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Manuscript Title: Further developments in the noniterative method of solving PDE's in electron scattering.
Authors: E.C. Sullivan, A. Temkin
Program title: SEPDE3
Catalogue identifier: ACJB_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 71(1992)319
Programming language: Fortran.
Computer: CRAY YMP.
Operating system: UNICOS, VMS.
RAM: 2000K words
Word size: 64
Peripherals: disc.
Keywords: General purpose, Atomic, Molecular, Electron scattering, Electron, Molecule, Scattering, Three dimensional Elliptic partial, Differential equations, Non-iterative solution, Large banded linear Systems, Rectangular boundary Conditions.
Classification: 4.3, 16.5.

Nature of problem:
The program will compute a noniterative solution of any two or three dimensional elliptic partial differential equation with rectangular boundary conditions. There are no restrictions on the type of differences used to represent the derivatives or the number of different boundary values that are specified. Coupled equations as well as equations using a higher order than second partial derivative operator, such as the biharmonic operator, also lend themselves to solution using this method. The main criterion is that the approximating linear system be representable by a banded block matrix. The program can also be used for computing two or three dimensional parabolic differential equations by solving an implicit system at a discrete value of the time variable, with repetitive use of the program to advance the solution in time. For a demonstration of the programs capability and proper execution, a block matrix evaluation routine is provided for a solvable 3 dimensional Schrodinger equation resulting from the hybrid-theory of electron- molecule scattering as applied to e-N2 scattering. The variables are the r and theta of the scattered electron and the internuclear separation R of the molecular target (N2).

Solution method:
The finite difference approximation to an elliptic partial differential equation in two or more dimensions may be represented as a linear system of equations that is naturally partitioned into block banded systems. The size of the blocks and the width of the band is dependent on the grid desired and the number of points used to approximate partial derivatives. The program will solve systems of arbitrary block size and band width subject, of course, to computer time, memory, and space constraints and obtain a noniterative solution. Since the method works with blocks comprising only portions of the complete system, the computer memory requirement has been significantly reduced.

The program is restricted to elliptic partial differential equations with rectangular boundary conditions. The memory requirements and temporary disk space required are dependent on the differential equation being solved and the grid size [e.g. a (23, 31, 79) grid will require 2 megawords of memory and 956 megabytes of disk space]. The user must provide a subroutine to evaluate the coefficients of the block matrices; similar to the way a derivative evaluation routine must be provided for ordinary differential equation solvers.

Running time:
A (5, 7, 19) grid with a block bandwidth of 3 takes 0.007 minutes while a (11, 15, 39) grid with a block bandwidth of 7 takes 82.37 minutes. Both cases are the times required on a Cray YMP computer.