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Collocation method

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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.

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Ordinary differential equations

Suppose that the ordinary differential equation

y ′ ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 ,

is to be solved over the interval [ t 0 , t 0 + 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 ( t 0 ) = y 0 , and the differential equation p ′ ( t k ) = f ( t k , p ( t k ) )

at all collocation points t k = t 0 + c k h for 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. The coefficient ck in the Butcher tableau of a Runge–Kutta method are the collocation points. However, not all implicit Runge–Kutta methods are collocation methods.

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 , 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 ) = α ( t − t 0 ) 2 + β ( t − t 0 ) + γ

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 .

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 ) ) ,

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

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. All Gauss–Legendre methods are A-stable.

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.

References

Collocation method Wikipedia
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