Computational implementation of integral equations: Difference between revisions

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Integral equations are solved numerically.
One has the [[Ornstein-Zernike relation]], <math>\gamma (12)</math>
and a closure relation, <math>c_2 (12)</math> (which
incorporates the [[bridge function]] <math>B(12)</math>).
The numerical solution is iterative;
# trial solution for  <math>\gamma (12)</math>
# calculate  <math>c_2 (12)</math>
# use the [[Ornstein-Zernike relation]] to generate a new  <math>\gamma (12)</math> ''etc.''
Note that the value of  <math>c_2 (12)</math> is '''local''', ''i.e.''
the  value of  <math>c_2 (12)</math> at a given point is given by
the value of  <math>\gamma (12)</math> at this point. However, the [[Ornstein-Zernike relation]] is '''non-local'''.
The way to convert the [[Ornstein-Zernike relation]] into a local equation
is to perform a [[Fast Fourier transform |(fast) Fourier transform]] (FFT).
Note: convergence is poor for liquid densities. (See Ref.s 1 to 6).
==Picard iteration==
==Picard iteration==
==Ng acceleration==
==Ng acceleration==
Line 5: Line 23:
==References==
==References==
#[http://dx.doi.org/10.1080/00268977900102861 M. J. Gillan "A new method of solving the liquid structure integral equations" Molecular Physics '''38''' pp. 1781-1794 (1979)]
#[http://dx.doi.org/10.1080/00268977900102861 M. J. Gillan "A new method of solving the liquid structure integral equations" Molecular Physics '''38''' pp. 1781-1794 (1979)]
#[http://dx.doi.org/10.1080/00268978500102651 Stanislav Labík,  Anatol Malijevský and Petr Voncaronka "A rapidly convergent method of solving the OZ equation" Molecular Physics '''56''' pp. 709-715 (1985)]
#[http://dx.doi.org/10.1080/00268978500102651 Stanislav Labík,  Anatol Malijevský and Petr Voncaronka "A rapidly convergent method of solving the OZ equation", Molecular Physics '''56''' pp. 709-715 (1985)]
#[http://dx.doi.org/10.1080/00268978200100202 F. Lado "Integral equations for fluids of linear molecules I. General formulation", Molecular Physics '''47''' pp. 283-298 (1982)]
#[http://dx.doi.org/10.1080/00268978200100212 F. Lado "Integral equations for fluids of linear molecules II. Hard dumbell solutions", Molecular Physics '''47''' pp. 299-311 (1982)]
#[http://dx.doi.org/10.1080/00268978200100222 F. Lado "Integral equations for fluids of linear molecules III. Orientational ordering", Molecular Physics '''47''' pp. 313-317 (1982)]
#[http://dx.doi.org/10.1080/00268978900101981 Enrique Lomba "An efficient procedure for solving the reference hypernetted chain equation (RHNC) for simple fluids" Molecular Physics '''68''' pp. 87-95 (1989)]
#[http://dx.doi.org/10.1080/00268978900101981 Enrique Lomba "An efficient procedure for solving the reference hypernetted chain equation (RHNC) for simple fluids" Molecular Physics '''68''' pp. 87-95 (1989)]

Revision as of 11:47, 30 May 2007

Integral equations are solved numerically. One has the Ornstein-Zernike relation, and a closure relation, (which incorporates the bridge function ). The numerical solution is iterative;

  1. trial solution for
  2. calculate
  3. use the Ornstein-Zernike relation to generate a new etc.

Note that the value of is local, i.e. the value of at a given point is given by the value of at this point. However, the Ornstein-Zernike relation is non-local. The way to convert the Ornstein-Zernike relation into a local equation is to perform a (fast) Fourier transform (FFT). Note: convergence is poor for liquid densities. (See Ref.s 1 to 6).


Picard iteration

Ng acceleration

References

  1. M. J. Gillan "A new method of solving the liquid structure integral equations" Molecular Physics 38 pp. 1781-1794 (1979)
  2. Stanislav Labík, Anatol Malijevský and Petr Voncaronka "A rapidly convergent method of solving the OZ equation", Molecular Physics 56 pp. 709-715 (1985)
  3. F. Lado "Integral equations for fluids of linear molecules I. General formulation", Molecular Physics 47 pp. 283-298 (1982)
  4. F. Lado "Integral equations for fluids of linear molecules II. Hard dumbell solutions", Molecular Physics 47 pp. 299-311 (1982)
  5. F. Lado "Integral equations for fluids of linear molecules III. Orientational ordering", Molecular Physics 47 pp. 313-317 (1982)
  6. Enrique Lomba "An efficient procedure for solving the reference hypernetted chain equation (RHNC) for simple fluids" Molecular Physics 68 pp. 87-95 (1989)