Parallel Implementation of the Bi-CGSTAB Method with Block Red–Black Gauss–Seidel Preconditioner Applied to the Hermite Collocation Discretization of Partial Differential Equations

We describe herein the parallel implementation of the Bi-CGSTAB method with a block red–black Gauss–Seidel (RBGS) preconditioner applied to the systems of linear algebraic equations that arise from the Hermite collocation discretization of partial differential equations in two spatial dimensions. The method is implemented on the Cray T3E, a parallel processing supercomputer. Speedup results are discussed

Development of a Parallel Computational Framework to Solve Flow and Transport in Integrated Surface-Subsurface Hydrologic Systems

HydroGeoSphere (HGS) is a 3D control-volume finite element hydrologic model describing fully-integrated surface-subsurface water flow and solute and thermal energy transport. Because the model solves tightly-coupled highly-nonlinear partial differential equations, often applied at regional and continental scales (for example, to analyze the impact of climate change on water resources), high performance computing (HPC) is essential. The target parallelization includes the composition of the Jacobian matrix for the iterative linearization method and the sparse-matrix solver, preconditioned BiCGSTAB. The Jacobian matrix assembly is parallelized by using a static scheduling scheme with taking account into data racing conditions, which may occur during the matrix construction. The parallelization of the solver is achieved by partitioning the domain into equal-size sub-domains, with an efficient reordering scheme. The computational flow of the BiCGSTAB solver is also modified to reduce the parallelization overhead and to be suitable for parallel architectures. The parallelized model is tested on several benchmark cases that include linear and nonlinear problems involving various domain sizes and degrees of hydrologic complexity. The performance is evaluated in terms of computational robustness and efficiency, using standard scaling performance measures. Simulation profiling results indicate that the efficiency becomes higher for three situations: 1) with an increasing number of nodes/elements in the mesh because the work load per CPU decreases with increasing the number of nodes, which reduces the relative portion of parallel overhead in total computing time., 2) for increasingly nonlinear transient simulations because this makes the coefficient matrix diagonal dominance, and 3) with domains of irregular geometry that increases condition number. These characteristics are promising for the large-scale analysis of water resource problems that involve integrated surface-subsurface flow regimes. Large-scale real-world simulations illustrate the importance of node reordering, which is associated with the process of the domain partitioning. With node reordering, super-scalarable parallel speedup was obtained when compared to a serial simulation performed with natural node ordering. The results indicate that the number of iterations increases as the number of threads increases due to the increased number of elements in the off-diagonal blocks in the coefficient matrix. In terms of the privatization scheme, the parallel efficiency with privatization was higher than that with the shared scheme for most of simulations performed

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described