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DOE OSTI2021

Multilevel Spectral Coarsening for Graph Laplacian Problems with Application to Reservoir Simulation

We extend previously developed two-level coarsening procedures for graph Laplacian problems written in a mixed saddle point form to the fully recursive multilevel case. The resulting hierarchy of discretizations gives rise to a hierarchy of upscaled models, in the sense that they provide approximation in the natural norms (in the mixed setting). This property enables us to utilize them in three applications: (i) as an accurate reduced model, (ii) as a tool in multilevel Monte Carlo simulations (in application to finite volume discretizations), and (iii) for providing a sequence of nonlinear operators in a full approximation scheme for solving nonlinear pressure equations discretized by the conservative two-point flux approximation. Finally, we illustrate the potential of the proposed multilevel technique in all three applications on a number of popular benchmark problems used in reservoir simulation.

multilevel Monte Carlo↗

DOE OSTI2021

Applying Time-Parallelization to Turbulent Flows

Parallelization of the temporal domain is explored for the solution of turbulent flows. Multigrid reduction-in-time (MGRIT) is used to advance the large-scale fluid dynamics in time sequentially on the coarsest space-time grid but propagate the information in time parallel on all other levels. The goal of this process is to accurately and efficiently resolve the coarse-scale turbulence structure and use that to drive the fine-scales of the turbulent flow. The extra forcing from nonlinear multigrid facilitates the coupling and interaction between fine and coarse scales, through which the multiscale nonlinear physics is properly captured. Adaptive mesh refinement is employed to finely resolve only the regions with strong gradients, which provides further computational efficiency. The underlying computational fluid dynamics solver is a fourth-order finite-volume scheme with the standard 4-stage Runge-Kutta method. An advanced approach is devised and implemented to enable MGRIT to solve highly turbulent flows successfully. Furthermore, the method is applied to solve a Taylor-Green vortex problem and a doubleshear-layer turbulent mixing flow. Results are promising, validating that MGRIT with the filtering approach has the potential to efficiently solve general turbulent flows.

Computational Fluid Dynamics↗

DOE OSTI2025

Numerical schemes for 3-wave kinetic equations: A complete treatment of the collision operator

In our previous work Walton and Tran (2023), numerical schemes for a simplified version of 3-wave kinetic equations, in which only the simple forward-cascade terms of the collision operators are kept, have been successfully designed, especially to capture the long time dynamics of the equation given the multiple blow-up time phenomenon. In this second work in the series, we propose numerical treatments for the complete 3-wave kinetic equations, in which the complete, much more complicated collision operators are fully considered based on a novel conservative form of the equation. Here we then derive an implicit finite volume scheme to solve the equation. The new discretization uses an adaptive time-stepping method which allows for the simulations to be carried to very long times. Our computed solutions are compared with previously derived long-time asymptotic estimates for the decay rate of total energy of time-dependent solutions of 3-wave kinetic equations and found to be in excellent agreement.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

A modern concept of Lagrangian hydrodynamics

Here, we offer a modern interpretation of Lagrangian hydrodynamics as employed in Lagrangian simulations of compressible fluid flow. Our main result is to show that artificial viscosity, traditionally viewed as a numerical artifice to control unphysical oscillations in flows with shocks, actually represents a physical process and is necessary to derive accurate simulations in any compressible flow. We begin by reviewing the origins of two numerical devices, artificial viscosity and finite-volume methods. We proceed to construct a mathematical (PDE) model that incorporates those numerics and in which a new length scale, the observer, arises representing the discretization. Associated with that length scale, there are new inviscid fluxes that are the artificial viscosity as first formulated by Richtmyer and an artificial heat flux postulated by Noh but typically not included in Lagrangian codes. We discuss the connection of our results to bivelocity hydrodynamics. We conclude with some speculation as to the direction of future developments in multidimensional Lagrangian codes as computers get faster and have larger memories.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Progress with the 5D full-F continuum gyrokinetic code COGENT

COGENT is an Eulerian gyrokinetic code being developed for edge plasma modelling. The code is distinguished by the use of a high-order finite-volume (conservative) discretization combined with mapped multi-block grid technology. Our recent work is focused on the development of a 5D full-F COGENT version.Anumerical algorithm utilizing locally a field-aligned multi-block coordinate system is implemented to facilitate simulations of highly anisotropic microturbulence in the presence of a strong magnetic shear. In this approach, the toroidal direction is divided into blocks such that,within each block, the cells are field-aligned and a non-matching (non-conformal) grid interface is allowed at the block boundaries. In this paper we report on details of the numerical implementation and present preliminary results of verification studies performed for the case of the ion temperature gradient (ITG) instability in a sheared toroidal annulus geometry.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2025

Adaptive Uncertainty Quantification for Stochastic Hyperbolic Conservation Laws

Here, we propose a predictor-corrector adaptive method for the study of hyperbolic partial differential equations (PDEs) under uncertainty. Constructed around the framework of stochastic finite volume (SFV) methods, our approach circumvents sampling schemes or simulation ensembles while also preserving fundamental properties, in particular hyperbolicity of the resulting systems and conservation of the discrete solutions. Furthermore, we augment the existing SFV theory with a priori convergence results for statistical quantities, in particular push-forward densities, which we demonstrate through numerical experiments. By linking refinement indicators to regions of the physical and stochastic spaces, we drive anisotropic refinements of the discretizations, introducing new degrees of freedom where deemed profitable. To illustrate our proposed method, we consider a series of numerical examples for nonlinear hyperbolic PDEs based on Burgers’ and Euler’s equations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Massively parallel axisymmetric fluid model for streamer discharges

A highly parallelizable fluid plasma simulation tool based upon the first-order drift-diffusion equations is discussed. Atmospheric pressure plasmas have densities and gradients that require small element sizes in order to accurately simulate the plasm resulting in computational meshes on the order of millions to tens of millions of elements for realistic size plasma reactors. To enable simulations of this nature, parallel computing is required and must be optimized for the particular problem. Here, a finite-volume, electrostatic drift-diffusion implementation for low-temperature plasma is discussed. The implementation is built upon the Message Passing Interface (MPI) library in C++ using Object Oriented Programming. The underlying numerical method is outlined in detail and benchmarked against simple streamer formation from other streamer codes. Electron densities, electric field, and propagation speeds are compared with the reference case and show good agreement. Convergence studies are also performed showing a minimal space step of approximately 4 μm required to reduce relative error to below 1% during early streamer simulation times and even finer space steps are required for longer times. Additionally, strong and weak scaling of the implementation are studied and demonstrate the excellent performance behavior of the implementation up to 100 million elements on 1024 processors. Lastly, different advection schemes are compared for the simple streamer problem to analyze the influence of numerical diffusion on the resulting quantities of interest.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2020

First implementation of gyrokinetic exact linearized Landau collision operator and comparison with models

Gyrokinetic simulations are fundamental to understanding and predicting turbulent transport in magnetically confined fusion plasmas. Previous simulations have used model collision operators with approximate field-particle terms of unknown accuracy and/or have neglected collisional finite Larmor radius (FLR) effects. We have implemented the linearized Fokker–Planck collision operator with exact field-particle terms and full FLR effects in a gyrokinetic code (GENE). The new operator, referred to as “exact” in this paper, allows the accuracy of model collision operators to be assessed. The conservative Landau form is implemented because its symmetry underlies the conservation laws and the H-theorem, and enables numerical methods to preserve this conservation, independent of resolution. The implementation utilizes the finite-volume method recently employed to discretize the Sugama collision model in GENE, allowing direct comparison between the two operators. Results show that the Sugama model appears accurate for the growth rates of trapped electron modes (TEMs) driven only by density gradients, but appreciably underestimates the growth rates as the collisionality and electron temperature gradient increase. The TEM turbulent fluxes near the nonlinear threshold using the exact operator are similar to the Sugama model for the n e = d ln T e /d ln n e = 0 case, but substantially larger than the Sugama model for the n e = 1 case. The FLR effects reduce the growth rates increasingly with wavenumber deepening a “valley” at the intermediate binormal wavenumber as the unstable mode extends from the TEM regime to the electron temperature gradient instability regime. Application to the Hinton–Rosenbluth problem shows that zonal flows decay faster as the radial wavenumber increases and the exact operator yields weaker decay rates.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1994

Parallel implicit unstructured grid Euler solvers

A mesh-vertex finite volume scheme for solving the Euler equations on triangular unstructured meshes is implemented on a multiple-instruction/multiple-data stream parallel computer. An explicit four-stage Runge-Kutta scheme is used to solve two-dimensional flow problems. A family of implicit schemes is also developed to solve these problems, where the linear system that arises at each time step is solved by a preconditioned GMRES algorithm. Two partitioning strategies are employed: one that partitions triangles and the other that partitions vertices. The choice of the preconditioner in a distributed memory setting is discussed. All of the methods are compared both in terms of elapsed times and convergence rates. It is shown that the implicit schemes offer adequate parallelism at the expense of minimal sequential overhead. The use of a global coarse grid to further minimize this overhead is also investigated. The schemes are implemented on a distributed memory parallel computer, the Intel iPSC/860.

TRT-THEORETICAL↗

DOE OSTI2025

A constrained-transport embedded boundary method for compressible resistive magnetohydrodynamics

Motivated by the increased interest in pulsed-power magneto-inertial fusion devices in recent years, we present a method for implementing an arbitrarily shaped embedded boundary on a Cartesian mesh while solving the equations of compressible resistive magnetohydrodynamics. The method is built around a finite volume formulation of the equations in which a Riemann solver is used to compute fluxes on the faces between grid cells, and a face-centered constrained transport formulation of the induction equation. The small time step problem associated with the cut cells is avoided by always computing fluxes on the faces and edges of the Cartesian mesh. We extend the method to model a moving interface between two materials with different properties using a ghost-fluid approach, and show some preliminary results including shock-wave-driven and magnetically-driven dynamical compressions of magnetohydrostatic equilibria. In conclusion, we present a thorough verification of the method and show that it converges at second order in the absence of discontinuities, and at first order with a discontinuity in material properties.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2020

Discretization Writeup for Grey Flux-Limited Radiation Diffusion

This report documents the time and space discretizations for grey flux-limited diffusion applied to the thermal radiative transfer (TRT) equations. We begin with a description of the physics being solved before moving into the diffusion approximation. Once we have the TRT system, we show a finite-volume-inspired discretization from Jim Morel (Texas A&M University, NUEN 627 class notes, lecture 8). As systems become hotter they emit more photons in the form of blackbody radiation. Because average photon energy of the blackbody source is proportional to the temperature of the system, we call these thermal photons or thermal radiation. As material temperatures increase, increasing fractions of the total energy in the system go into the radiation field. In addition, radiation can deposit energy and momentum non-locally, making it an important phenomenon for heating and impulse. In order to accurately study systems at high temperatures, we wish to add the physics of thermal radiation to our hydrodynamic system. In practice, coupling radiation and hydrodynamics is often done by operator-splitting each timestep into two consecutive, non-overlapping phases: (1) update the hydrodynamics for a fixed radiation state (2) update the radiation and internal energy for an otherwise fixed hydrodynamic state. Because of this clean separation of physics updates, in this report we show only the latter phase, which involves solely the TRT equations.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2022

Modelling of electrostatic ion-scale turbulence in divertor tokamaks with the gyrokinetic code $\mathrm{COGENT}$

Here, continuum gyrokinetic simulations of electrostatic ion scale turbulence are presented for the case of a diverted (single-null) tokamak geometry. The simulation model, implemented in the finite-volume code COGENT, solves the long-wavelength limit of the full-F gyrokinetic equation for ion species coupled to a vorticity equation for electrostatic potential variations, where a fluid model is used for an electron response. The model describes the ion scale ion temperature gradient (ITG) and resistive drift modes as well as neoclassical ion physics effects. Different turbulence regimes are observed depending on the plasma profiles, and the roles of a self-consistent background electric field and an X-point geometry are explored. In particular, increasing the pedestal density gradient and the corresponding radial electric field is demonstrated to suppress the ITG turbulence, whereas the same edge plasma background can still be destabilized by the resistive modes when the plasma resistivity is increased. The effects of X-point geometry are assessed by comparing cross-separatrix simulations with counterpart calculations performed for a toroidal annulus geometry. For the simulation parameters considered, similar global behaviour is observed in both cases, whereas strong local suppression of turbulence fluctuations is demonstrated near the X-point for the case of a single-null geometry.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2024

A new re-redistribution scheme for weighted state redistribution with adaptive mesh refinement

State redistribution (SRD) is a recently developed technique for stabilizing cut cells that result from finite-volume embedded boundary methods. SRD has been successfully applied to a variety of compressible and incompressible flow problems. When used in conjunction with adaptive mesh refinement (AMR), additional steps are needed to preserve the accuracy and conservation properties of the solution if the embedded boundary is not restricted to a single level of the mesh hierarchy. In this work, we extend the weighted state redistribution algorithm to cases where cut cells live at or near a coarse-fine interface within the domain. Here, we present numerical results that demonstrate that the algorithm is conservative when the coarse-fine interface intersects the embedded boundary. Additionally we compare the numerical solution of the Sod shock tube problem in an inclined cylinder with the analytic solution, and we compare the simulation of a shock hitting a cylindrical obstacle with experimental data. Finally we demonstrate the methodology for simulation of the multicomponent compressible Navier-Stokes equations in a piston-bowl geometry, and discuss the computational efficiency gained by not requiring the entire embedded boundary to be defined at the finest level.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Accelerating high-order continuum kinetic plasma simulations using multiple GPUs

Kinetic plasma simulations solve the Vlasov-Poisson or Vlasov-Maxwell equations to evolve scalar-variable distribution functions in position-velocity phase space and vector-variable electromagnetic fields in configuration space. The immense computational cost of evolving high-dimensional variables, and their large number of degrees of freedom, often limits the utility of continuum kinetic simulations and presents a challenge when it comes to accurately simulating real-world physical phenomena. To address this challenge, we present techniques that accelerate and minimize the computational work required for a scalable Vlasov-Poisson solver. We show theoretical hardware compute and communication bounds for solving a fourth-order finite-volume Vlasov-Poisson system. These bounds are then used to inform and evaluate the design of performance portable algorithms for a multiple graphics processing unit (GPU) accelerated version of the Vlasov-Poisson solver VCK-CPU [1]. We demonstrate that the multi-GPU Vlasov solver implementation, VCK-GPU, simultaneously minimizes required inter-process data transfer while also being bounded by the machine network performance limits. This results in an overall strong scaling speedup per timestep of up to 40x in three-dimensional phase space (one position, two velocity coordinates) and 54x in four dimensional phase space (two position, two velocity coordinates) and a 341x increase in simulation throughput of the GPU accelerated code over the existing CPU code. The GPU code is also able to weak scale up to 256 compute nodes and 1024 GPUs. In conclusion, we demonstrate that the improved compute performance enables exploring configurations which were previously computationally infeasible, including resolving fine-scale distribution function filamentation and multi-species dynamics with realistic electron-proton mass ratios.

Continuum kinetics↗

NASA NTRSDate not supplied

Assessment of Edge-Based Viscous Method for Corner-Flow Solutions on Graphics Processing Units

A highly efficient, edge-based viscous (EBV) discretization method has been recently implemented in a practical, unstructured-grid, node-centered, finite-volume flow solver and evaluated for Reynolds-averaged Navier-Stokes (RANS) formulations. In comparison to a well-established cell-based viscous (CBV) method, the EBV method has demonstrated multifold acceleration of all viscous-kernel computations on general unstructured mixed-element grids. The viscous kernels include evaluation of viscous fluxes, diffusion terms in turbulence models, and the corresponding Jacobian terms. In this paper, an EBV implementation of a nonlinear extension of the Spalart-Allmaras turbulence model, SA-neg-QCR2000, is presented and verified. The SA-neg-QCR2000 model is used for simulating turbulent corner flows. Previously reported EBV computations have been conducted on traditional computing architectures based on central processing units (CPU). This paper assesses benefits of the EBV method on modern high-performance computing architectures based on graphics processing units (GPU). The GPU implementations of the CBV and EBV methods are verified by comparing solutions and iterative convergence with those observed in CPU computations on the same grids. A comprehensive assessment of the EBV speedup on CPU and GPU architectures is presented for established benchmark corner flows, namely, a supersonic flow through a long square duct and a subsonic flow around a NASA juncture flow model.

CFD↗

NASA NTRSDate not supplied

Assessment of Edge-Based Viscous Method for Corner-Flow Solutions on Graphics Processing Units

CFD↗

DOE OSTI2021

SOMAFOAM: An OpenFOAM based solver for continuum simulations of low-temperature plasmas

Here, we report the development of SOMAFOAM, a finite volume framework for performing continuum simulations of low-temperature plasmas. The primary goal of this work is to discuss the features of SOMAFOAM along with representative results provided as examples for a range of operating conditions and geometries. This includes plasma and plasma–dielectric systems operating in direct current, radio frequency, and microwave regimes from pressures as low as 100 mTorr to atmospheric pressure. The code has several useful features including the ability to run massively parallel simulations using arbitrary geometries, structured/unstructured meshes, choice of models such as drift–diffusion/full-momentum at runtime, and species-dependent timesteps to name a few. The verification/validation studies presented include comparison with previously published continuum simulations (low-pressure direct current and radio frequency plasma), with experiments (Gaseous Electronics Conference Reference Cell and microwave microplasma ignited in a split ring resonator), and previously published kinetic simulations (low-pressure radio frequency plasma). Other examples provided include a direct current atmospheric pressure microplasma bounded by dielectric sidewalls and a helium–nitrogen plasma ignited using a needle electrode facing a dielectric. The performance of the code is also discussed with serial and distributed memory parallel runs demonstrated up to 512 cores. The design and implementation of the code in a modular object-oriented framework allows for easy extension and seamless coupling with other codes and can be expected to play an important role in both academia and industry.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

Investigating Inherent Numerical Stabilization for the Moist, Compressible, Non‐Hydrostatic Euler Equations on Collocated Grids

This study investigates inherent numerical dissipation due to upwind fluxes and reconstruction strategies for collocated Finite-Volume integration of the Euler equations. Idealized supercell simulations are used without any explicit dissipation. Flux terms are split into: mass flux, pressure, and advected quantities. They are computed with the following upwind strategies: central, advectively upwind, and acoustically upwind. This is performed for third and ninth-order-accurate reconstructions with and without Weighted Essentially Non-Oscillatory limiting. Acoustic-only upwinding for pressure and mass flux terms and advective-only upwinding for advected quantities is the most flexible simplification found. It reduces data movement and computations. Assuming a constant speed of sound in acoustic upwinding gives similar results to using the true speed of sound. Dissipation from upwind adapts automatically to grid spacing, time step, reconstruction accuracy, and flow smoothness. While stability is maintained even at 21st-order spatial accuracy, there is a limit to the spatial order of accuracy for which upwinding alone can create a realizable solution in the conditions of this study. Convex combinations of upwind and central solutions for flux terms also reduced dissipation, but as the central proportion grows, solutions become physically unrealizable. The range of length scales of the kinetic energy spectra can be extended along k −5/3 to smaller spatial scales by reducing dissipation either with higher-order reconstructions or using convex combinations of upwind and central fluxes. However, not all extensions of the length scale range along k −5/3 exhibit physically realizable solutions, even though the spectra appear to be physical.

54 ENVIRONMENTAL SCIENCES↗