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Manuscript Title: CPDES2: a preconditioned conjugate gradient solver for linear asymmetric matrix equations arising from coupled partial differential equations in two dimensions.
Authors: D.V. Anderson, A.E. Koniges, D.E. Shumaker
Program title: CPDES2
Catalogue identifier: ABFD_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 51(1988)391
Programming language: Fortran.
Computer: CRAY-2.
Operating system: CTSS.
Word size: 64
Keywords: General purpose, Matrix, Coupled partial, Partial differential Equations, Elliptic, Parabolic, Two-dimensional, Plasma physics, Implicit, Preconditioned, Conjugate gradient, Biconjugate gradient, Sparse asymmetric, Matrix.
Classification: 4.8.

Nature of problem:
Certain coupled elliptic and parabolic partial differential equations that arise in plasma physics and other applications are to be solved in two dimensions. The implicit solution techniques used for these equations give rise to a system of linear equations whose matrix operator is sparse (with a complicated subband structure) and generally asymmetric. We provide a fully vectorized algorithm for their solution.

Solution method:
In some earlier matrix solver packages we used an incomplete LU decomposition preconditioning with a conjugate gradient iteration. Alternatively, we now offer the same preconditioning in conjunction with the biconjugate gradient (BCG) method. BCG enjoys faster convergence in most cases but in some instances diverges. Then CG iterations must be used. Here these same methods are applied to the complicated sparse matrix obtained from coupled partial differential equations in two dimensions. The sparsity patterns and coupling stencils, which were previously determined manually, are now generated automatically. The resulting algorithm has gather-scatter constructs which are fully vectorized in our Cray-2 implementation.

The discretization of the coupled two-dimensional PDE's and their boundary conditions must result in an operator stencil that fit into the Cray-2 memory. The matrix must possess a reasonable amount of diagonal dominance for the preconditioning technique to be effective.

Unusual features:
This code relies on the vectorization of gather-scatter procedures. Thus it will be relatively efficient only on computers which posses gather-scatter hardware.

Running time:
These are problem dependent because ill-conditioned matrices require more iterations than well-conditioned ones. The test problems typically converged in a minute or less on the Cray-2 to relative errors of 1.0 * 10**-10.