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| In mathematics, a '''collocation method''' is a method for the [[numerical analysis|numerical]] solution of [[ordinary differential equation]]s, [[partial differential equation]]s and [[integral equation]]s. The idea is to choose a finite-dimensional space of candidate solutions (usually, [[polynomial]]s 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.
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| == Ordinary differential equations ==
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| Suppose that the [[ordinary differential equation]]
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| :<math> y'(t) = f(t,y(t)), \quad y(t_0)=y_0, </math> | |
| is to be solved over the interval [''t''<sub>0</sub>, ''t''<sub>0</sub> + ''h'']. Choose 0 ≤ ''c''<sub>1</sub>< ''c''<sub>2</sub>< … < ''c''<sub>''n''</sub> ≤ 1.
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| The corresponding (polynomial) collocation method approximates the solution ''y'' by the polynomial ''p'' of degree ''n'' which satisfies the initial condition ''p''(''t''<sub>0</sub>) = ''y''<sub>0</sub>, and the differential equation ''p''<nowiki>'</nowiki>(''t'') = ''f''(''t'',''p''(''t'')) at all points, called the '''collocation points,''' ''t'' = ''t''<sub>0</sub> + ''c''<sub>''k''</sub>''h'' where ''k'' = 1, …, ''n''. This gives ''n'' + 1 conditions, which matches the ''n'' + 1 parameters needed to specify a polynomial of degree ''n''.
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| All these collocation methods are in fact implicit [[Runge–Kutta methods]]. The coefficient ''c''<sub>''k''</sub> in the Butcher tableau of a Runge–Kutta method are the collocation points. However, not all implicit Runge–Kutta methods are collocation methods.
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| <ref>{{harvnb|Ascher|Petzold|1998}}; {{harvnb|Iserles|1996|pp=43–44}}</ref>
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| === Example: The trapezoidal rule ===
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| Pick, as an example, the two collocation points ''c''<sub>1</sub> = 0 and ''c''<sub>2</sub> = 1 (so ''n'' = 2). The collocation conditions are
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| :<math> p(t_0) = y_0, \, </math> | |
| :<math> p'(t_0) = f(t_0, p(t_0)), \, </math>
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| :<math> p'(t_0+h) = f(t_0+h, p(t_0+h)). \, </math>
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| There are three conditions, so ''p'' should be a polynomial of degree 2. Write ''p'' in the form
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| :<math> p(t) = \alpha (t-t_0)^2 + \beta (t-t_0) + \gamma \, </math>
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| to simplify the computations. Then the collocation conditions can be solved to give the coefficients
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| :<math>
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| \begin{align}
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| \alpha &= \frac{1}{2h} \Big( f(t_0+h, p(t_0+h)) - f(t_0, p(t_0)) \Big), \\
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| \beta &= f(t_0, p(t_0)), \\
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| \gamma &= y_0.
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| \end{align}
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| </math>
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| The collocation method is now given (implicitly) by
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| :<math> y_1 = p(t_0 + h) = y_0 + \frac12h \Big (f(t_0+h, y_1) + f(t_0,y_0) \Big), \, </math>
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| where ''y''<sub>1</sub> = ''p''(''t''<sub>0</sub> + ''h'') is the approximate solution at ''t'' = ''t''<sub>0</sub> + ''h''.
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| This method is known as the "[[trapezoidal rule (differential equations)|trapezoidal rule]]" for differential equations. Indeed, this method can also be derived by rewriting the differential equation as
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| :<math> y(t) = y(t_0) + \int_{t_0}^t f(\tau, y(\tau)) \,\textrm{d}\tau, \, </math>
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| and approximating the integral on the right-hand side by the [[trapezoidal rule]] for integrals. | |
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| === Other examples ===
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| The [[Gauss–Legendre method]]s use the points of [[Gauss–Legendre quadrature]] as collocation points. The Gauss–Legendre method based on ''s'' points has order 2''s''.<ref>{{harvnb|Iserles|1996|pp=47}}</ref> All Gauss–Legendre methods are [[A-stability|A-stable]].<ref>{{harvnb|Iserles|1996|pp=63}}</ref>
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| == Notes ==
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| {{Reflist}}
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| == References ==
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| * {{Citation | last1=Ascher | first1=Uri M. | last2=Petzold | first2=Linda R. | title=Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations | publisher=[[Society for Industrial and Applied Mathematics]] | location=Philadelphia | isbn=978-0-89871-412-8 | year=1998}}.
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| * {{Citation | last1=Hairer | first1=Ernst | last2=Nørsett | first2=Syvert Paul | last3=Wanner | first3=Gerhard | title=Solving ordinary differential equations I: Nonstiff problems | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-56670-0 | year=1993}}.
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| * {{Citation | last1=Iserles | first1=Arieh | author1-link=Arieh Iserles | title=A First Course in the Numerical Analysis of Differential Equations | publisher=[[Cambridge University Press]] | isbn=978-0-521-55655-2 | year=1996}}.
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| {{Numerical PDE}}
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| {{DEFAULTSORT:Collocation Method}}
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| [[Category:Numerical differential equations]]
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