Collocation Methods
Solving differential equations by requiring the approximate solution to satisfy the equation exactly at a chosen set of points.
Enforce the equation at points
A collocation method represents the solution as a combination of basis functions with unknown coefficients, then requires the equation to hold exactly at a finite set of collocation points. This gives as many equations as unknown coefficients, which are solved simultaneously. Unlike the Galerkin approach, which enforces the equation in an integral (weak) sense, collocation enforces it pointwise, making it simpler to implement because no integrals of the residual are needed.
The choice of points matters
The accuracy of collocation hinges on where the points are placed. For high-order spectral collocation, the points are chosen at the roots of orthogonal polynomials, the Gauss or Gauss-Lobatto points, which yield spectral accuracy and connect collocation to Chebyshev and Legendre methods. Poorly chosen (for example equally spaced) points degrade accuracy and can trigger instability, mirroring the interpolation issues seen with barycentric interpolation.
Orthogonal collocation and finite elements
- Global collocation: one high-order basis over the whole domain, spectral accuracy for smooth solutions
- Orthogonal collocation on finite elements: local high-order bases per element, combining flexibility and accuracy
- Superconvergence: at special (Gauss) points the solution is accurate beyond the nominal order
Collocation for time integration
Collocation is not only a spatial method; many implicit Runge-Kutta methods are collocation methods in time, requiring the polynomial that interpolates the solution to satisfy the ODE at the stage points. The Gauss collocation methods derived this way are symplectic and A-stable, linking collocation to the stiff solvers and symplectic integrators used for hard time-dependent problems.
Where it is used
Collocation is popular for boundary value problems and optimal control, where robust codes discretize the whole interval by collocation on a mesh of elements. Its pointwise simplicity and high accuracy make it attractive for one-dimensional profile and transport equations, and for the discretization of trajectory optimization problems.