In mathematics, a collocation method is a method for the
numerical solution of
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
s,
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s and
integral equation
In mathematical analysis, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,\ldots,x_n ; u(x_1,x_2 ...
s. The idea is to choose a finite-dimensional space of candidate solutions (usually
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
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.
Ordinary differential equations
Suppose that the
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
:
is to be solved over the interval
1< ''c''
2< ... < ''c''
''n'' ≤ 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, \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
In numerical analysis, the Runge–Kutta methods ( ) are a family of Explicit and implicit methods, implicit and explicit iterative methods, List of Runge–Kutta methods, which include the Euler method, used in temporal discretization for the a ...
. The coefficients ''c''
''k'' 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 ''c''
1 = 0 and ''c''
2 = 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
:
\begin
\alpha &= \frac \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
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 ''y''
1 = ''p''(''t''
0 + ''h'') is the approximate solution at ''t'' = ''t''
1 = ''t''
0 + ''h''.
This method is known as the "
trapezoidal rule
In calculus, the trapezoidal rule (or trapezium rule in British English) is a technique for numerical integration, i.e., approximating the definite integral:
\int_a^b f(x) \, dx.
The trapezoidal rule works by approximating the region under the ...
" for differential equations. Indeed, this method can also be derived by rewriting the differential equation as
:
y(t) = y(t_0) + \int_^t f(\tau, y(\tau)) \,\textrm\tau, \,
and approximating the integral on the right-hand side by the
trapezoidal rule
In calculus, the trapezoidal rule (or trapezium rule in British English) is a technique for numerical integration, i.e., approximating the definite integral:
\int_a^b f(x) \, dx.
The trapezoidal rule works by approximating the region under the ...
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 2''s''. 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.
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
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...
.
Notes
References
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{{DEFAULTSORT:Collocation Method
Curve fitting
Numerical differential equations