Numerical solution for 3D Poisson equation in circular cylindrical coordinates : Cohl et. al. (
Green's function for 3D Laplace equation or 3D Poisson Equation
Gravitational Potential, Quadrics, Cyclides, Heine identity : Cohl et. al. (
Cylindrical, toroidal, oblate and prolate spheroidal, parabolic, bispherical coordinate systems : Cohl et. al.
New addition theorems for rotationally invariant coordinate systems which R-separate 3D Laplace equation: Cohl et. al.
3D biharmonic, 3D triharmonic, and 3D higher harmonic Green's functions : Cohl (2002)
Spherical coordinate system : Cohl et. al.
Coulomb direct (classical) and exchange (quantum) integrals/interactions : Cohl et. al.
Two-electron interactions : Cohl et. al.
Spherical azimuthal and separation angle Fourier expansions : Cohl et. al.
Magnetic field of an infinitesimally thin circular current loop : Cohl & Tohline
Symmetry properties of associated Legendre/toroidal functions : Cohl et. al.
Whipple formulae for toroidal/associated Legendre functions : Cohl et. al.
New addition theorem for spherical coordinates : Cohl et. al.
Solar White Light Flares : Neidig et. al.
1997 - Cohl, H. S., Xian-He Sun and J. E. Tohline
"Parallel Implementation of a Data-Transpose Technique
for the Solution of Poisson's Equation in Cylindrical Coordinates"
Proceedings of the 8th SIAM Conference on Parallel Processing
for Scientific Computing, Minneapolis, Minnesota, March.
1999a - Cohl, H. S.
"On the numerical solution of the cylindrical Poisson
equation for isolated self-gravitating systems"
The Louisiana State University
and Agricultural and Mechanical College, 122 pages
1999b - Cohl, H. S. and J. E. Tohline
"A Compact Cylindrical Green's Function
Expansion for the Solution of Potential Problems"
The Astrophysical Journal, 527, 86-101.
2000 - Cohl, H. S., J. E. Tohline, A. R. P. Rau, H. M. Srivastava
"Developments in determining the gravitational potential using toroidal functions"
Astronomische Nachrichten, 321, 5/6, 363-372.
2001 - Cohl, H. S., Rau, A. R. P., Tohline, J. E., Browne, D. A., Cazes, J. E. and Barnes, E. I.
"Useful alternative to the multipole expansion of 1/r potentials"
Physical Review A: Atomic and Molecular Physics and Dynamics, 64, 5, 52509.
2002 - Cohl, H. S.
"Portent of Heine's Reciprocal Square Root Identity"
Proceedings of the 3D Stellar Evolution Workshop, ed. R. Cavallo,
S. Keller, S. Turcotte, Livermore, California
Howard S. Cohl - Research Interests
We compute axisymmetric and non-axisymmetric self-gravitating pressure-balanced
equilibria for stellar and protostellar fluid configurations. In order to
compute proper equilibria, special algorithmic treatments are used in 2D
and 3D. Axisymmetric equilibria are obtained with the Hachisu self-consistent
field technique. Non-axisymmetric equilibria can be obtained with Andalib's
self-consistent field technique which is capable of producing flows with
non-trivial internal motions. General self-gravitational non-axisymmetric
azimuthal eigenmode linear stability analysis of unstable dynamic fluid
equilibria are obtained for axisymmetric and non-axisymmetric configurations.
This re-investigation of self-gravitating fluid systems is inspired by the
advent of compact Fourier representation for the Green's functions of
mathematical physics and geometrically describe infinite meridional
contributions to these Green's functions. One may compute Newtonian or
Coulomb or Yukawa boundary potentials by solving the inhomogeneous Laplace or Poisson
equation. One may also express the potential in terms of an inhomogeneous
biharmonic equation whose solution is expressible in terms of an integral of
it's Green's function, |x-x'|, convolved with the Laplacian of the source
density. The Green's function for the biharmonic equation is a much better
behaved integration kernel than that for Poisson's equation, 1/|x-x'|. We
have recently generalized these compact Fourier representations for 1/|x-x'|
to |x-x'| and is amenable in analytic and computational physics applications on
high-performance-computing architectures. Numerically, boundary values are
computed along an arbitrarily chosen z-axis and at values chosen to lie within
the outer-most extent of a chosen volumetric region. We adopt orthogonal
curvilinear or partially structured meshes on modern high-performance
computing architectures to solve these problems. We guarantee appropriate
physical solution of the interior Poisson solve by proper boundary value
treatment as above. Accelerations are obtained from the potential by
performing a precise numerical gradient on the given mesh. In the case of
cylindrical and spherical coordinates, the 3D Poisson solve can be further
facilitated through the use of a discrete azimuthal Fourier transform. This
decouples the 3D Poisson problem into a set of decoupled 2D problems which can
be solved with second order accurate finite differencing using either direct
or iterative methods. I am extremely interested in algorithmic development
and optimization issues in code. I have emphasized on elliptic solvers, but
have given a great deal of attention to general CFD methods. I am particularly
interested in hierarchical programming and memory management strategies for
implementing parallel algorithms on n-way symmetric multiprocessor
architectures. I have experience on many SIMD and MIMD HPC systems. In
terms of astrophysics issues I collaborate with Joel Tohline, LSU, Juhan Frank, LSU, Jim Imamura, University of Oregon, Norman Lebovitz, University of Chicago,
Richard Durisen, Indiana University, and Peter Bodenheimer, UC Santa Cruz. I
collaborate with David Dearborn, Peter Eggleton, John Castor, and Doug Peters, LLNL,
on single and double stellar evolution and pulsation studies with djehuty. I
collaborate with Omar Hurricane, LLNL, and K.S. Balasubramaniam, NSO Sac Peak
on coronal physics.
My work in mathematics focuses on the the different infinite domain
expressions for the Green's functions of the 3D linear PDE's of
mathematical physics. Single, double, and triple integration and summation
expressions for these Green's functions are obtained using R-separation of
variables which allows one to separate a linear 3D PDE into a set of three
ordinary differential equations, thus solving the system analytically. Green's
functions represent the integral inverses of the differential operator which
satisfy inhomogeneous partial differential equations. All properties of
separability will be reflected in the Green's functions and their expansions
in R-separable coordinate systems for these partial differential equations.
There are many known R-separable coordinate systems for these PDE's which are
equivalent to within a conformal symmetry. Application of a conformal
transformation to the solution of these PDE's produces other R-separable
coordinate systems. The 3D linear PDE's I have been focusing on are the
Laplace equation, the Helmholtz equation, the biharmonic equation, the wave
equation, and the Schroedinger equation. It is now clear that there must
exist new compact single summation representations for these Green's functions.
In the rotationally invariant coordinate systems which R-separably solve
these 3D PDE's, new compact Fourier representations exist for the Green's
functions and are given in terms of the azimuthal and separation angles.
The coefficients of the discrete Fourier representation of these Green's
functions are given in terms of identifiable infinite series expressions
whose transcendental function description lead to further understanding of
implied hierarchical geometry. Further understanding of the implied
symmetries can be described through pairs of commuting and anti-commuting
operators which represent the coordinate systems in a Lie algebraic and group
formalism. The fundamental mathematical tools required in order to complete
this investigation are commonly available in the mathematics and physics
literature. I am collaborating on these issues with Hari Srivastava,
University of Victoria, Canada, Willard Miller, University of Minnesota,
and Ernie Kalnins, University of Waikato, New Zealand.
Proper analytic and algorithmic implementation of compact representations for
the Green's functions of mathematical physics significantly improves the
accuracy of the solution for the PDE's. Toroidal expressions for the
reciprocal distance between two points |x-x'|-1
lead to new classical and quantum energy theorems which allow for rapid and
precise evaluation of the Coulomb direct and exchange interactions. In
quantum physics this is accomplished through the use of the azimuthal
selection rule for the self-energies, namely for the direct and exchange
Hamiltonian elements only the m=0 and m=m1-m2 terms survive respectively.
The application of the selection rule allows for exact evaluation of the
Hamiltonian matrix elements for two-electron interactions in atomic physics,
molecular physics, condensed matter physics, physical chemistry and biology.
We are constructing 3D parallel Coulomb Hamiltonian solvers in the above
mentioned mesh geometries for use in these fields. Two-electron interactions
are critically important in obtaining opacities and correct equations of state
in astrophysically dense atomic and molecular fluid media.
Precise Coulomb and Yukawa energies in the nuclear Hamiltonian allow for a
higher degree of precision in obtaining nuclear structure. We perform
computational nuclear structure in order to reproduce and explore basic
periodic table properties such as the stability and energies of nuclear
bound states. Finally, we will search the high-Z areas where magic numbers
are expected to lie, and map these areas for bound states isotopes. In a
parallel computing exercise, I explore the Ziff-Gulari-Barshad model for
the carbon-monoxide oxidation reaction using 2D Monte Carlo. In quantum
physics I collaborate with Dana A. Browne, LSU, A.R.P. Rau, LSU, and Vijay
Sonnad, LLNL. In molecular physics and quantum chemistry I collaborate with
Devashis Majumdar, UC Davis. By using Heine methods in chemistry we study
the nature of bonding and chemical reactions. In nuclear physics, I
collaborate with Sait Umar and Volker E. Oberacker, Vanderbilt University
with specific application on high-Z nuclei. I collaborate with Omar Hurricane
on precise vector potential solutions to MHD problems by expressing the MHD
equations in terms of a vector Poisson equation. MHD problems can now be
easily handled with Fourier expansions for the Green's function of the
biharmonic equation. By expressing the equations of fluid dynamics of a
compressible media in terms of a velocity potential and a vector Poisson
equation, one may compute precise velocity boundary values analytically in
vortex and shock flow regions (Lamb 1945) using Poisson problems. Radiation
transport can be also facilitated through compact Poisson formulation of the
Green's function for the diffusion equation. Inversion methods and data
analysis for helioseismology and classical and quantum scattering theory
using compact Fourier expansions will gain further insight with experimental data.