An official website of the United States government
Abstract Numerical modeling of radiative transfer in nongray reacting media is a challenging problem in computational science and engineering. The choice of radiation models is important for accurate and efficient high-fidelity combustion simulations. Different applications usually involve different degrees of complexity, so there is yet no consensus in the community. In this paper, the performance of different radiative transfer equation (RTE) solvers and spectral models for a turbulent piloted methane/air jet flame are studied. The flame is scaled from the Sandia Flame D with a Reynolds number of 22,400. Three classes of RTE solvers, namely the discrete ordinates method, spherical harmonics method, and Monte Carlo method, are examined. The spectral models include the Planck-mean model, the full-spectrum k-distribution (FSK) method, and the line-by-line (LBL) calculation. The performances of different radiation models in terms of accuracy and computational cost are benchmarked. The results have shown that both RTE solvers and spectral models are critical in the prediction of radiative heat source terms for this jet flame. The trade-offs between the accuracy, the computational cost, and the implementation difficulty are discussed in detail. The results can be used as a reference for radiation model selection in combustor simulations.
more »
« less
Fast Poisson solvers for spectral methods
Abstract Poisson’s equation is the canonical elliptic partial differential equation. While there exist fast Poisson solvers for finite difference (FD) and finite element methods, fast Poisson solvers for spectral methods have remained elusive. Here we derive spectral methods for solving Poisson’s equation on a square, cylinder, solid sphere and cube that have optimal complexity (up to polylogarithmic terms) in terms of the degrees of freedom used to represent the solution. Whereas FFT-based fast Poisson solvers exploit structured eigenvectors of FD matrices, our solver exploits a separated spectra property that holds for our carefully designed spectral discretizations. Without parallelization we can solve Poisson’s equation on a square with 100 million degrees of freedom in under 2 min on a standard laptop.
more »
« less
- Award ID(s):
- 1818757
- PAR ID:
- 10125836
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- IMA Journal of Numerical Analysis
- ISSN:
- 0272-4979
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
The Whitney forms on a simplex T admit high-order generalizations that have received a great deal of attention in numerical analysis. Less well-known are the shadow forms of Brasselet, Goresky, and MacPherson. These forms generalize the Whitney forms, but have rational coefficients, allowing singularities near the faces of T. Motivated by numerical problems that exhibit these kinds of singularities, we introduce degrees of freedom for the shadow k-forms that are well-suited for finite element implementations. In particular, we show that the degrees of freedom for the shadow forms are given by integration over the k-dimensional faces of the blow-up T̃ of the simplex T. Consequently, we obtain an isomorphism between the cohomology of the complex of shadow forms and the cellular cohomology of T̃, which vanishes except in degree zero. Additionally, we discover a surprising probabilistic interpretation of shadow forms in terms of Poisson processes. This perspective simplifies several proofs and gives a way of computing bases for the shadow forms using a straightforward combinatorial calculation.more » « less
-
We consider large-scale implicit solvers for the numerical solution of partial differential equations (PDEs). The solvers require the high-bandwith networks of an HPC system for a fast time to solution. The increasing variability in performance of the HPC systems, most likely caused by variable communication latencies and network congestion, however, makes the execution time of solver algorithms unpredictable and hard to measure. In particular, the performance variability of the underlying system makes the reliable comparison of different algorithms and implementations difficult or impossible on HPC. We propose the use of statistical methods relying on hidden Markov models (HMM) to separate variable performance data into regimes corresponding to different levels of system latency. This allows us to, for ex- ample, identify and remove time periods when extremely high system latencies throttle application performance and distort performance measurements. We apply HMM to the careful analysis of implicit conjugate gradient solvers for finite-element discretized PDE, in particular comparing several new communication hiding methods for matrix-free operators of a PDE, which are critical for achieving peak performance in state-of-the-art PDE solvers. The HMM analysis allows us to overcome the strong performance variability in the HPC system. Our performance results for a model PDE problem discretized with 135 million degrees of freedom parallelized over 7168 cores of the Anvil supercomputer demonstrate that the communication hiding techniques can achieve up to a 10% speedup for the matrix-free matrix-vector product.more » « less
-
Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity p o l y ( 1 / ϵ ) , where ϵ is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be p o l y ( d , log ( 1 / ϵ ) ) , where d is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations.more » « less
-
Abstract The Poisson–Boltzmann (PB) model is a widely used electrostatic model for biomolecular solvation analysis. Formulated as an elliptic interface problem, the PB model can be numerically solved on either Eulerian meshes using finite difference/finite element methods or Lagrangian meshes using boundary element methods. Molecular surface generators, which produce the discretized dielectric interfaces between solutes and solvents, are critical factors in determining the accuracy and efficiency of the PB solvers. In this work, we investigate the utility of the Eulerian Solvent Excluded Surface (ESES) software for rendering conjugated Eulerian and Lagrangian surface representations, which enables us to numerically validate and compare the quality of Eulerian PB solvers, such as the MIBPB solver, and the Lagrangian PB solvers, such as the TABI‐PB solver. Furthermore, with the ESES software and its associated PB solvers, we are able to numerically validate an interesting and useful but often neglected source‐target symmetric property associated with the linearized PB model.more » « less
