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.

Ordinary differential equations

Suppose that the ordinary differential equation

 y'(t) = f(t,y(t)), \quad y(t_0)=y_0,

is to be solved over the interval [t0t0 + h]. Choose 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(t0) = y0, and the differential equation p'(t) = f(t,p(t)) at all points, called the collocation points, t = t0 + ckh 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 implicit Runge–Kutta methods are collocation methods. [1]


Pick, as an example, the two collocation points c1 = 0 and c2 = 1 (so n = 2). The collocation conditions are

 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)). \,

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

   \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. 

The collocation method is now given (implicitly) by

 y_1 = p(t_0 + h) = y_0 + \frac12h \Big (f(t_0+h, y_1) + f(t_0,y_0) \Big), \,

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

 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.


  1. ^ Ascher, Uri M.; Linda R. Petzold (1998). Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Philadelphia, USA: SIAM. ISBN 0-89871-412-5. 
  • 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).

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Method of lines — The method of lines (MOL, NMOL, NUMOL) (Schiesser, 1991; Hamdi, et al., 2007; Schiesser, 2009 ) is a technique for solving partial differential equations (PDEs) in which all but one dimension is discretized. MOL allows standard, general purpose… …   Wikipedia

  • Collocation (remote sensing) — Collocation is a procedure used in remote sensing to match measurements from two or more different instruments. This is done for two main reasons: for validation purposes when comparing measurements of the same variable, and to relate… …   Wikipedia

  • Collocation extraction — is the task of extracting collocations automatically from a corpus using a computer. Within the area of corpus linguistics, collocation is defined as a sequence of words or terms which co occur more often than would be expected by chance. Crystal …   Wikipedia

  • Gauss pseudospectral method — The Gauss Pseudospectral Method (abbreviated GPM ) is a direct transcription method for discretizing a continuous optimal control problem into a nonlinear program (NLP). The Gauss pseudospectral method differs from several other pseudospectral… …   Wikipedia

  • Spectral method — Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain Dynamical Systems, often involving the use of the Fast Fourier Transform. Where applicable, spectral methods have… …   Wikipedia

  • Crank–Nicolson method — In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations.[1] It is a second order method in time, implicit in time, and is numerically …   Wikipedia

  • Multigrid method — Multigrid (MG) methods in numerical analysis are a group of algorithms for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in (but not… …   Wikipedia

  • Schwarz alternating method — In mathematics, the Schwarz alternating method, named after Hermann Schwarz, is an iterative method to find the solution of a partial differential equations on a domain which is the union of two overlapping subdomains, by solving the equation on… …   Wikipedia

  • Discontinuous Galerkin method — Discontinuous Galerkin methods (DG methods) in mathematics form a class of numerical methods for solving partial differential equations. They combine features of the finite element and the finite volume framework and have been successfully… …   Wikipedia

  • Neumann–Dirichlet method — In mathematics, the Neumann–Dirichlet method is a domain decomposition preconditioner which involves solving Neumann boundary value problem on one subdomain and Dirichlet boundary value problem on another, adjacent across the interface between… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.