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Ordinary differential equations

Suppose that the ordinary differential equation

y ′ ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 , {\displaystyle y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0},}

is to be solved over the interval [ t 0 , t 0 + h ] {\displaystyle [t_{0},t_{0}+h]} . Choose c k {\displaystyle c_{k}} from 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 ( t 0 ) = y 0 {\displaystyle p(t_{0})=y_{0}} , and the differential equation p ′ ( t k ) = f ( t k , p ( t k ) ) {\displaystyle p'(t_{k})=f(t_{k},p(t_{k}))} at all collocation points t k = t 0 + c k h {\displaystyle t_{k}=t_{0}+c_{k}h} for k = 1 , … , n {\displaystyle k=1,\ldots ,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 coefficients ck in the Butcher tableau of a Runge–Kutta method are the collocation points. However, not all implicit Runge–Kutta methods are collocation methods. ¹

Example: The trapezoidal rule

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 , {\displaystyle p(t_{0})=y_{0},\,} p ′ ( t 0 ) = f ( t 0 , p ( t 0 ) ) , {\displaystyle p'(t_{0})=f(t_{0},p(t_{0})),\,} p ′ ( t 0 + h ) = f ( t 0 + h , p ( t 0 + h ) ) . {\displaystyle 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 ) = α ( t − t 0 ) 2 + β ( t − t 0 ) + γ {\displaystyle 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

α = 1 2 h ( f ( t 0 + h , p ( t 0 + h ) ) − f ( t 0 , p ( t 0 ) ) ) , β = f ( t 0 , p ( t 0 ) ) , γ = y 0 . {\displaystyle {\begin{aligned}\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}.\end{aligned}}}

The collocation method is now given (implicitly) by

y 1 = p ( t 0 + h ) = y 0 + 1 2 h ( f ( t 0 + h , y 1 ) + f ( t 0 , y 0 ) ) , {\displaystyle y_{1}=p(t_{0}+h)=y_{0}+{\frac {1}{2}}h{\Big (}f(t_{0}+h,y_{1})+f(t_{0},y_{0}){\Big )},\,}

where y1 = p(t0 + h) is the approximate solution at t = t1 = t0 + h.

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

y ( t ) = y ( t 0 ) + ∫ t 0 t f ( τ , y ( τ ) ) d τ , {\displaystyle 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.

Other examples

The Gauss–Legendre methods use the points of Gauss–Legendre quadrature as collocation points. The Gauss–Legendre method based on s points has order 2s.² All Gauss–Legendre methods are A-stable.³

In fact, one can show that the order of a collocation method corresponds to the order of the quadrature rule that one would get using the collocation points as weights.

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 polynomial basis, such as the Legendre polynomials.

Notes

• Ascher, Uri M.; Petzold, Linda R. (1998), Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-412-8.
• Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag, ISBN 978-3-540-56670-0.
• Iserles, Arieh (1996), A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, ISBN 978-0-521-55655-2.
• Wang, Yingwei; Chen, Suqin; Wu, Xionghua (2009), "A rational spectral collocation method for solving a class of parameterized singular perturbation problems", Journal of Computational and Applied Mathematics, 233 (10): 2652–2660, doi:10.1016/j.cam.2009.11.011.

References

1. Ascher & Petzold 1998; Iserles 1996, pp. 43–44 - Ascher, Uri M.; Petzold, Linda R. (1998), Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-412-8 ↩

2. Iserles 1996, pp. 47 - Iserles, Arieh (1996), A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, ISBN 978-0-521-55655-2 https://books.google.com/books?id=7Zofw3SFTWIC&q=%22Collocation+method%22 ↩

3. Iserles 1996, pp. 63 - Iserles, Arieh (1996), A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, ISBN 978-0-521-55655-2 https://books.google.com/books?id=7Zofw3SFTWIC&q=%22Collocation+method%22 ↩