (hardback), ISBN 0-521-55655-4 (paperback).