A method to parallelize tridiagonal solvers

Müller, Silvia M.; Scheerer, Dieter

doi:10.1016/s0167-8191(05)80104-8

Cited by 43 publications

(14 citation statements)

References 2 publications

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“…These equations were numerically solved using a fourth-order explicit Runge-Kutta time integration in conjunction with a sixth-order compact scheme with eighth-order filter for spatial derivatives (Kennedy & Carpenter 1994); the filtering (applied at interior points only) is required to maintain numerical stability for long-time integrations but since it acts only on the shortest waves that can be resolved on the grid, it does not act as a turbulence model allowing underresolved computations. A parallelization using three-dimensional domain decomposition and message passing, and an efficient parallel tridiagonal solver (Muller & Scheerer 1991) were used.…”

Section: Description Of the Dns Databasementioning

confidence: 99%

Subgrid-scale models and large-eddy simulation of oxygen stream disintegration and mixing with a hydrogen or helium stream at supercritical pressure

Taskinoglu

Bellan

2011

J. Fluid Mech.

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For flows at supercritical pressure, p, the large-eddy simulation (LES) equations consist of the differential conservation equations coupled with a real-gas equation of state, and the equations utilize transport properties depending on the thermodynamic variables. Compared to previous LES models, the differential equations contain not only the subgrid-scale (SGS) fluxes but also new SGS terms, each denoted as a 'correction'. These additional terms, typically assumed null for atmospheric pressure flows, stem from filtering the differential governing equations and represent differences, other than contributed by the convection terms, between a filtered term and the same term computed as a function of the filtered flow field. In particular, the energy equation contains a heat-flux correction (q-correction) which is the difference between the filtered divergence of the molecular heat flux and the divergence of the molecular heat flux computed as a function of the filtered flow field. We revisit here a previous a priori study where we only had partial success in modelling the q-correction term and show that success can be achieved using a different modelling approach. This a priori analysis, based on a temporal mixing-layer direct numerical simulation database, shows that the focus in modelling the q-correction should be on reconstructing the primitive variable gradients rather than their coefficients, and proposes the approximate deconvolution model (ADM) as an effective means of flow field reconstruction for LES molecular heat-flux calculation. Furthermore, an a posteriori study is conducted for temporal mixing layers initially containing oxygen (O) in the lower stream and hydrogen (H) or helium (He) in the upper stream to examine the benefit of the new model. Results show that for any LES including SGS-flux models (constant-coefficient gradient or scale-similarity models; dynamic-coefficient Smagorinsky/Yoshizawa or mixed Smagorinsky/Yoshizawa/gradient models), the inclusion of the q-correction in LES leads to the theoretical maximum reduction of the SGS molecular heat-flux difference; the remaining error in modelling this new subgrid term is thus irreducible. The impact of the q-correction model first on the molecular heat flux and then on the mean, fluctuations, second-order correlations and spatial distribution of dependent variables is also demonstrated. Discussions on the utilization of the models in general LES are presented.

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Section: Description Of the Dns Databasementioning

confidence: 99%

Subgrid-scale models and large-eddy simulation of oxygen stream disintegration and mixing with a hydrogen or helium stream at supercritical pressure

Taskinoglu

Bellan

2011

J. Fluid Mech.

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“…23 The computations were parallelized using three-dimensional domain decomposition and message passing interface; the tridiagonal solver for the spatial discretization scheme was efficiently parallelized. 24 p and T were computed from the EOS and the known functional dependence of e on (T, p) using an iterative scheme. The grid spacing, ∆x, was uniform and was selected to ensure that the smallest scales relevant to dissipation were resolved.…”

Section: Description Of the Dns Databasementioning

confidence: 99%

A priori and a posteriori investigations for developing large eddy simulations of multi-species turbulent mixing under high-pressure conditions

Borghesi

Bellan

2015

Physics of Fluids

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A Direct Numerical Simulation (DNS) database was created representing mixing of species under high-pressure conditions. The configuration considered is that of a temporally evolving mixing layer. The database was examined and analyzed for the purpose of modeling some of the unclosed terms that appear in the Large Eddy Simulation (LES) equations. Several metrics are used to understand the LES modeling requirements. First, a statistical analysis of the DNS-database large-scale flow structures was performed to provide a metric for probing the accuracy of the proposed LES models as the flow fields obtained from accurate LESs should contain structures of morphology statistically similar to those observed in the filtered-and-coarsened DNS (FC-DNS) fields. To characterize the morphology of the large-scales structures, the Minkowski functionals of the iso-surfaces were evaluated for two different fields: the second-invariant of the rate of deformation tensor and the irreversible entropy production rate. To remove the presence of the small flow scales, both of these fields were computed using the FC-DNS solutions. It was found that the large-scale structures of the irreversible entropy production rate exhibit higher morphological complexity than those of the second invariant of the rate of deformation tensor, indicating that the burden of modeling will be on recovering the thermodynamic fields. Second, to evaluate the physical effects which must be modeled at the subfilter scale, an a priori analysis was conducted. This a priori analysis, conducted in the coarse-grid LES regime, revealed that standard closures for the filtered pressure, the filtered heat flux, and the filtered species mass fluxes, in which a filtered function of a variable is equal to the function of the filtered variable, may no longer be valid for the high-pressure flows considered in this study. The terms requiring modeling are the filtered pressure, the filtered heat flux, the filtered pressure work, and the filtered species mass fluxes. Improved models were developed based on a scale-similarity approach and were found to perform considerably better than the classical ones. These improved models were also assessed in an a posteriori study. Different combinations of the standard models and the improved ones were tested. At the relatively small Reynolds numbers achievable in DNS and at the relatively small filter widths used here, the standard models for the filtered pressure, the filtered heat flux, and the filtered species fluxes were found to yield accurate results for the morphology of the large-scale structures present in the flow. Analysis of the temporal evolution of several volume-averaged quantities representative of the mixing layer growth, and of the cross-stream variation of homogeneous-plane averages and second-order correlations, as well as of visualizations, indicated that the models performed equivalently for the conditions of the simulations. The expectation is that at the much larger Reynolds numbers and much larger filter widths used in practical applications, the improved models will have much more accurate performance than the standard one.

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“…Notable among these are cyclic reduction [6,16], recursive doubling [27], and partition methods [22,32]. These algorithms have typically targeted vector supercomputers such as the Illiac IV, CDC STAR-100 and Cray-1.…”

Section: Introductionmentioning

confidence: 99%

Fast tridiagonal solvers on the GPU

2010

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We study the performance of three parallel algorithms and their hybrid variants for solving tridiagonal linear systems on a GPU: cyclic reduction (CR), parallel cyclic reduction (PCR) and recursive doubling (RD). We develop an approach to measure, analyze, and optimize the performance of GPU programs in terms of memory access, computation, and control overhead. We find that CR enjoys linear algorithm complexity but suffers from more algorithmic steps and bank conflicts, while PCR and RD have fewer algorithmic steps but do more work each step. To combine the benefits of the basic algorithms, we propose hybrid CR+PCR and CR+RD algorithms, which improve the performance of PCR, RD and CR by 21%, 31% and 61% respectively. Our GPU solvers achieve up to a 28x speedup over a sequential LAPACK solver, and a 12x speedup over a multi-threaded CPU solver.

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A method to parallelize tridiagonal solvers

Cited by 43 publications

References 2 publications

Subgrid-scale models and large-eddy simulation of oxygen stream disintegration and mixing with a hydrogen or helium stream at supercritical pressure

Subgrid-scale models and large-eddy simulation of oxygen stream disintegration and mixing with a hydrogen or helium stream at supercritical pressure

A priori and a posteriori investigations for developing large eddy simulations of multi-species turbulent mixing under high-pressure conditions

Fast tridiagonal solvers on the GPU

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