# Polynomial collocation method

I was reading this article (https://pdfs.semanticscholar.org/6b11/ff0dcf14991f7be50e354d85c932409a01d9.pdf) on colloaction method, where author has considered the IVP $$y'(t)=f(t,y(t))~~,~~t \in I=[0,T]~~,~~y(0)=y_0$$ Assume $$f:I \times \Omega \subset \mathbb{R} \to \mathbb{R}$$ is Lipschitz. Now define a mesh $$I_h=\{t_n : 0=t_0 and $$h_n=t_{n+1}-t_n$$ and $$h=\text{max} \{h_n: 0 \le n \le N-1 \}$$ which is the step size. Now the solution will be approximated by an element $$u_h$$ of the polynomial space $$S^{(0)}_m(I_h)=\{v \in C(I): v|_{[t_n,t_{n+1}]} \in \pi_m\}$$ Where $$\pi_m$$denotes the space of all real polynomials of degree less than or equal to $$m$$. Then author has written that $$dim\,S^{(0)}_m(I_h)\,=Nm+1$$ , which im having a little difficulty to understand.Another doubt is does $$u_h$$ is getting approximated as a polynomial in every single intervals in the mesh or over the whole range ?

Also the author has defined a set of collocation points $$X_h=\{t=t_n+c_ih_n:0 \le c_1 \le \ldots and its cardinality as \begin{align}|X_h|&=Nm ~\text{if}~0 < c_1 < \ldots I'm also having difficulty to understand $$(*)$$. Also in general what is the difference between RK methods and Collocation , what extra advantage collocation provide over RK methods ?

Gauss-Legendre Runge-Kutta methods, for example, are collocation methods based on Gauss-Legendre quadrature points. Their advantages are multiple. For example, they have high order ($$2s$$ for a method on $$s$$ stages), and they have great properties in geometric problems (conservation of quadratic invariants, symplecticity...). Another class of collocation methods are the Radau methods, which are $$A$$ and $$L$$-stable, making them an extremely good candidate for integrating stiff equations (order $$2s-1$$ for $$s$$ stages).
Regarding the dimension of $$S_m^{(0)}(I_h)$$, you have $$N$$ intervals, and on each interval a polynomial of degree $$m$$. In total, therefore, you have $$N\cdot (m+1)$$ degrees of freedom, which give the dimension of the space. Or does it? You have to impose continuity at the internal extrema of the intervals, therefore you have $$N-1$$ constraints (free ends): therefore $$\dim{S_m^{(0)}(I_h)} = Nm + N - N + 1 = Nm+1.$$ The only doubt I have about this, is that if you impose the initial condition you get an additional constraint, which would make the dimension just $$Nm$$. A similar reasoning yields the cardinality of the set $$X_h$$.