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Many-Body Numeric Integration Algorithm

  1. Aug 4, 2008 #1
    That is to say, how does one go about it for a non-separable partial differential equation?

    Let me preface by saying that I am not asking for an answer of perturbation theory, variational theory, mean-field theory, or some sort of self-consistent guess-and-check method (i.e., coupled cluster). I know and acknowledge that coupled cluster is king of the hill for numeric methods of analysis for nuclear and chemical systems. My question is geared more toward the pure math of it...at the risk of possibly showing ignorance of the math behind using coupled cluster.

    Here is what I think is what works, please tell me if this is on the money.

    Normal methods of numeric quadrature are just fine for all of the ODE terms, it is the non-separable terms that represent a bit more of a challenge. Thus, one creates a system of equations, one for each body. One does numeric integration on the first body in the system with all other coordinates not of that body held constant, then the second equation of the system is numerically integrated on the second set of variables, etc., until one has a matrix which has as many rows as bodies and as many columns as terms in the Hamiltonian.

    Perhaps in more explicit terms...if I am in 3 dimensions, and if I have 3 bodies in question, letting the bodies be 1,2,3 with variables x1,y1,z1; x2,y2,z2; x3,y3z3...
    I will take the non-separable PDE, begin numeric integration with x2,y2,z2 and x3,y3,z3 held constant, generating the first matrix equation, then the next two matrix equations will be the PDE numerically integrated with first x1,y1,z1 and x3,y3,z3 constant, then x1,y1,z1 x2,y2,z2 held constant?

    Thanks for your time,
  2. jcsd
  3. Aug 5, 2008 #2
    For a sparse system of linear equations I think it's using
    Pre-conditioned Conjugate Gradient or Multigrid methods.
    For non-sparse and non-linear systems then I'm not sure.
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