Roland A. Sweet

Algorithm 541: Efficient Fortran Subprograms for the Solution of Separable Elliptic Partial Differential Equations [D3]

Computer models of geophysical processes often require the numerical solution of elliptic partial differential equations. This is particularly true for models that make use of stream functions, velocity potentials, or vorticity equations and for models that compute the pressure of an incompressible fluid. The numerical solution of elliptic equations can be a formidable programming task. Moreover, the equations are often time dependent, requiring repeated solutions and, hence, considerable computing resources. Recent advances in computing methods [1, 2] made it possible to solve a very large class of elliptic equations (the separable ones) rapidly and with minimal storage. And as a result of work on singular problems [6, 8], this class is free of special cases for which solutions cannot be obtained numerically. This paper describes a package of computer programs that make use of current methods for solving elliptic partial differential equations. The package is fully documented in [7]. We fLrst became involved in implementing the Buneman algorithm [2] and its extensions via the capacitance matrix approach [1] for solving Poissons equation

A Cyclic Reduction Algorithm for Solving Block Tridiagonal Systems of Arbitrary Dimension

A generalization of the Buneman variant of cyclic odd–even reduction algorithm for solving finite difference approximations to Poisson’s equations is presented. This generalization places no restriction on the block size, n, of the system and computes the solution in

A direct method for the solution of poisson's equation with neumann boundary conditions on a staggered grid of arbitrary size

O(n^2 \log _2 n)

A Generalized Cyclic Reduction Algorithm

operations.

Efficient FORTRAN subprograms for the solution of elliptic partial differential equations (abstract)

A method based on cyclic reduction is described for the solution of the discrete Poisson equation on a rectangular two-dimensional staggered grid with an arbitrary number of grid points in each direction. Neumann boundary conditions are assumed in one direction and any boundary condition may be used in the other direction. The coefficients of the equation can be functions of the latter direction so that, e.g., non-equidistant grid spacings or non-Cartesian coordinates can be used. Poissons equation with these boundary conditions describes, e.g., the pressure field of an incompressible fluid flow within rigid boundaries. Numerical results are reported for a FORTRAN subroutine using the method. For an M × N grid the operation count is proportional to MN log2 N, and about MN storage locations are required.

A Parallel and Vector Variant of the Cyclic Reduction Algorithm

For the discrete Poisson equation, a generalization of cyclic reduction to arbitrary block size is given. A stable algorithm for accumulating the right sides is presented and results are given concerning optimal block size. Numerical experiments are also reported.

Efficient FORTRAN subprograms for the solution of elliptic partial differential equations.

The purpose of this paper is to describe technical note, NCAR TN/IA-109, which is intended to provide scientists with a package of computer programs which make use of current methods for solving elliptic partial differential equations. Computer models of geophysical processes often require the numerical solution of elliptic partial differential equations. This is particularly true for models which make use of stream functions, velocity potentials, or vorticity equations, or in which the pressure of an incompressible fluid is computed. The numerical solution of elliptic equations can be a formidable programming task. Also, the equations are often time-dependent, requiring repeated solutions and, hence, considerable computing resources. Efficient, reliable, and well-documented computer subroutines for solving such equations are needed. With recent advances in computing methods, it became apparent to the authors that a very large class of elliptic equations (separable) could be solved rapidly and with minimal storage. Of particular importance was the fact that, as a result of work on singular problems, this class was free of special cases for which solutions could not be obtained numerically.

The Direct Solution of the Discrete Poisson Equation on a Disk

The Buneman variant of the block cyclic reduction algorithm begins as a highly parallel algorithm, but collapses with each reduction to a very serial one. Using partial fraction expansions of rational matrix functions, it is shown how to regain the parallelism. The resulting algorithm using

Vector and parallel methods for the direct solution of Poisson's equation

n^2

Mesh refinements for parabolic equations

processors runs in