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A collocation method chooses an approximate solution from a finite-dimensional trial space and forces the residual of a differential equation to vanish at selected nodes. Collocation Runge-Kutta methods apply this principle to polynomial approximations on each time step.

Given distinct nodes $c_i$, a collocation Runge-Kutta method uses the Lagrange basis ${\ell }_j$ with $a_{ij}={\int }_0^{c_i}{\ell }_j(\tau )d\tau$ and $b_j={\int }_0^1{\ell }_j(\tau )d\tau$. Its stages enforce the differential equation at the collocation nodes.

This applies the collocation method to the time-step polynomial of an ordinary differential equation.

Integrate the ordinary differential equation across one step, using the fundamental theorem of calculus:The midpoint quadrature rule approximates the integral by $hf(x_n+h/2,y(x_n+h/2))$. Approximate the unknown midpoint state by the midpoint value of linear interpolation between the endpoint states:Combining these approximations gives the implicit midpoint ruleIt is an implicit time-stepping method because its new state occurs inside the evaluation of $f$. An equivalent derivation uses a straight-line approximation on the step and imposes the ordinary differential equation at its midpoint, a one-stage collocation method.

A spectral method approximates an unknown function by a finite expansion in global basis functions, such as Fourier modes or orthogonal polynomials. A Galerkin method makes the residual orthogonal to the retained trial space; a collocation method instead makes it vanish at selected nodes. The expansion coefficients are the numerical unknowns.

For the $2\pi$-periodic heat equation, a Fourier series truncationgives $a_m^{\prime }(t)=-m^2{a}_m(t)$ under Fourier–Galerkin method projection. Each retained mode decays independently, and differentiation becomes multiplication of its coefficient by $im$. This turns a partial differential equation into finitely many ordinary differential equations.

Rapid convergence comes from rapid coefficient decay. If $|a_m|\le C{e}^{-\sigma |m|}$ with $\sigma >0$, the omitted tail has an exponentially small uniform norm bound by a geometric series. Limited smoothness or discontinuities can prevent that rate. A large number of retained modes cannot justify exponential convergence without the needed solution regularity.