Green's function for 3D Laplace equation or 3D Poisson Equation

Gravitational Potential, Quadrics, Cyclides, Heine identity : Cohl et. al. ( 1999a, 1999b, 2000, 2001, 2002)

Cylindrical, toroidal, oblate and prolate spheroidal, parabolic, bispherical coordinate systems : Cohl et. al. (1999a, 2000)

New addition theorems for rotationally invariant coordinate systems which R-separate 3D Laplace equation: Cohl et. al. (1999a, 2000)

3D biharmonic, 3D triharmonic, and 3D higher harmonic Green's functions : Cohl (2002)

Spherical coordinate system : Cohl et. al. (2001)

Coulomb direct (classical) and exchange (quantum) integrals/interactions : Cohl et. al. (2001)

Two-electron interactions : Cohl et. al. (2001)

Spherical azimuthal and separation angle Fourier expansions : Cohl et. al. (2001)

Magnetic field of an infinitesimally thin circular current loop : Cohl & Tohline (1999)

Symmetry properties of associated Legendre/toroidal functions : Cohl et. al. (2000)

Whipple formulae for toroidal/associated Legendre functions : Cohl et. al. (2000)

New addition theorem for spherical coordinates : Cohl et. al. (2001)

Solar White Light Flares : Neidig et. al. (1993)

"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,

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,

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,

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

Astrophysics:

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.

Mathematics:

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.

Theoretical Physics:

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'|