I'm trying to numerically solve a pair of coupled second-order nonlinear ODEs. I was previously trying to do it using regular polynomials but found this to be very inefficient. So, I decided I'd try to use orthogonal polynomials to make the computation time shorter. As I understand it (and please correct me if I'm wrong), the orthogonality of a polynomial dictates that a fitted coefficient of order n-1 will be unaffected when increasing the order of the fit from n-1 to n. Basically, this allows me to incrementally increase the polynomial degree and only force my poor, five-year-old computer to vary one of the coefficients at a time.
So, assuming the above is correct, how would I go about selecting an appropriate polynomial? The problem I'm solving is an axisymmetric shell problem in elasticity (geometrically nonlinear, linear material). I'm currently working with Legendre polynomials, but I'm not sure this is the correct choice.
One more question, while I'm at it, regarding mapping of the polynomials. Legendre polynomials are orthogonal on -1<=x<=1. My domain is really 0<=t<=pi/2. Thus, I let x=4t/pi-1, which is just a simple linear mapping. Does this mapping ensure that the polynomial in t is orthogonal on [0,pi/2]?
So, assuming the above is correct, how would I go about selecting an appropriate polynomial? The problem I'm solving is an axisymmetric shell problem in elasticity (geometrically nonlinear, linear material). I'm currently working with Legendre polynomials, but I'm not sure this is the correct choice.
One more question, while I'm at it, regarding mapping of the polynomials. Legendre polynomials are orthogonal on -1<=x<=1. My domain is really 0<=t<=pi/2. Thus, I let x=4t/pi-1, which is just a simple linear mapping. Does this mapping ensure that the polynomial in t is orthogonal on [0,pi/2]?
Facebook
Twitter