A Conservative Six-Equation Multi-Material Method for Compact Finite Differences
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Engineering - Mechanical and civil engineering↗
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Machine Learning for Conservative-to-Primitive in Relativistic Hydrodynamics
The numerical solution of relativistic hydrodynamics equations in conservative form requires root-finding algorithms that invert the conservative-to-primitive variables map. These algorithms employ the equation of state of the fluid and can be computationally demanding for applications involving sophisticated microphysics models, such as those required to calculate accurate gravitational wave signals in numerical relativity simulations of binary neutron stars. This work explores the use of machine learning methods to speed up the recovery of primitives in relativistic hydrodynamics. Artificial neural networks are trained to replace either the interpolations of a tabulated equation of state or directly the conservative-to-primitive map. The application of these neural networks to simple benchmark problems shows that both approaches improve over traditional root finders with tabular equation-of-state and multi-dimensional interpolations. In particular, the neural networks for the conservative-to-primitive map accelerate the variable recovery by more than an order of magnitude over standard methods while maintaining accuracy. Neural networks are thus an interesting option to improve the speed and robustness of relativistic hydrodynamics algorithms.
79 ASTRONOMY AND ASTROPHYSICS↗
A two-fluid single-column model of turbulent shallow convection. Part II: Single-column model formulation and numerics
The two-fluid single-column model of Thuburn et al. (Quart. J. R. Meteorol. Soc., 2019, 145, 1535–1550) is extended to include moisture and horizontal wind shear. Turbulent kinetic energy is introduced as a prognostic variable, dependence on a diagnosed boundary-layer height is removed, and subfilter fluxes are approximated using a two-fluid version of a Mellor–Yamada scheme. Three mechanisms for entrainment and detrainment processes are introduced, which represent entrainment of unstable air at the surface, forced detrainment of air at the top of the boundary/cloud layers, and turbulent mixing that relaxes the convective fluid to a reference profile. A semi-implicit Eulerian discretization replaces the semi-implicit semi-Lagrangian implementation of Thuburn et al. (Quart. J. R. Meteorol. Soc., 2019, 145, 1535–1550) to improve numerical stability and conservation. The equations for the implicit time step are solved using a quasi-Newton method, which is shown to perform well in numerical tests for conservation and convergence. The two-fluid single-column model presented in this article will be applied to simulations of shallow cumulus convection in Part III.
54 ENVIRONMENTAL SCIENCES↗
Simple and General Unitarity Conserving Numerical Real-Time Propagators of the Time-Dependent Schrödinger Equation Based on Magnus Expansion
Magnus expansion (ME) provides a general way to expand the real-time propagator of a time-dependent Hamiltonian within the exponential such that the unitarity is satisfied at any order. Here, we use this property and explicit integration of Lagrange interpolation formulas for the time-dependent Hamiltonian within each time interval and derive approximations that preserve unitarity for the differential time evolution operators of general time-dependent Hamiltonians. The resulting second-order approximation is the same as using the average of Hamiltonians for two end points of time. We identify three fourth-order approximations involving commutators of Hamiltonians at different times and also derive a sixth-order expression. A test of these approximations along with other available expressions for a two-state time-dependent Hamiltonian with sinusoidal time dependences provides information on the relative performance of these approximations and suggests that the derived expressions can serve as useful numerical tools for time evolution in time-resolved spectroscopy, quantum control, quantum sensing, real-time ab initio quantum dynamics, and open system quantum dynamics.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Second-order invariant domain preserving approximation of the compressible Navier–Stokes equations
Here, we present a fully discrete approximation technique for the compressible Navier–Stokes equations that is second-order accurate in time and space, semi-implicit, and guaranteed to be invariant domain preserving. The restriction on the time step is the standard hyperbolic CFL condition, i.e. τ ≲ O ( h ) / V where V is some reference velocity scale and h the typical meshsize.
42 ENGINEERING↗
Discontinuous Galerkin sparse grids methods for time domain Maxwell’s equations
In this paper, we present a time-explicit sparse grid discontinuous Galerkin method for solving the three-dimensional time-domain Maxwell equations. The conservation properties and convergence rates are established for different choices of numerical fluxes. The convergence rates are proved theoretically and then verified by several numerical examples. Even though our scheme does not preserve the divergence, but by implying the higher order polynomial, one can observe the same convergence rate as numerical solution. Several numerical tests are presented to validate these conclusions.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Feasibility and performance of the staged Z-pinch: A one-dimensional study with FLASH and MACH2
Z-pinch platforms constitute a promising pathway to fusion energy research. Here, we present a one-dimensional numerical study of the staged Z-pinch (SZP) concept using the FLASH and MACH2 codes. We discuss the verification of the codes using two analytical benchmarks that include Z-pinch-relevant physics, building confidence on the codes' ability to model such experiments. Then, FLASH is used to simulate two different SZP configurations: a xenon gas-puff liner (SZP1*) and a silver solid liner (SZP2). The SZP2 results are compared against previously published MACH2 results, and a new code-to-code comparison on SZP1* is presented. Using an ideal equation of state and analytical transport coefficients, FLASH yields a fuel convergence ratio (CR) of approximately 39 and a mass-averaged fuel ion temperature slightly below 1 keV for the SZP2 scheme, significantly lower than the full-physics MACH2 prediction. For the new SZP1* configuration, full-physics FLASH simulations furnish large and inherently unstable CRs (>300) but achieve fuel ion temperatures of many kilo-electron volts. While MACH2 also predicts high temperatures, the fuel stagnates at a smaller CR. The integrated code-to-code comparison reveals how magnetic insulation, heat conduction, and radiation transport affect platform performance and the feasibility of the SZP concept.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Assessing physics of ion temperature gradient turbulence via hierarchical reduced-model representations
In this work, the saturation physics of ion temperature gradient (ITG) turbulence is probed by studying how amplitudes and scalings with key parameters vary across a hierarchy of reduced models. The models derive from nonlinear fluid equations for toroidal ITG turbulence under approximations to the mode coupling interactions in wavenumber space and the representation of turbulent decorrelation. Mode coupling approximations include local-in-wavenumber treatments like the spectral density of flux in quasilinear theory, a truncation to three nonlinearly interacting waves, and the interactions in a cascade to high radial wavenumber mediated by a single zonal flow. Turbulent decorrelation treatments are based on the triplet correlation time with and without eddy damping. Model fidelity is assessed by the scalings and magnitudes of the squared amplitudes of unstable mode, stable mode, and zonal flow with respect to the flow-damping rate and temperature gradient. It is shown that all models reproduce fundamental scalings, provided they incorporate the coupling of unstable mode, stable mode, and zonal flow. Accurate amplitude prediction requires eddy damping in the triplet correlation time and proper representation of the zonal-flow drive by interactions associated with the radial wavenumber cascade.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Validating corrosion models: Influence of physical properties
In chloride containing environments, two metals in physical contact can undergo galvanic corrosion limiting the lifetime of components. Being able to accurately predict galvanic corrosion damage distributions over time has not been widely presented in literature. Therefore, Finite Element Method (FEM) corrosion models were experimentally validated for two galvanic couples as a function of governing equations, environment, anode material, anode:cathode ratio, and time. Carbon steel/stainless steel (CS/SS) and zinc/stainless steel (Zn/SS) galvanic couples were exposed to NaCl solutions at room temperatures for up to 14 days. For the galvanic couples and environments, the Laplace equation with variable conductivity and reactions was sufficient to model the experimental corrosion damage. Increasing the chloride concentration for the CS/SS galvanic couple, regardless of the anode:cathode ratio, decreased the observed and modeled corrosion damage. Decreasing the anode:cathode ratio (i.e., increasing the cathode length), increased the experimentally observed and modeled corrosion damage. For small anode:cathode ratios, the governing equations deviate over long time periods with the Nernst-Plank equation being conservative. For the Zn/SS galvanic couple, the cathode length controls the dominant cathodic reduction reaction. For small cathode lengths, the hydrogen evolution reaction is dominant. For larger cathode lengths, the oxygen reduction reaction is dominant. The results are discussed with regard to the influence of solution chemistry, ohmic drop, and governing reactions. Overall, validated FEM models were presented, and the resultant models, physics, and mechanisms can be applied to other corrosion scenarios with confidence.
Carbon steel↗
A general metriplectic framework with application to dissipative extended magnetohydrodynamics
General equations for conservative yet dissipative (entropy producing) extended magnetohydrodynamics are derived from two-fluid theory. Keeping all terms generates unusual cross-effects, such as thermophoresis and a current viscosity that mixes with the usual velocity viscosity. While the Poisson bracket of the ideal version of this model has already been discovered, we determine its metriplectic counterpart that describes the dissipation. This is done using a new and general thermodynamic point of view to derive dissipative brackets, a means of derivation that is natural for understanding and creating dissipative dynamics without appealing to underlying kinetic theory orderings. Finally, the formalism is used to study dissipation in the Lagrangian variable picture where, in the context of extended magnetohydrodynamics, non-local dissipative brackets naturally emerge.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Theory of ignition and burn propagation in inertial fusion implosions
A detailed analytic model is presented here to investigate the physics of burn propagation in inertially confined plasmas. The onset of ignition and burn propagation occurs when alpha heating of the hot spot causes rapid ablation of shell mass into the hot spot. This allows large energy gains to be achieved since most of the fuel mass is located in the shell. Here, we first present a comprehensive review of previous analytic models that have been used to describe the physics of hot-spot evolution and ignition; we then show that a proper description of a propagating burn wave requires a comprehensive model of hot spot and shell evolution that includes proper mass conservation in the shell, fusion reactivity, and fuel depletion. The analytic theory is in good agreement with detailed radiation-hydrodynamic simulations that predict the onset of burn propagation as occurring when the yield enhancement caused by alpha heating is between 15- and 25-fold, $f_α$ ~ 1.4, where $f_α$ = alpha energy deposited/hot-spot energy at bang time, and the hot-spot burnup fraction is approximately 2%. We show that the definition of ignition is not sensitive to the alpha-particle stopping power nor asymmetries provided that the absorbed fraction of alpha particles $θ_α$ is correctly accounted for. Finally, we use the results of 2-D simulations to show that even when $θ_α$ is small and unknown (as is true in hot spots with mid modes that have significant leakage of alpha particles into the surrounding cold bubbles), one can still relate the experimentally measureable parameter $\chi^{53}_α$ to the yield amplification and the burning-plasma parameter $Q^{\text{hs}}_α$ = alpha energy deposited/total input work delivered to the hot spot
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Extending the Heat Index
The Heat Index is a widely used measure of apparent temperature that accounts for the effects of humidity using Steadman’s model of human thermoregulation. Steadman’s model, however, gives unphysical results when the air is too hot and humid or too cold and dry, leading to an undefined Heat Index. For example, at a relative humidity of 80%, the Heat Index is only defined for temperatures in the range of 288-304 K (59-88 degrees Fahrenheit). Here, Steadman’s thermoregulation model is extended to define the Heat Index for all combinations of temperature and humidity, allowing for an assessment of Earth’s future habitability. The extended Heat Index can be mapped onto physiological responses of an idealized human, such as heat exhaustion, heat stroke, and even heat death, providing an indication of regional health outcomes for different degrees of global warming.
54 ENVIRONMENTAL SCIENCES↗
Theoretical Understanding of the Linear Relationship between Convective Updrafts and Cloud-Base Height for Shallow Cumulus Clouds. Part II: Continental Conditions
This is the Part II of a two-part study that seeks a theoretical understanding of an empirical relationship for shallow cumulus clouds: subcloud updraft velocity covaries linearly with the cloud-base height. This work focuses on continental cumulus clouds that are more strongly forced by surface fluxes and more deviated from equilibrium than those over oceans (Part I). We use a simple analytical model for shallow cumulus that is well tested against a high-resolution (25 m in the horizontal) large-eddy simulation model. Consistent with a conventional idea, we find that surface Bowen ratio is the key variable that regulates the covariability of both parameters: under the same solar insolation, a drier surface allows for stronger buoyancy flux, triggering stronger convection that deepens the subcloud layer. We find that the slope of the Bowen-ratio-regulated relationship between the two parameters (defined as λ) is dependent on both the local time and the stability of the lower free atmosphere. The value of λ decreases with time exponentially from sunrise to early afternoon and linearly from early afternoon to sunset. The value of λ is larger in a more stable atmosphere. In addition, continental λ in the early afternoon more than doubles the oceanic λ. Validation of the theoretical results against ground observations over the Southern Great Plains shows a reasonable agreement. Physical mechanisms underlying the findings are explained from the perspective of different time scales at which updrafts and cloud-base height respond to a surface flux forcing.
54 ENVIRONMENTAL SCIENCES↗
Generalized Lapse Rate Formulas for Use in Entraining CAPE Calculations
Abstract Common assumptions in temperature lapse rate formulas for lifted air parcels include neglecting mixing, hydrostatic balance, the removal of all condensate once it forms (pseudoadiabatic), and/or the retention of all condensate within the parcel (adiabatic). These formulas are commonly derived from the conservation of entropy, which leads to errors when nonequilibrium mixed-phase condensate is present. To evaluate these assumptions, a new general lapse rate formula is derived from an expression for energy conservation, rather than entropy conservation. This new formula incorporates mixing of the parcel with its surroundings, relaxes the hydrostatic assumption, allows for nonequilibrium mixed-phase condensate, and can be formulated for pseudoadiabatic or adiabatic ascent. The new formula is shown to exactly conserve entropy for reversible ascent. Predictions by the new formula are compared to that of older and less general formulas. The errors in previous formulas arise from the assumption of hydrostatic balance, which results in considerable warm biases due to the neglect of the energy sink from buoyancy. Predictions of ascent with entrainment using the new formula are then compared to parcel properties along trajectories in large eddy simulations. Simulated parcel properties are better predicted by the formula using a diluted analogy to adiabatic ascent, wherein condensate is diluted at the same rate as other parcel properties, than by the diluted analogy to pseudoadiabatic ascent, wherein all condensate is removed. These results suggest that CAPE should be computed with adiabatic, rather than pseudoadiabatic, parcel ascent.
54 ENVIRONMENTAL SCIENCES↗
Multidisciplinary benchmarks of a conservative spectral solver for the nonlinear Boltzmann equation
The Boltzmann equation describes the evolution of the phase-space probability distribution of classical particles under binary collisions. Approximations to it underlie the basis for several scholarly fields, including aerodynamics and plasma physics. While these approximations are appropriate in their respective domains, they can be violated in niche but diverse applications which require direct numerical solution of the original nonlinear Boltzmann equation. An expanded implementation of the Galerkin–Petrov conservative spectral algorithm is employed to study a wide variety of physical problems. Enabled by distributed precomputation, solutions of the spatially homogeneous Boltzmann equation can be achieved in seconds on modern personal hardware, while spatially-inhomogeneous problems are solvable in minutes. Here, several benchmarks are presented focusing on accuracy compared to both analytic theoretical predictions and other Boltzmann solvers. These benchmarks span several physical domains including weakly ionized plasma, gaseous fluids, and atomic-plasma interaction.
97 MATHEMATICS AND COMPUTING↗
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↗
Empirical Model of Puncture Energy for Metals
The purpose of this work is to fit a previously developed empirical equation for puncture energy to simulation data. The conservative puncture energy equation could be used to expedite the process of performing calculations in the development of safety measures, avoiding the need to create complex finite element models for specific puncture scenarios. A total of 108 simulations are developed by varying coupon thickness, coupon material, probe shape, and probe diameter. The simulations are comprised of a low-velocity probe puncturing the coupons, from which the probe kinetic energy change is calculated. The empirical equation is fit to the dimensions, material properties, and energy results using a non-linear least-squares regression method within Python, which determines the two constant parameters for each fit. More statistically significant fit results are achieved by separating the data by probe shape and coupon material.
42 ENGINEERING↗