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At least 19 records

SpF: Enabling Petascale Performance for Pseudospectral Dynamo Models

Pseudospectral (PS) methods possess a number of characteristics (e.g., efficiency, accuracy, natural boundary conditions) that are extremely desirable for dynamo models. Unfortunately, dynamo models based upon PS methods face a number of daunting challenges, which include exposing additional parallelism, leveraging hardware accelerators, exploiting hybrid parallelism, and improving the scalability of global memory transposes. Although these issues are a concern for most models, solutions for PS methods tend to require far more pervasive changes to underlying data and control structures. Further, improvements in performance in one model are difficult to transfer to other models, resulting in significant duplication of effort across the research community.We have developed an extensible software framework for pseudospectral methods called SpF that is intended to enable extreme scalability and optimal performance. High-level abstractions provided by SpF unburden applications of the responsibility of managing domain decomposition and load balance while reducing the changes in code required to adapt to new computing architectures. The key design concept in SpF is that each phase of the numerical calculation is partitioned into disjoint numerical kernels that can be performed entirely in-processor. The granularity of domain-decomposition provided by SpF is only constrained by the data-locality requirements of these kernels. SpF builds on top of optimized vendor libraries for common numerical operations such as transforms, matrix solvers, etc., but can also be configured to use open source alternatives for portability. SpF includes several alternative schemes for global data redistribution and is expected to serve as an ideal testbed for further research into optimal approaches for different network architectures.In this presentation, we will describe the basic architecture of SpF as well as preliminary performance data and experience with adapting legacy dynamo codes. We will conclude with a discussion of planned extensions to SpF that will provide pseudospectral applications with additional flexibility with regard to time integration, linear solvers, and discretization in the radial direction.

Pseudospectral (PS)↗

Using SpF to Achieve Petascale for Legacy Pseudospectral Applications

Pseudospectral (PS) methods possess a number of characteristics (e.g., efficiency, accuracy, natural boundary conditions) that are extremely desirable for dynamo models. Unfortunately, dynamo models based upon PS methods face a number of daunting challenges, which include exposing additional parallelism, leveraging hardware accelerators, exploiting hybrid parallelism, and improving the scalability of global memory transposes. Although these issues are a concern for most models, solutions for PS methods tend to require far more pervasive changes to underlying data and control structures. Further, improvements in performance in one model are difficult to transfer to other models, resulting in significant duplication of effort across the research community. We have developed an extensible software framework for pseudospectral methods called SpF that is intended to enable extreme scalability and optimal performance. Highlevel abstractions provided by SpF unburden applications of the responsibility of managing domain decomposition and load balance while reducing the changes in code required to adapt to new computing architectures. The key design concept in SpF is that each phase of the numerical calculation is partitioned into disjoint numerical kernels that can be performed entirely inprocessor. The granularity of domain decomposition provided by SpF is only constrained by the datalocality requirements of these kernels. SpF builds on top of optimized vendor libraries for common numerical operations such as transforms, matrix solvers, etc., but can also be configured to use open source alternatives for portability. SpF includes several alternative schemes for global data redistribution and is expected to serve as an ideal testbed for further research into optimal approaches for different network architectures. In this presentation, we will describe our experience in porting legacy pseudospectral models, MoSST and DYNAMO, to use SpF as well as present preliminary performance results provided by the improved scalability.

DYNAMO↗

A pseudospectral Legendre method for hyperbolic equations with an improved stability condition

A new pseudospectral method is introduced for solving hyperbolic partial differential equations. This method uses different grid points than previously used pseudospectral methods: in fact the grid are related to the zeroes of the Legendre polynomials. The main advantage of this method is that the allowable time step is proportional to the inverse of the number of grid points 1/N rather than to 1/n(2) (as in the case of other pseudospectral methods applied to mixed initial boundary value problems). A highly accurate time discretization suitable for these spectral methods is discussed.

Tal-Ezer, H.↗

A pseudospectral Legendre method for hyperbolic equations with an improved stability condition

A new pseudospectral method is introduced for solving hyperbolic partial differential equations. This method uses different grid points than previously used pseudospectral methods: in fact the grid points are related to the zeroes of the Legendre polynomials. The main advantage of this method is that the allowable time step is proportional to the inverse of the number of grid points 1/N rather than to 1/n(2) (as in the case of other pseudospectral methods applied to mixed initial boundary value problems). A highly accurate time discretization suitable for these spectral methods is discussed.

Tal-Ezer, Hillel↗

Pseudospectral method for problems with nonperiodic boundary conditions

The capability of the pseudospectral technique and the time integration scheme for the time dependent nonperiodic problem is demonstrated. Significant improvements are shown in the accuracy for calculation of the two dimensional time dependent advection diffusion equation. The pseudospectral method is a viable alternative to the finite difference methods for the equation of the spatial deriviations.

Lee, H. N.↗

The stability of pseudospectral-Chebyshev methods

The pseudospectral-Chebyshev methods are shown to be convergent in variable coefficient problems and, in some cases, hyperbolic problems. The analysis demonstrates that the rate of convergence is greater for finite difference methods or the finite element method. For a single first-order hyperbolic equation, the method is seen as remaining stable even when the coefficient changes sign, although in this case it is specified that care must be taken to have adequate spatial resolution. It is noted that this fact, combined with the fact that collocation methods are easy to apply in the nonlinear case, shows that the pseudospectral method is in general preferable to the Galerkin or Tau methods.

Gottlieb, D.↗

Stability of pseudospectral and finite-difference methods for variable coefficient problems

It is shown that pseudospectral approximation to a special class of variable coefficient one-dimensional wave equations is stable and convergent even though the wave speed changes sign within the domain. Computer experiments indicate similar results are valid for more general problems. Similarly, computer results indicate that the leapfrog finite-difference scheme is stable even though the wave speed changes sign within the domain. However, both schemes can be asymptotically unstable in time when a fixed spatial mesh is used.

Gottlieb, D.↗

Pseudospectral solution of two-dimensional gas-dynamic problems

Chebyshev pseudospectral methods are used to compute two dimensional smooth compressible flows. Grid refinement tests show that spectral accuracy can be obtained. Filtering is not needed if resolution is sufficiently high and if boundary conditions are carefully prescribed.

Kopriva, D. A.↗

Modeling of unsteady small disturbance transonic flow using parametric differentiation, pseudospectral analysis and finite-differencing

A procedure for solving the nonlinear unsteady small disturbance transonic equation is formulated. This procedure is a synthesis of a pseudospectral method, parametric differentiation, and finite differencing. It is equally applicable to lifting and nonlifting airfoils. The procedure is particularly well suited to aeroelastic stability studies that may require a set of affine solutions.

Brown, M.↗

The eigenvalues of the pseudospectral Fourier approximation to the operator sin (2x) d/dx

It is shown that the eigenvalues Z sub i of the pseudospectral Fourier approximation to the operator sin(2x) curly d/curly dx satisfy (R sub e) (Z sub i) = + or - 1 or (R sub e)(Z sub I) = 0. Whereas this does not prove stability for the Fourier method, applied to the hyperbolic equation U sub t = sin (2x)(U sub x) - pi x pi; it indicates that the growth in time of the numerical solution is essentially the same as that of the solution to the differential equation.

Tal-Ezer, H.↗

Parallel pseudospectral domain decomposition techniques

The influence of interface boundary conditions on the ability to parallelize pseudospectral multidomain algorithms is investigated. Using the properties of spectral expansions, a novel parallel two domain procedure is generalized to an arbitrary number of domains each of which can be solved on a separate processor. This interface boundary condition considerably simplifies influence matrix techniques.

Gottlieb, David↗

A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations

A new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested. This method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions. Stability and convergence results are proven for the Chebyshev approximation of linear scalar hyperbolic equations. The eigenvalues of this method applied to parabolic equations are shown to be real and negative.

Funaro, D.↗

Convergence results for pseudospectral approximations of hyperbolic systems by a penalty type boundary treatment

A new method of imposing boundary conditions in the pseudospectral approximation of hyperbolic systems of equations is proposed. It is suggested to collocate the equations, not only at the inner grid points, but also at the boundary points and use the boundary conditions as penalty terms. In the pseudo-spectral Legrendre method with the new boundary treatment, a stability analysis for the case of a constant coefficient hyperbolic system is presented and error estimates are derived.

Funaro, Daniele↗

Modified Chebyshev pseudospectral method with O(N exp -1) time step restriction

The extreme eigenvalues of the Chebyshev pseudospectral differentiation operator are O(N exp 2) where N is the number of grid points. As a result of this, the allowable time step in an explicit time marching algorithm is O(N exp -2) which, in many cases, is much below the time step dictated by the physics of the partial differential equation. A new set of interpolating points is introduced such that the eigenvalues of the differentiation operator are O(N) and the allowable time step is O(N exp -1). The properties of the new algorithm are similar to those of the Fourier method. The new algorithm also provides a highly accurate solution for non-periodic boundary value problems.

Kosloff, Dan↗

Preconditioning matrices for the pseudospectral approximation of first-order operators

The behavior of the eigenvalues of preconditioning matrices for the pseudospectral approximation to the derivative operator has been analyzed in one and two dimensions. The one-dimensional analysis resulted in real and positive eigenvalues for the selected tridiagonal matrices. In the two-dimensional analysis, the eigenvalues of the selected block-diagonal matrices behaved well, but the preconditioner is full and therefore not suitable for applications. The Richardson scheme has been applied in the unpreconditioned as well as the preconditioned version to find the solution of the model problem.

Funaro, D.↗

Parallel pseudospectral domain decomposition techniques

The influence of interface boundary conditions on the ability to parallelize pseudospectral multidomain algorithms is investigated. Using the properties of spectral expansions, a novel parallel two domain procedure is generalized to an arbitrary number of domains each of which can be solved on a separate processor. This interface boundary condition considerably simplifies influence matrix techniques.

Gottlieb, David↗

A Fourier-Chebyshev pseudospectral method for solving steady 3-D Navier-Stokes and heat equations in cylindrical cavities

A Fourier-Chebyshev pseudospectral method for solving steady 3D Navier-Stokes equations in cylindrical cavities is presented and discussed. The general method is pseudo-unsteady and uses a semi-implicit finite difference scheme for the time discretization. The generalized ADI (Alternating Direction Implicit) procedure is then applied to reduce the problem to a successive solution of one-dimensional problems. The spatial approximation uses a Fourier-Galerkin approximation in the periodic direction and a Chebyshev-collocation approximation in the other directions. Difficulties related to the pressure are surmounted by using the artifical compressibility method. A suitable variable change was chosen to avoid the problem of singularity at the axis generated by cylindrical coordinates. The method is first tested on an advection-diffusion equation and then on the Navier-Stokes equations. Finally, the method is illustrated by a convection problem of a differentially heated fluid.

Pulicani, J. P.↗