Showing papers by "Paul Fischer published in 1998"

Direct Numerical Simulation of Three-Dimensional Flow and Augmented Heat Transfer in a Grooved Channel

Abstract: Direct numerical simulations of three-dimensional flow and augmented convective heat transfer in a transversely grooved channel are presented for the Reynolds number range 140 < Re < 2000. These calculations employ the spectral element technique. Multiple flow transitions are documented as the Reynolds number increases, from steady two-dimensional flow through broad-banded unsteady three-dimensional mixing. Three-dimensional simulations correctly predict the Reynolds-number-independent friction factor behavior of this flow and quantify its heat transfer to within 16 percent of measured values. Two-dimensional simulations, however, incorrectly predict laminar-like friction factor and heat transfer behaviors

An overlapping Schwarz method for spectral element simulation of three-dimensional incompressible flows.

Abstract: As the sound speed is infinite for incompressible flows, computation of the pressure constitutes the stiffest component in the time advancement of unsteady simulations. For complex geometries, efficient solution is dependent upon the availability of fast solvers for sparse linear systems. In this paper we develop a Schwarz preconditioner for the spectral element method using overlapping subdomains for the pressure. These local subdomain problems are derived from tensor products of one-dimensional finite element discretizations and admit use of fast diagonalization methods based upon matrix-matrix products. In addition, we use a coarse grid projection operator whose solution is computed via a fast parallel direct solver. The combination of overlapping Schwarz preconditioning and fast coarse grid solver provides as much as a fourfold reduction in simulation time over previously employed methods based upon deflation for parallel solution of multi-million grid point flow problems.

Algorithms for large-scale parallel simulation of unsteady incompressible flows in three-dimensional complex geometries

Abstract: We develop a fast direct solver for parallel solution of "coarse grid" problems, Ax = b, such as arise when domain decomposition or multigrid methods are applied to elliptic partial differential equations in d space dimensions. The approach is based upon a (quasi-) sparse factorization of the inverse of A. If A is $n\times n,$ P is the number of processors, and $\gamma\equiv {{d-1}\over d},$ then each solve requires $O(n\sp\gamma\ \log\sb2\ P)$ time for communication and $O(n\sp{1+\gamma}/P)$ time for computation. Results from a 512 node Intel Paragon show that our algorithm compares favorably to more commonly used approaches which require $O(n\ \log\sb2\ P)$ time for communication and $O(n\sp{1+\gamma})$ or $O(n\sp2/P)$ time for computation. Moreover, for leading edge multicomputer systems with thousands of processors and $n = P$ (i.e., communication dominated solves), we expect our algorithm to he markedly superior as it achieves substantially reduced message volume and arithmetic complexity compared to competing methods while retaining minimal message startup cost. As the characteristic propagation speed is infinite for unsteady incompressible flows, solving for the pressure operator is the most computationally challenging phase of flow simulation. Efficient solution is dependent upon the availability of fast solvers for sparse linear systems. In this paper we develop an Schwarz preconditioner using overlapping subdomains for the pressure. These local subdomain problems are derived from tensor products of one-dimensional finite element discretizations and admit use of fast diagonalization methods based upon matrix-matrix products. In addition, we use a coarse grid projection operator applied directly to the pressure of the interior Gauss points whose solution is computed via a fast parallel direct solver. The combination of overlapping Schwarz preconditioning and coarse grid solver provides as much as a four-fold reduction in simulation time over previously employed methods based upon deflation for parallel solution of multi-million grid point flow problems.

Modified Conjugate Gradient Method for the Solution of A {\underline x} = {\underline b}

Abstract: In this note, we examine a modified conjugate gradient procedure for solving A {\underline x} = {\underline b} in which the approximation space is based upon the Krylov space (({\mathscr K}^k_{\sqrt {A}, {\underline b}})) associated with sqrt {A} and {\underline b}. We show that, given initial vectors {\underline b} and \sqrt {A} \, {\underline b} (possibly computed at some expense), the best fit solution in {\mathscr K}^k_{\sqrt {A}, {\underline b}} can be computed using a finite-term recurrence requiring only one multiplication by A per iteration. The initial convergence rate appears, as expected, to be twice as fast as that of the standard conjugate gradient method, but stability problems cause the convergence to be degraded.

High Order Accuracy Computational Methods for Long Time Integration of Nonlinear PDEs in Complex Domains

Abstract: : The overarching goal of this research was to construct stable, robust and efficient high order accurate computational methods for long time integration of nonlinear partial differential equations High order accuracy methods (Spectral, Finite Difference and Finite Elements) for the numerical simulations of flows with discontinuities, in complex geometries were developed In particular applications in supersonic combustion were emphasized Specific research subjects included: Robust high order compact difference schemes, ENO and WENO schemes, discontinuous Galerkin methods, the resolution of the Gibbs phenomenon, parallel computing and high order accurate boundary conditions In order to overcome the difficulties stemming from complicated geometries, we have developed multidomain techniques as well as spectral methods on arbitrary grids Several multidimensional codes for supersonic reactive flows had been constructed as well as a library of spectral codes (Pseudopack)