Browsing by Author "Pitts, George G."

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    Domain decomposition and high order discretization of elliptic partial differential equations
    Pitts, George G. (Virginia Tech, 1994)
    Numerical solutions of partial differential equations (PDEs) resulting from problems in both the engineering and natural sciences result in solving large sparse linear systems Au = b. The construction of such linear systems and their solutions using either direct or iterative methods are topics of continuing research. The recent advent of parallel computer architectures has resulted in a search for good parallel algorithms to solve such systems, which in turn has led to a recent burgeoning of research into domain decomposition algorithms. Domain decomposition is a procedure which employs subdivision of the solution domain into smaller regions of convenient size or shape and, although such partitionings have proven to be quite effective on serial computers, they have proven to be even more effective on parallel computers. Recent work in domain decomposition algorithms has largely been based on second order accurate discretization techniques. This dissertation describes an algorithm for the numerical solution of general two-dimensional linear elliptic partial differential equations with variable coefficients which employs both a high order accurate discretization and a Krylov subspace iterative solver in which a preconditioner is developed using domain decomposition. Most current research into such algorithms has been based on symmetric systems; however, variable PDE coefficients generally result in a nonsymmetric A, and less is known about the use of preconditioned Krylov subspace iterative methods for the solution of nonsymmetric systems. The use of the high order accurate discretization together with a domain decomposition based preconditioner results in an iterative technique with both high accuracy and rapid convergence. Supporting theory for both the discretization and the preconditioned iterative solver is presented. Numerical results are given on a set of test problems of varying complexity demonstrating the robustness of the algorithm. It is shown that, if only second order accuracy is required, the algorithm becomes an extremely fast direct solver. Parallel performance of the algorithm is illustrated with results from a shared memory multiprocessor.
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    Hodiex: A Sixth Order Accurate Method for Solving Elliptical PDEs
    Pitts, George G.; Ribbens, Calvin J. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1993)
    This paper describes a method for discretizing general linear two dimensional elliptical PDEs with variable coefficients, Lu=g, which achieves high orders of accuracy on an extended range of problems. The method can be viewed as an extension of the ELLPACK6 discretization module HODIE ("High Order Difference Approximation with Identity Expansion"), which achieves high orders of accuracy on a more limited class of problems. We thus call this method HODIEX. An advantage of HODIEX methods, including the one described here, is that they are based on a compact 9-point stencil which yields linear systems with a smaller bandwidth than if a larger stencil were used to achieve higher accuracy.
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    Parallel ELLPACK for Shared Memory Multiprocessors
    Ribbens, Calvin J.; Pitts, George G.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1992)
    This paper describes a parallel version of ELLPACK for shared memory multiprocessors. ELLPACK is a system for numerically solving elliptic PDEs. It consists of a very high level language for defining PDE problems and selecting methods of solution, and a library of approximately fifty problem solving modules. Earlier work considered three discretization modules (five point star, hodie, and hermite collocation), two linear system solution modules (linpack spd band and jacobi cg), and a triple module (hodie fft) which includes both discretization and solution, all for rectangular domains and simple boundary conditions. Here we describe parallel versions of six additional modules (hermite collocation, hodie helmholtz, five point star, band ge, sor, symmetric sor cg) for general boundary conditions and domains, and discuss modifications to the ELLPACK preprocessor, the tool that translates an ELLPACK "program" into FORTRAN.
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    Strategies for Parallelizing PDE Software
    Ribbens, Calvin J.; Pitts, George G. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1992)
    Three strategies for parallelizing components of the mathematical software package ELLPACK are considered: an explicit approach using compiler directives available only on the target machine, an automatic approach using an optimizing and parallelizing precompiler, and a two-level approach based on extensive use of a set of low level computational kernels. Each approach to parallelization is described in detail, along with a discussion of the effort involved. In connection with the third strategy, a set of computational kernels useful for PDE solving is proposed. We describe our experience in parallelizing six problem solving components of ELLPACK using each of the three strategies and give performance results for a shared memory multiprocessor. Our results suggest that the two-level strategy allows the best balance among programmer effort, portability, and parallel performance.