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All sourcesNASA NTRSDOE OSTI
NASA NTRS1986

Entropy conservative finite element schemes

The question of entropy stability for discrete approximations to hyperbolic systems of conservation laws is studied. The amount of numerical viscosity present in such schemes is quantified and related to their entropy stability by means of comparison. To this end, two main ingredients are used: entropy variables and the construction of certain entropy conservative schemes in terms of piecewise-linear finite element approximations. It is then shown that conservative schemes are entropy stable, if and (for three-point schemes) only if, they contain more numerical viscosity than the abovementioned entropy conservation ones.

Tadmor, E.↗
DOE OSTI2026

Axisymmetric hydrodynamics in numerical relativity: treating coordinate singularity, artificial heating and modeling MHD instabilities

Two-dimensional axisymmetric simulations of binary neutron star (BNS) merger remnant are a cheap alternative to 3D simulations. To maintain realism for secular timescales, simulations must avoid accumulated errors from drifts in conserved quantities and artificial heating, and they must model turbulent transport in a way that remains plausible throughout the evolution. It is also crucial to avoid numerical artifacts due to the polar coordinate axis singularity. Methods that behave well near the axis often break flux-conservative form of the hydrodynamic equations, resulting in significant drifts in conserved quantities. We present a flux-conservative scheme that maintains smoothness near the axis without sacrificing conservative formulation of the equations or incurring drifts in conserved global quantities. We compare the numerical performance of different treatments of the hydrodynamic equations when evolving a hypermassive neutron star resembling the remnant of a BNS merger. These simulations demonstrate that the new scheme combines the axis smoothness of non-conservative methods with the mass and angular momentum conservation of other conservative methods on $\sim 10^2$ ms timescales of viscous and neutrino-driven evolution. Because fluid profiles remain smooth in the remnant interior, it is possible to remove artificial heating by evolving the entropy density. We show how physical heating and cooling terms can be easily calculated from source terms of the conservative evolution variables and demonstrate our implementation. Finally, we discuss and implement improvements to the effective viscosity scheme to better model the effect of magnetohydrodynamic instabilities as the remnant evolves.

axisymmetry↗
NASA NTRS1975

Numerical solution of the Navier-Stokes equations with topography

A finite difference scheme for solving the equations of fluid motion in a generalized coordinate system has been constructed. The scheme conserves mass and all the first integral moments of the motion. The scheme also advectively 'almost conserves' second moments, in that the magnitude of implicit numerical smoothing is typically about an order smaller than explicit viscosity and diffusion. Calculations with the model support the theoretical conjecture that the difference scheme is stable whenever the analogous Cartesian scheme is stable. The scheme has been used to calculate dry atmospheric convection due to differential heating between top and bottom of mountainous terrain. The general small-scale characteristics of mountain up-slope winds have been simulated. In addition, the results have demonstrated the crucial role played by the eddy diffusivities and the environmental stability, in determining both the quantitative and the qualitative features of the circulation.

Gal-Chen, T.↗
NASA NTRS1995

Convergence properties of finite-difference hydrodynamics schemes in the presence of shocks

We investigate the asymptotic convergence of finite-difference schemes for the Euler equations when the limiting solution contains shocks. The Lax-Wendroff theorem guarantees that certain conservative schemes converge to correct, physically valid solutions. We focus on two one-dimensional operator-split schemes with explicit artificial-viscosity terms. One, an internal-energy scheme, does not satisfy the assumptions of Lax-Wendroff; the other, a conservative total-energy scheme, does. With viscous lengths chosen proportional to the grid size, we find that both schemes converge to their zero-grid-size limits at the theoretically expected rate, but only the conversative scheme converges toward correct solutions of the inviscid fluid equations. We show that the difference in their behaviors results directly from the presence of shocks in the limiting solution. Empirically, we find that when the viscous lenghts tend toward zero more slowly than the grid size, however the nonconservative scheme also converges toward correct solutions. We characterize the asymptotic behavior of the total-energy scheme in a particular problem in which a shock forms. As the grid is refined, a Cauchy error approaches the expected rate of change slowly. We show that the changes in the artificial viscosity alter the diffusion of small-amplitude waves. The differences associated with such waves make the dominant contribution to the Cauchy error. We formulate an analytic model to relate the rate of approach to the effect of varying diffusion in waves and find quantitative agreement with our numerical results.

Kimoto, Paul A.↗
NASA NTRS2017

On High-Order Upwind Methods for Advection

In the fourth installment of the celebrated series of five papers entitled "Towards the ultimate conservative difference scheme", Van Leer (1977) introduced five schemes for advection, the first three are piecewise linear, and the last two, piecewise parabolic. Among the five, scheme I, which is the least accurate, extends with relative ease to systems of equations in multiple dimensions. As a result, it became the most popular and is widely known as the MUSCL scheme (monotone upstream-centered schemes for conservation laws). Schemes III and V have the same accuracy, are the most accurate, and are closely related to current high-order methods. Scheme III uses a piecewise linear approximation that is discontinuous across cells, and can be considered as a precursor of the discontinuous Galerkin methods. Scheme V employs a piecewise quadratic approximation that is, as opposed to the case of scheme III, continuous across cells. This method is the basis for the on-going "active flux scheme" developed by Roe and collaborators. Here, schemes III and V are shown to be equivalent in the sense that they yield identical (reconstructed) solutions, provided the initial condition for scheme III is defined from that of scheme V in a manner dependent on the CFL number. This equivalence is counter intuitive since it is generally believed that piecewise linear and piecewise parabolic methods cannot produce the same solutions due to their different degrees of approximation. The finding also shows a key connection between the approaches of discontinuous and continuous polynomial approximations. In addition to the discussed equivalence, a framework using both projection and interpolation that extends schemes III and V into a single family of high-order schemes is introduced. For these high-order extensions, it is demonstrated via Fourier analysis that schemes with the same number of degrees of freedom 𝐾 per cell, in spite of the different piecewise polynomial degrees, share the same sets of eigenvalues and thus, have the same stability and accuracy. Moreover, these schemes are accurate to order 2𝐾−1, which is higher than the expected order of 𝐾.

high-order methods↗
NASA NTRS1977

Computational design of the basic dynamical processes of the UCLA general circulation model

The 12-layer UCLA general circulation model encompassing troposphere and stratosphere (and superjacent 'sponge layer') is described. Prognostic variables are: surface pressure, horizontal velocity, temperature, water vapor and ozone in each layer, planetary boundary layer (PBL) depth, temperature, moisture and momentum discontinuities at PBL top, ground temperature and water storage, and mass of snow on ground. Selection of space finite-difference schemes for homogeneous incompressible flow, with/without a free surface, nonlinear two-dimensional nondivergent flow, enstrophy conserving schemes, momentum advection schemes, vertical and horizontal difference schemes, and time differencing schemes are discussed.

Arakawa, A.↗
DOE OSTI2020

Conservative discontinuous Galerkin scheme of a gyro-averaged Dougherty collision operator

Here, a conservative discontinuous Galerkin scheme for a non-linear Dougherty collision operator in full-f long-wavelength gyrokinetics is presented. Analytically this model operator has the advective-diffusive form of Fokker-Planck operators, it has a non-decreasing entropy functional, and conserves particles, momentum and energy. Discretely these conservative properties are maintained exactly as well, independent of numerical resolution. In this work the phase space discretization is performed using a novel version of the discontinuous Galerkin scheme, carefully constructed using concepts of weak equality and recovery. Discrete time advancement is carried out with an explicit time-stepping algorithm, whose stability limits we explore. The formulation and implementation within the long-wavelength gyrokinetic solver of Gkeyll are validated with relaxation tests, collisional Landau-damping benchmarks and the study of 5D gyrokinetic turbulence on helical, open field lines.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1987

Comparison of horizontal difference schemes for the shallow water equations on a sphere

The accuracy of horizontal difference schemes used in the hydrodynamics parts of General Circulation Models are compared by means of numerical experiments for the shallow water equations on a sphere. As expected, the phase lag of moving waves decreases as the order of accuracy of a scheme increases or as the grid resolution increases. Overall, Takano and Wurtele's partial fourth order energy and potential enstrophy conserving scheme on the C grid is most accurate. It is clearly superior to the other schemes for the Rossby-Haurwitz wave number 6 initial conditions for coarse grid resolution.

Russell, Gary L.↗
NASA NTRS1990

Triangle based TVD schemes for hyperbolic conservation laws

A triangle based total variation diminishing (TVD) scheme for the numerical approximation of hyperbolic conservation laws in two space dimensions is constructed. The novelty of the scheme lies in the nature of the preprocessing of the cell averaged data, which is accomplished via a nearest neighbor linear interpolation followed by a slope limiting procedures. Two such limiting procedures are suggested. The resulting method is considerably more simple than other triangle based non-oscillatory approximations which, like this scheme, approximate the flux up to second order accuracy. Numerical results for linear advection and Burgers' equation are presented.

Durlofsky, Louis J.↗
DOE OSTI2025

A Mass Conservation Relaxed (MCR) LSTM Model for Streamflow Simulation Across CONUS

The recent development of the physics-aware Mass-Conserving Long Short-Term Memory network (MC-LSTM) provides an alternative to other data-driven Deep Learning (DL) models in hydrology. Mass-Conserving Long Short-Term Memory incorporates mass conservation directly into the LSTM architecture. Despite the theoretical advancements, studies have reported a surprisingly limited performance of the MC-LSTM in streamflow simulation. We hypothesize that such a limitation is due to the unrealistic mass conservation scheme in MC-LSTM, which overlooks unobserved incoming water fluxes beyond precipitation. As an attempt to verify this hypothesis, we propose a Mass Conservation Relaxed LSTM (MCR-LSTM), which incorporates a bi-directional mass relaxation (MR) component to account for potential incoming water fluxes beyond precipitation. We train and test the proposed MCR-LSTM model across 531 watersheds in the contiguous United States (CONUS) against three baseline models: the Sacramento Soil Moisture Accounting, LSTM, and MC-LSTM. Our results show that MCR-LSTM outperforms MC-LSTM despite its underperformance compared to LSTM. Specifically, MCR-LSTM's advantage over MC-LSTM is mainly seen in the Plains and Western U.S., where the newly incorporated MR component better simulates water loss and suggests the likely existence of additional incoming water fluxes beyond precipitation, respectively. The novelty and contribution of this study are twofold: firstly, it introduces an alternative physics-aware DL tool (i.e., MCR-LSTM) in hydrology with higher accuracy in specific regions compared to MC-LSTM. Secondly, it provides a diagnosis of regions where strict, precipitation-based mass conservation constraints may be unrealistic in streamflow simulation.

deep learning↗
NASA NTRS2013

Comparative Study on High-Order Positivity-preserving WENO Schemes

The goal of this study is to compare the results obtained by non-positivity-preserving methods with the recently developed positivity-preserving schemes for representative test cases. In particular the more di cult 3D Noh and Sedov problems are considered. These test cases are chosen because of the negative pressure/density most often exhibited by standard high-order shock-capturing schemes. The simulation of a hypersonic nonequilibrium viscous shock tube that is related to the NASA Electric Arc Shock Tube (EAST) is also included. EAST is a high-temperature and high Mach number viscous nonequilibrium ow consisting of 13 species. In addition, as most common shock-capturing schemes have been developed for problems without source terms, when applied to problems with nonlinear and/or sti source terms these methods can result in spurious solutions, even when solving a conservative system of equations with a conservative scheme. This kind of behavior can be observed even for a scalar case (LeVeque & Yee 1990) as well as for the case consisting of two species and one reaction (Wang et al. 2012). For further information concerning this issue see (LeVeque & Yee 1990; Griffiths et al. 1992; Lafon & Yee 1996; Yee et al. 2012). This EAST example indicated that standard high-order shock-capturing methods exhibit instability of density/pressure in addition to grid-dependent discontinuity locations with insufficient grid points. The evaluation of these test cases is based on the stability of the numerical schemes together with the accuracy of the obtained solutions.

Kotov, Dmitry V.↗
NASA NTRS2001

High Order Difference Method for Low Mach Number Aeroacoustics

A high order finite difference method with improved accuracy and stability properties for computational aeroacoustics (CAA) at low Mach numbers is proposed. The Euler equations are split into a conservative and a symmetric non- conservative portion to allow the derivation of a generalized energy estimate. Since the symmetrization is based on entropy variables, that splitting of the flux derivatives is referred to as entropy splitting. Its discretization by high order central differences was found to need less numerical dissipation than conventional conservative schemes. Owing to the large disparity of acoustic and stagnation quantities in low Mach number aeroacoustics, the split Euler equations are formulated in perturbation form. The unknowns are the small changes of the conservative variables with respect to their large stagnation values. All nonlinearities and the conservation form of the conservative portion of the split flux derivatives can be retained, while cancellation errors are avoided with its discretization opposed to the conventional conservative form. The finite difference method is third-order accurate at the boundary and the conventional central sixth-order accurate stencil in the interior. The difference operator satisfies the summation by parts property analogous to the integration by parts in the continuous energy estimate. Thus, strict stability of the difference method follows automatically. Spurious high frequency oscillations are suppressed by a characteristic-based filter similar to but without limiter. The time derivative is approximated by a 4-stage low-storage second-order explicit Runge-Kutta method. The method has been applied to simulate vortex sound at low Mach numbers. We consider the Kirchhoff vortex, which is an elliptical patch of constant vorticity rotating with constant angular frequency in irrotational flow. The acoustic pressure generated by the Kirchhoff vortex is governed by the 2D Helmholtz equation, which can be solved analytically using separation of variables.

Mueller, B.↗
NASA NTRS1982

High-resolution schemes for hyperbolic conservation laws

A class of new explicit second order accurate finite difference schemes for the computation of weak solutions of hyperbolic conservation laws is presented. These highly nonlinear schemes are obtained by applying a nonoscillatory first order accurae scheme to an appropriately modified flux function. The so derived second order accurate schemes achieve high resolution while preserving the robustness of the original nonoscillatory first order accurate scheme.

Harten, A.↗
NASA NTRS1983

High resolution schemes for hyperbolic conservation laws

A class of new explicit second order accurate finite difference schemes for the computation of weak solutions of hyperbolic conservation laws is presented. These highly nonlinear schemes are obtained by applying a nonoscillatory first order accurate scheme to an appropriately modified flux function. The so-derived second order accurate schemes achieve high resolution while preserving the robustness of the original nonoscillatory first order accurate scheme. Numerical experiments are presented to demonstrate the performance of these new schemes.

Harten, A.↗
NASA NTRS1994

Three-Dimensional Navier-Stokes Method with Two-Equation Turbulence Models for Efficient Numerical Simulation of Hypersonic Flows

A new computational efficient 3-D compressible Reynolds-averaged implicit Navier-Stokes method with advanced two equation turbulence models for high speed flows is presented. All convective terms are modeled using an entropy satisfying higher-order Total Variation Diminishing (TVD) scheme based on implicit upwind flux-difference split approximations and arithmetic averaging procedure of primitive variables. This method combines the best features of data management and computational efficiency of space marching procedures with the generality and stability of time dependent Navier-Stokes procedures to solve flows with mixed supersonic and subsonic zones, including streamwise separated flows. Its robust stability derives from a combination of conservative implicit upwind flux-difference splitting with Roe's property U to provide accurate shock capturing capability that non-conservative schemes do not guarantee, alternating symmetric Gauss-Seidel 'method of planes' relaxation procedure coupled with a three-dimensional two-factor diagonal-dominant approximate factorization scheme, TVD flux limiters of higher-order flux differences satisfying realizability, and well-posed characteristic-based implicit boundary-point a'pproximations consistent with the local characteristics domain of dependence. The efficiency of the method is highly increased with Newton Raphson acceleration which allows convergence in essentially one forward sweep for supersonic flows. The method is verified by comparing with experiment and other Navier-Stokes methods. Here, results of adiabatic and cooled flat plate flows, compression corner flow, and 3-D hypersonic shock-wave/turbulent boundary layer interaction flows are presented. The robust 3-D method achieves a better computational efficiency of at least one order of magnitude over the CNS Navier-Stokes code. It provides cost-effective aerodynamic predictions in agreement with experiment, and the capability of predicting complex flow structures in complex geometries with good accuracy.

Bardina, J. E.↗
NASA NTRS2019

Entropy Stable Method for the Euler Equations Revisited: Central Differencing via Entropy Splitting and SBP

The two decades old high order central differencing via entropy splitting and summation-by-parts (SBP) difference boundary closure of Olsson & Oliger, Gerritsen & Olsson, and Yee et al. (15, 7, 37) is revisited. The objective of this paper is to prove for the first time that the entropy split scheme is an entropy stable method for central differencing with SBP operators for both periodic and non-periodic boundary conditions for nonlinear Euler equations. Standard high order spatial central differencing as well as high order central spatial DRP (dispersion relation preserving) spatial differencing is part of the entropy stable methodology framework. The proof is to replace the spatial derivatives by summation-by-parts (SBP) difference operators in the entropy split form of the equations using the physical entropy of the Euler equations. The numerical boundary closure follows directly from the SBP operator. No additional numerical boundary procedure is required. In contrast, Tadmor-type entropy conserving schemes (31) using mathematical entropies and more recently in (35], do not naturally come with a numerical boundary closure and a generalized SBP operator has to be developed (18). Long time integration of 2D and 3D test cases is included to show the comparison of this efficient entropy stable method with the Tadmor-type of entropy conservative methods. Studies also include the comparison among the three skew-symmetric splittings on their nonlinear stability and accuracy performance without added numerical dissipations for smooth flows. These are, namely, entropy splitting, Ducros et al. splitting and the Kennedy & Grubber splitting.

Sjogreen, Bjorn↗