Legendre Gauss Radau collocation points

Rhaello · August 21, 2014, 14:49

Hello there, I'm currently dealing with the discretization of an optimal control problem to a NLP problem, I'm a newbie in this subject so please keep that in mind

. I'm using orthogonal collocation on finite elements to approximate the states of the system to a set of polynomials, I'm using several low order piecewise lagrange polynomials for this.
Now I'm reading about the schemes used to determine the set of collocation points, in particular the legendre-gauss-radau collocation points, they describe the collocation points as being the roots of the legendre polynomial.
My questions are the following:
Why exactly do they use these collocation points? I've read it is to minimize the error due to the runge phenomenon but I'm not sure if this is the only reason.
Does it make a difference to use this points if I use several low order polynomials instead of one high order polynomial? (say 5 polynomials of 3rd degree).
Thank you for your time.

cfdnewbie · August 22, 2014, 05:39

These points have good interpolation properties (low Lebesque constant) and are much better at interpolating functions than equispaced points (which give you large oscillations when interpolating, especially for more complex functions, e.g. Runge's function).
Plus, they should have a good Gauss-integration rule associated with them, so if you are thinking from switching from collocation to projection, that would help.
The only thing that is unknown to me is why they are using the Radau type: usually, this means that one end of the domain has a node, the other one not. This could make sense for thinks like global hyperbolic problems, but that's problem specific.

Rhaello · August 22, 2014, 09:17

Thanks for the answer.
It's not really decided to use radau collocation points, I'm just reading about them given that I'm not familiarized with such schemes, I could also use legendre gauss lobatto nodes so that they include both endpoints.
I've read they are specially useful for high degree polynomials, so I guess my question would be if it makes any difference in using these nodes when using piecewise low order polynomials, and if using a single high order polynomial with the legendre nodes offers any advantage over the piecewise low order polynomials.

August 21, 2014, 14:49	Legendre Gauss Radau collocation points	#1
Rhaello New Member Rhaello Join Date: Aug 2014 Posts: 2 Rep Power: 0	Hello there, I'm currently dealing with the discretization of an optimal control problem to a NLP problem, I'm a newbie in this subject so please keep that in mind . I'm using orthogonal collocation on finite elements to approximate the states of the system to a set of polynomials, I'm using several low order piecewise lagrange polynomials for this. Now I'm reading about the schemes used to determine the set of collocation points, in particular the legendre-gauss-radau collocation points, they describe the collocation points as being the roots of the legendre polynomial. My questions are the following: Why exactly do they use these collocation points? I've read it is to minimize the error due to the runge phenomenon but I'm not sure if this is the only reason. Does it make a difference to use this points if I use several low order polynomials instead of one high order polynomial? (say 5 polynomials of 3rd degree). Thank you for your time.

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August 22, 2014, 05:39		#2
cfdnewbie Senior Member cfdnewbie Join Date: Mar 2010 Posts: 557 Rep Power: 20	These points have good interpolation properties (low Lebesque constant) and are much better at interpolating functions than equispaced points (which give you large oscillations when interpolating, especially for more complex functions, e.g. Runge's function). Plus, they should have a good Gauss-integration rule associated with them, so if you are thinking from switching from collocation to projection, that would help. The only thing that is unknown to me is why they are using the Radau type: usually, this means that one end of the domain has a node, the other one not. This could make sense for thinks like global hyperbolic problems, but that's problem specific.

August 22, 2014, 09:17		#3
Rhaello New Member Rhaello Join Date: Aug 2014 Posts: 2 Rep Power: 0	Thanks for the answer. It's not really decided to use radau collocation points, I'm just reading about them given that I'm not familiarized with such schemes, I could also use legendre gauss lobatto nodes so that they include both endpoints. I've read they are specially useful for high degree polynomials, so I guess my question would be if it makes any difference in using these nodes when using piecewise low order polynomials, and if using a single high order polynomial with the legendre nodes offers any advantage over the piecewise low order polynomials.