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Suppose that the [[ordinary differential equation]]
:<math> y'(t) = f(t,y(t)), \quad y(t_0)=y_0, </math>
is to be solved over the interval <math> [t_0,t_0+
The corresponding (polynomial) collocation method approximates the solution ''y'' by the polynomial ''p'' of degree ''n'' which satisfies the initial condition <math>p(t_0) = y_0</math>, and the differential equation <math>p'(t_k) = f(t_k,p(t_k)) </math>
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== Orthogonal collocation method ==
In direct collocation method, we are essentially performing variational calculus with the finite-dimensional subspace of piecewise linear functions (as in trapezoidal rule), or cubic functions, or other piecewise polynomial functions. In orthogonal collocation method, we instead use the finite-dimensional subspace spanned by the first N vectors in some [[Orthogonal polynomials|orthogonal polynomial]] basis, such as the [[Legendre polynomials]].
== Applications ==
=== Motorsport ===
Top F1 teams began switching from Quasi-Static simulation to collocation methods in 2010s to simulate the time it takes for a car to go around a circuit. It is thought that Sauber were one of the first teams to make this transition. Traditional Quasi-Static simulation involves the construction of a gLat-gLong-vCar performance envelope for the car, then starting at each apex (minimum car speed) and accelerating forwards, and backwards through the braking zone using this envelope to stitch a lap together. It is a gross simplification of the problem because the envelope is steady state so ignores any of the dynamics that occur, for example, the car can switch instantaneously between understeer and oversteer.
The switch to collocation methods in simulation involved casting the entire lap as an optimisation problem, broken down by distance steps around the lap which describe the car physics at each point. The objective is to minimise laptime and error in the physics at each point. Once the optimisation is complete, the collocation method finds the minimum laptime for a particular car setup around a circuit by varying brake/throttle and steering wheel angle while obeying the physics at every point. Additional constraints can be added to the objective function alongside minimisation of laptime, such as energy constraints (fuel, electrical, tyre sliding, brakes), and temperature constraints (tyres, battery temperature), and additional controls, such as multiple throttle pedals controlling power to all four wheels can be added to the physics. This allows for extremely complex problems to be solved optimally.
== Notes ==
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* {{Citation | last1=Ascher | first1=Uri M. | last2=Petzold | first2=Linda R. |author2-link=Linda Petzold| 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}}.
* {{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}}.
* {{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| bibcode=1996fcna.book.....I |url=https://books.google.com/books?id=7Zofw3SFTWIC&q=%22Collocation+method%22}}.
* {{Citation | last1=Wang | first1=Yingwei | last2=Chen|first2=Suqin|last3=Wu|first3=Xionghua| title=A rational spectral collocation method for solving a class of parameterized singular perturbation problems|date=2009|journal= Journal of Computational and Applied Mathematics|volume=233|issue=10|pages=2652–2660|doi=10.1016/j.cam.2009.11.011|doi-access=free}}.
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