Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “NUMERICAL FLUX FUNCTION”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

Some notes on shock resolving flux functions. Part 1: Stationary characteristics

Numerical flux functions for solving the Euler equations using exact and/or approximate solutions of the Riemann problem of gasdynamics are discussed. Under certain restrictive conditions, schemes using these flux functions produce systems of equations which can exhibit a single degree of freedom. In some instances, the solutions represented by this degree of freedom are unstable to perturbations. This local instability can seriously degrade the temporal convergence of numerical schemes. This point is demonstrated by numerical example.

Barth, Timothy J.↗
NASA NTRS1981

On the relation between upwind-differencing schemes of Godunov, Engquist-Osher and Roe

The upwind differencing first order schemes of Godunov, Engquist-Osher and Roe are discussed on the basis of the inviscid Burgers equations. The differences between the schemes are interpreted as differences between the approximate Riemann solutions on which their numerical flux functions are based. Special attention is given to the proper formulation of these schemes when a source term is present. Second order two step schemes, based on the numerical flux functions of the first order schemes are also described. The schemes are compared in a numerical experiment, and recommendations on their use are included.

Vanleer, B.↗
DOE OSTI2021

Well-balanced treatment of gravity in astrophysical fluid dynamics simulations at low Mach numbers

We report accurate simulations of flows in stellar interiors are crucial to improving our understanding of stellar structure and evolution. Because the typically slow flows are merely tiny perturbations on top of a close balance between gravity and the pressure gradient, such simulations place heavy demands on numerical hydrodynamics schemes. We demonstrate how discretization errors on grids of reasonable size can lead to spurious flows orders of magnitude faster than the physical flow. Well-balanced numerical schemes can deal with this problem. Three such schemes were applied in the implicit, finite-volume SEVEN-LEAGUE HYDRO code in combination with a low-Mach-number numerical flux function. We compare how the schemes perform in four numerical experiments addressing some of the challenges imposed by typical problems in stellar hydrodynamics. We find that the α-β and deviation well-balancing methods can accurately maintain hydrostatic solutions provided that gravitational potential energy is included in the total energy balance. They accurately conserve minuscule entropy fluctuations advected in an isentropic stratification, which enables the methods to reproduce the expected scaling of convective flow speed with the heating rate. The deviation method also substantially increases accuracy of maintaining stationary orbital motions in a Keplerian disk on long timescales. The Cargo–LeRoux method fares substantially worse in our tests, although its simplicity may still offer some merits in certain situations. Overall, we find the well-balanced treatment of gravity in combination with low Mach number flux functions essential to reproducing correct physical solutions to challenging stellar slow-flow problems on affordable collocated grids.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1993

Grid-independent upwind scheme for multidimensional flow

Recent advances in the development of a grid-independent finite volume scheme for the Euler equations of gas dynamics are described. In the proposed method, flowfield gradient data are reconstructed locally (on a triangle) using five elementary planar waves, and an upwind numerical flux function for grid-oblique waves is developed to model the effect of the passage of these waves on the data in a cell. Numerical examples for several two-dimensional test problems are included. These results show high wave resolution and nearly monotone strong-wave transitions.

Parpia, Ijaz H.↗
DOE OSTI2019

Analysis and Entropy Stability of the Line-Based Discontinuous Galerkin Method

We develop a discretely entropy-stable line-based discontinuous Galerkin method for hyperbolic conservation laws based on a flux differencing technique. By using standard entropy-stable and entropy-conservative numerical flux functions, this method guarantees that the discrete integral of the entropy is non-increasing. This nonlinear entropy stability property is important for the robustness of the method, in particular when applied to problems with discontinuous solutions or when the mesh is under-resolved. This line-based method is significantly less computationally expensive than a standard DG method. Numerical results are shown demonstrating the effectiveness of the method on a variety of test cases, including Burgers’ equation and the Euler equations, in one, two, and three spatial dimensions.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1983

Numerical approximations to nonlinear conservation laws with locally varying time and space grids

Numerical approximations to the initial value problem for nonlinear systems of conservation laws are considered. The considered system is said to be hyperbolic when all eigenvalues of every real linear combination of the Jacobian matrices are real. Solutions may develop discontinuities in finite time, even when the initial data are smooth. In the investigation, explicit finite difference methods which use locally varying time grids are considered. The global CFL restriction is replaced by a local restriction. The numerical flux function is studied from a finite volume viewpoint, and a differencing technique is developed at interface points between regions of distinct time increments.

Osher, S.↗
NASA NTRS1985

On the implementation of a class of upwind schemes for system of hyperbolic conservation laws

The relative computational effort among the spatially five point numerical flux functions of Harten, van Leer, and Osher and Chakravarthy is explored. These three methods typify the design principles most often used in constructing higher than first order upwind total variation diminishing (TVD) schemes. For the scalar case the difference in operation count between any two algorithms may be very small and yet the operation count for their system counterparts might be vastly different. The situation occurs even though one starts with two different yet equivalent representations for the scalar case.

Yee, H. C.↗
NASA NTRS1989

Sonic-point capturing

A prototype scheme that produces perfectly smooth transonic solutions to nozzle-flow problems is derived and tested. The basic upwind scheme is described as well as satisfying the entropy condition, treatment of the source term, and numerical verification. The analysis yielded a numerical flux function for use near a sonic point, which is based on a full model of a transonic expansion wave, and a matched treatment for the source term.

Van Leer, Bram↗
NASA NTRS1991

Numerical flux formulas for the Euler and Navier-Stokes equations. 2: Progress in flux-vector splitting

The accuracy of various numerical flux functions for the inviscid fluxes when used for Navier-Stokes computations is studied. The flux functions are benchmarked for solutions of the viscous, hypersonic flow past a 10 degree cone at zero angle of attack using first order, upwind spatial differencing. The Harten-Lax/Roe flux is found to give a good boundary layer representation, although its robustness is an issue. Some hybrid flux formulas, where the concepts of flux-vector and flux-difference splitting are combined, are shown to give unsatisfactory pressure distributions; there is still room for improvement. Investigations of low diffusion, pure flux-vector splittings indicate that a pure flux-vector splitting can be developed that eliminates spurious diffusion across the boundary layer. The resulting first-order scheme is marginally stable and not monotone.

Coirier, William J.↗
NASA NTRS1991

Numerical flux formulas for the Euler and Navier-Stokes equations. II - Progress in flux-vector splitting

The accuracy of various numerical flux functions for the inviscid fluxes when used for Navier-Stokes computations is studied. The flux functions are benchmarked for solutions of the viscous, hypersonic flow past a 10 degree cone at zero angle of attack using first order, upwind spatial differencing. The Harten-Lax/Roe flux is found to give a good boundary layer representation, although its robustness is an issue. Some hybrid flux formulas, where the concepts of flux-vector and flux-difference splitting are combined, are shown to give unsatisfactory pressure distributions; there is still room for improvement. Investigations of low diffusion, pure flux-vector splittings indicate that a pure flux-vector splitting can be developed that eliminates spurious diffusion across the boundary layer. The resulting first-order scheme is marginally stable and not monotone.

Coirier, William J.↗
NASA NTRS2001

Energy Stable Flux Formulas For The Discontinuous Galerkin Discretization Of First Order Nonlinear Conservation Laws

We consider the discontinuous Galerkin (DG) finite element discretization of first order systems of conservation laws derivable as moments of the kinetic Boltzmann equation. This includes well known conservation law systems such as the Euler For the class of first order nonlinear conservation laws equipped with an entropy extension, an energy analysis of the DG method for the Cauchy initial value problem is developed. Using this DG energy analysis, several new variants of existing numerical flux functions are derived and shown to be energy stable.

Barth, Timothy↗
NASA NTRS2015

Development of the US3D Code for Advanced Compressible and Reacting Flow Simulations

Aerothermodynamics and hypersonic flows involve complex multi-disciplinary physics, including finite-rate gas-phase kinetics, finite-rate internal energy relaxation, gas-surface interactions with finite-rate oxidation and sublimation, transition to turbulence, large-scale unsteadiness, shock-boundary layer interactions, fluid-structure interactions, and thermal protection system ablation and thermal response. Many of the flows have a large range of length and time scales, requiring large computational grids, implicit time integration, and large solution run times. The University of Minnesota NASA US3D code was designed for the simulation of these complex, highly-coupled flows. It has many of the features of the well-established DPLR code, but uses unstructured grids and has many advanced numerical capabilities and physical models for multi-physics problems. The main capabilities of the code are described, the physical modeling approaches are discussed, the different types of numerical flux functions and time integration approaches are outlined, and the parallelization strategy is overviewed. Comparisons between US3D and the NASA DPLR code are presented, and several advanced simulations are presented to illustrate some of novel features of the code.

CFD↗
DOE OSTI2017

An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly curved, quadrilateral meshes. The scheme is derived from an equivalent flux differencing formulation of the split form of the equations. We prove that this discretization exactly preserves the local mass and momentum. Furthermore, combined with a special numerical interface flux function, the method exactly preserves the mathematical entropy, which is the total energy for the shallow water equations. By adding a specific form of interface dissipation to the baseline entropy conserving scheme we create a provably entropy stable scheme. That is, the numerical scheme discretely satisfies the second law of thermodynamics. Finally, with a particular discretization of the bathymetry source term we prove that the numerical approximation is well-balanced. We provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
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↗
NASA NTRS1986

Thermal balance of the atmospheres of Jupiter and Uranus

Two-dimensional, radiative-convective-dynamical models of the visible atmospheres of Jupiter and Uranus are presented. Zonally-averaged temperatures and heat fluxes are calculated numerically as functions of pressure and latitude. In addition to radiative heat fluxes, the dynamical heat flux due to large-scale baroclinic eddies is included and is parametrized using a mixing length theory which gives heat fluxes similar to those of Stone. The results for Jupiter indicate that the internal heat flow is non-uniform in latitude and nearly balances the net radiative flux leaving the atmosphere. The thermal emission is found to be uniform in latitude in agreement with Pioneer and Voyager observations. Baroclinic eddies are calculated to transport only a small amount of the meridional heat flow necessary to account for the uniformity of thermal emission with latitude. The bulk of the meridional heat transfer is found to occur very deep in the stable interior of Jupiter as originally proposed by Ingersoll and Porco. The relative importance of baroclinic eddies vs. internal heat flow in the thermal balance of Uranus depends on the ratio of emitted thermal power to absorbed solar power. The thermal balance of Uranus is compared to that of Jupiter for different values of this ratio.

Friedson, A. J.↗
DOE OSTI2022

Agglomeration-based geometric multigrid solvers for compact discontinuous Galerkin discretizations on unstructured meshes

Here, we present a geometric multigrid solver for the Compact Discontinuous Galerkin method through building a hierarchy of coarser meshes using a simple agglomeration method which handles arbitrary element shapes and dimensions. The method is easily extendable to other discontinuous Galerkin discretizations, including the Local DG method and the Interior Penalty method. We demonstrate excellent solver performance for Poisson's equation, provided a flux formulation is used for the operator coarsening and a suitable switch function chosen for the numerical fluxes.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1984

A shock-capturing finite element method

Finite element methods are developed for a one-dimensional singular perturbation problem involving a nonlinear flux function. In the limit of vanishing dissipation, the numerical flux is found to be almost identical to that due to Engquist and Osher, the only difference being in the treatment of transonic compression. This facilitates sharper resolution of stationary shocks, but is accompanied by loss of smoothness and monotonicity.

Hughes, T. J. R.↗
DOE OSTI2015

ON MAGNETIC EQUILIBRIA IN BAROTROPIC STARS

Upper main-sequence stars, white dwarfs, and neutron stars are known to possess stable, large-scale magnetic fields. Numerical works have confirmed that stable magnetohydrodynamic equilibria can exist in non-barotropic, stably stratified stars. On the other hand, it is unclear whether stable equilibria are possible in barotropic stars, although the existing evidence suggests that they are all unstable. This work aims to construct barotropic equilibria in order to study their properties, as a first step to test their stability. We have assumed that the star is a perfectly conducting, axially symmetric fluid, allowing for both poloidal and toroidal components of the magnetic field. In addition, we made the astrophysically justified assumption that the magnetic force has a negligible influence on the fluid structure, in which case the equilibrium is governed by the Grad–Shafranov equation, involving two arbitrary functions of the poloidal flux. We built a numerical code to solve this equation, allowing for an arbitrary prescription for these functions. Taking particularly simple, but physically reasonable choices for these functions with a couple of adjustable parameters, all of the equilibria found present only a small (≲10%) fraction of the magnetic energy stored in the toroidal component, confirming previous results. We developed an analytical model in order to study in more detail the behavior of the magnetic energy over the full range of parameters. The model confirms that the toroidal fraction of the energy and the ratio of toroidal to poloidal flux are bounded from above for the whole range of parameters.

79 ASTRONOMY AND ASTROPHYSICS↗