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I'm studying numerical integration for solution of second order ivp. But I don't seem to get the concept of collocation and interpolation points. How are they being assigned? What are some things to consider? I need a layman intuition on this topic.

OdunayoO
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    Intuitively it's not much different from adaptive step size selection in Runge-Kutta and similar methods. You choose a type of piecewise polynomial to work with, you choose a mesh. You plug in the ODE and the boundary conditions at the mesh points and use that to get a system of algebraic equations for the polynomial coefficients. You solve that system. Then you compute local error estimates, and refine the mesh where these estimates are not satisfactory. – Ian May 16 '17 at 11:04
  • If you want a method giving higher regularity, then you will also have continuity equations for the derivatives at each mesh point, exactly analogous to cubic spline interpolation. If you prefer to do things non-adaptively, then you may as well choose the nodes to be the same nodes as in your favorite method for numerical integration of a given function (e.g. trapezoidal rule, Gauss-Legendre quadrature, etc.) – Ian May 16 '17 at 11:07
  • If this answers your question (I'm not sure whether it will), then please leave a comment so that I can convert it into an answer. – Ian May 16 '17 at 11:11
  • To get a good answer, I think you need to narrow down your question. In particular, are you confused about collocation vs interpolation points, as the title suggests? Is your question specific to time-stepping for 2nd order IVP? What time-stepping schemes are you looking at? – user66081 May 16 '17 at 11:16

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