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91
The PERFECT Club Benchmarks: Effective Performance Evaluation of Supercomputers
 International Journal of Supercomputer Applications
, 1988
"... This report consists of two major portions. First is the presentation of a methodology for measuring the performance of supercomputers. This includes a set of thirteen Fortran programs that total well over 50,000 lines of source code. They represent applications in a number of areas of engineering a ..."
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Cited by 210 (4 self)
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This report consists of two major portions. First is the presentation of a methodology for measuring the performance of supercomputers. This includes a set of thirteen Fortran programs that total well over 50,000 lines of source code. They represent applications in a number of areas of engineering and scientific computing, and in many cases they represent codes that are currently used by a number of computational research and development groups. We also present the PERFECT Fortran standard which is simply a set of guidelines that allow portability to a number of types of machines. Furthermore, we present some performance measures and a methodology for recording and sharing results among a group of diverse users on different machines. The second portion of the paper presents some of the results we have obtained over the past year and a half. The results should not be used to compare machines, except in a very preliminary sense. Rather, the results are presented to show how the methodolo...
Analysis and Design of Numerical Schemes for Gas Dynamics 1 Artificial Diffusion, Upwind Biasing, Limiters and Their Effect on Accuracy and Multigrid Convergence
 INTERNATIONAL JOURNAL OF COMPUTATIONAL FLUID DYNAMICS
, 1995
"... The theory of nonoscillatory scalar schemes is developed in this paper in terms of the local extremum dimin ishing (LED) principle that maxima should not increase and minima should not decrease. This principle can be used for multidimensional problems on both structured and unstructured meshes, w ..."
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Cited by 108 (46 self)
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The theory of nonoscillatory scalar schemes is developed in this paper in terms of the local extremum dimin ishing (LED) principle that maxima should not increase and minima should not decrease. This principle can be used for multidimensional problems on both structured and unstructured meshes, while it is equivalent to the total variation diminishing (TVD) principle for onedimensional problems. A new formulation of symmetric limited positive (SLIP) schemes is presented, which can be generalized to produce schemes with arbitrary high order of accuracy in regions where the solution contains no extrema, and which can also be implemented on multidimensional unstructured meshes. Systems of equations lead to waves traveling with distinct speeds and possibly in opposite directions. Alternative treatments using characteristic splitting and scalar diffusive fluxes are examined, together with a modification of the scalar diffusion through the addition of pressure differences to the momentum equations to produce full upwinding in supersonic flow. This convective upwind and split pressure (CUSP) scheme exhibits very rapid convergence in multigrid calculations of transonic flow, and provides excellent shock resolution at very high Mach numbers.
Multigrid Solution for HighOrder Discontinuous Galerkin . . .
, 2004
"... A highorder discontinuous Galerkin finite element discretization and pmultigrid solution procedure for the compressible NavierStokes equations are presented. The discretization has an elementcompact stencil such that only elements sharing a face are coupled, regardless of the solution space. Thi ..."
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Cited by 34 (13 self)
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A highorder discontinuous Galerkin finite element discretization and pmultigrid solution procedure for the compressible NavierStokes equations are presented. The discretization has an elementcompact stencil such that only elements sharing a face are coupled, regardless of the solution space. This limited coupling maximizes the effectiveness of the pmultigrid solver, which relies on an elementline Jacobi smoother. The elementline Jacobi smoother solves implicitly on lines of elements formed based on the coupling between elements in a p = 0 discretization of the scalar transport equation. Fourier analysis of 2D scalar convectiondiffusion shows that the elementline Jacobi smoother as well as the simpler element Jacobi smoother are stable independent of p and flow condition. Mesh refinement studies for simple problems with analytic solutions demonstrate that the discretization achieves optimal order of accuracy of O(h p+1). A subsonic, airfoil test case shows that the multigrid convergence rate is independent of p but weakly dependent on h. Finally, higherorder is shown to outperform grid refinement in terms of the time required to reach a desired accuracy level.
Positive schemes and shock modelling for compressible flows
 International Journal for Numerical Methods in Fluids
, 1995
"... A unified theory of nonoscillatory finite volume schemes for both structured and unstructured meshes is developed in two parts. In the first part, a theory of local extremum diminishing (LED) and essentially local extremum diminishing (ELED) schemes is developed for scalar conservation laws. This l ..."
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Cited by 33 (2 self)
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A unified theory of nonoscillatory finite volume schemes for both structured and unstructured meshes is developed in two parts. In the first part, a theory of local extremum diminishing (LED) and essentially local extremum diminishing (ELED) schemes is developed for scalar conservation laws. This leads to symmetric and upstream limited positive (SLIP and USLIP) schemes which can be formulated on either structured or unstructured meshes. The second part examines the application of similar ideas to the treatment of systems of conservation laws. An analysis of discrete shock structure leads to conditions on the numerical flux such that stationary discrete shocks can contain a single interior point. The simplest formulation which meets these conditions is a convective upwind and split pressure (CUSP) scheme, in which the coefficient of the pressure differences is fully determined by the coefficient of convective diffusion. Numerical results are presmted which confirm the properties of these schemes. KEY WORDS computational aerodynamics; shock capturing; positive schemes 1.
Realcoded adaptive range genetic algorithm applied to transonic wing optimization
 Applied Soft Computing
, 2001
"... applied to a practical threedimensional shape optimization for aerodynamic design of an aircraft wing. The realcoded ARGAs possess both advantages of the binarycoded ARGAs and the floatingpoint representation to overcome the problems of having a large search space that requires continuous sampli ..."
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Cited by 18 (8 self)
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applied to a practical threedimensional shape optimization for aerodynamic design of an aircraft wing. The realcoded ARGAs possess both advantages of the binarycoded ARGAs and the floatingpoint representation to overcome the problems of having a large search space that requires continuous sampling. The results confirm that the realcoded ARGAs consistently find better solutions than the conventional realcoded Genetic Algorithms do. 1
Recent Improvements in Efficiency, Accuracy and Convergence for Implicit Approximate Factorization Algorithms," AIAA Paper
, 1985
"... and three ' dimensions have been developed at NASA Ames and have been widely distributed since their introduction in 1977 and 1978. These codes, now referred to ARC2D and ARC3D, can run either in inviscid or viscous mode for steady or unsteady flow. They use general coordinate systems and can ..."
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Cited by 18 (0 self)
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and three ' dimensions have been developed at NASA Ames and have been widely distributed since their introduction in 1977 and 1978. These codes, now referred to ARC2D and ARC3D, can run either in inviscid or viscous mode for steady or unsteady flow. They use general coordinate systems and can be run on Bny smoothly varying curvilinear mesh, even a mesh that is quite skew. Because they use well ordered finite difference grids, the codes can take advant.age of vectorized computer processors and have been implemented for the Control Data 205 and the CRAY IS and XMP. On a single processor of the XMP a vectorized version of the code runs approximately 20 times faster than the original code which was written for the Control Data 7600. Traditionally gains in Computational efficiency due to improved numerical algorithms have kept pace with gains due to increased computer power. Since the ARC2D and ARC3D codes were introduced, a variety of algorithmic changes have been individually tested and have been shown to improve overall computational efficiency. These include use of a spatially varying time step ( A t) , use of a sequence of mesh refinements t o establish approximate solutions, implementation of various ways to reduce inversion work, improved numerical dissipation terms, and more implicit treatment of terms. Although the
A highorder discontinuous Galerkin multigrid solver for . . .
, 2004
"... Results are presented from the development of a highorder discontinuous Galerkin finite element solver using pmultigrid with line Jacobi smoothing. The line smoothing algorithm is presented for unstructured meshes, and pmultigrid is outlined for the nonlinear Euler equations of gas dynamics. Anal ..."
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Cited by 15 (2 self)
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Results are presented from the development of a highorder discontinuous Galerkin finite element solver using pmultigrid with line Jacobi smoothing. The line smoothing algorithm is presented for unstructured meshes, and pmultigrid is outlined for the nonlinear Euler equations of gas dynamics. Analysis of 2D advection shows the improved performance of line implicit versus block implicit relaxation. Through a mesh refinement study, the accuracy of the discretization is determined to be the optimal O(h p+1) for smooth problems in 2D and 3D. The multigrid convergence rate is found to be independent of the interpolation order but weakly dependent on the grid size. Timing studies for each problem indicate that higher order is advantageous over grid refinement when high accuracy is required. Finally, parallel versions of the 2D and 3D solvers demonstrate close to ideal coarsegrain scalability.
Multigrid Acceleration of an Upwind Euler Solver on Unstructured Meshes
, 1995
"... Multigrid acceleration has been implemented for an upwind flow solver on unstructured meshes. The flow solver is a straightforward implementation of Barth and Jespersen's unstructured scheme, with leastsquares linear reconstruction and a directional implementation of Venkatakrishnan 's li ..."
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Cited by 14 (9 self)
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Multigrid acceleration has been implemented for an upwind flow solver on unstructured meshes. The flow solver is a straightforward implementation of Barth and Jespersen's unstructured scheme, with leastsquares linear reconstruction and a directional implementation of Venkatakrishnan 's limiter. The multigrid scheme itself is designed to work on mesh systems which are not nested, allowing great flexibility in generating coarse meshes and in adapting fine meshes. A new scheme for automatically generating coarse unstructured meshes from fine ones is presented. A subset of the fine mesh vertices are selected for retention in the coarse mesh. The coarse mesh is generated incrementally from the fine mesh by removing one rejected vertex at a time. In this way, a valid coarse mesh triangulation is guaranteed. Factors affecting multigrid convergence rate for inviscid flow are thoroughly examined, including the effect of the number of coarse meshes used; the type of multigrid cycle employed; th...
An Efficient NewtonGMRES Solver for Aerodynamic Computations
 Proceedings of the 13th AIAA CFD Conference, Snowmass
, 1997
"... An efficient inexactNewtonKrylov algorithm is presented for the computation of steady aerodynamic flows. The algorithm uses preconditioned, restarted GMRES in matrixfree form to solve the linear system arising at each Newton iteration. The preconditioner is formed using an ILU(2) factorization of ..."
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Cited by 13 (4 self)
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An efficient inexactNewtonKrylov algorithm is presented for the computation of steady aerodynamic flows. The algorithm uses preconditioned, restarted GMRES in matrixfree form to solve the linear system arising at each Newton iteration. The preconditioner is formed using an ILU(2) factorization of an approximate Jacobian matrix after applying the Reverse CuthillMcKee reordering. The algorithm has been successfully applied to a wide range of test cases which include inviscid, laminar, and turbulent aerodynamic flows. In all cases except one, convergence of the residual to 10^12 is achieved with a CPU cost equivalent to fewer than 1200 function evaluations. The sole exception is a low Mach number case where some form of local preconditioning is needed. Several other efficient implicit solvers have been applied to the same test cases, and the matrixfree inexactNewtonGMRES algorithm is seen to be the fastest and most robust of the methods studied. Hence this strategy is an excellent option for flow computations in which memory use is not critical, such as twodimensional applications.
Multigrid Methods For Differential Equations With Highly Oscillatory Coefficients
 In Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods
, 1993
"... this paper we analyse the convergence of multigrid methods for equation (1) by introducing new coarse grid operators, based on local or global homogenized form of the equation. We consider only two level multigrid methods. For full multigrid or with more general coefficients the homogenized operator ..."
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Cited by 12 (2 self)
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this paper we analyse the convergence of multigrid methods for equation (1) by introducing new coarse grid operators, based on local or global homogenized form of the equation. We consider only two level multigrid methods. For full multigrid or with more general coefficients the homogenized operator can be numerically calculated from the finer grids based on local solution of the so called cell problem [2]. In a number of numerical tests we compare the convergence rate for different choices of parameter and coarse grid operators applied to a two dimensional elliptic model problem. The convergence rate is also analyzed theoretically for a one dimensional problem. If, for example, the oscillatory coefficient is replaced by its average, the direct estimate for multigrid convergence rate is not asymptotically better than just using the damped Jacobi smoothing operator. The homogenized coefficient reduces the number of smoothing operation from O(h