A compact artificial viscosity equivalent to a tensor viscosity

We present a compact artificial viscosity for staggered grid Lagrangian hydrodynamics on polygonal cells in two Cartesian dimensions and using a decomposition into triangles we show that this viscosity is equivalent to a tensor viscosity of Campbell and Shashkov for quadrilaterals.     

Mimetic finite difference method for the Stokes problem on polygonal meshes

Various approaches to extend finite element methods to non-traditional elements (general polygons, pyramids, polyhedra, etc.) have been developed over the last decade. The construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis. The mimetic finite difference (MFD) method works on general polygonal meshes and has many similarities with low-order finite element methods. Both schemes try to preserve the fundamental properties of the underlying physical and mathematical models. The essential difference between the two schemes is that the MFD method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices. Since no extension of basis functions inside the mesh elements is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and non-convex elements. In this article, we present a new MFD method for the Stokes problem on arbitrary polygonal meshes and analyze its stability. The method is developed for the general case of tensor coefficients, which allows us to apply it to a linear elasticity problem, as well. Numerical experiments show, for the velocity variable, second-order convergence in a discrete L² norm and first-order convergence in a discrete H¹ norm. For the pressure variable, first-order convergence is shown in the L² norm.     

IDeC(k): A new velocity reconstruction algorithm on arbitrarily polygonal staggered meshes

We propose a novel algorithm for velocity reconstruction from staggered data on arbitrary polygonal staggered meshes. The formulation of the new algorithm is based on a constant polynomial reconstruction approach in conjunction with an iterative defect correction method and is referred to as the IDeC(k) reconstruction. The algorithm is designed for second order accuracy of the reconstructed velocity field and also leads to a consistent estimate of velocity gradients. Accuracy, convergence and robustness of the new algorithm are studied on different mesh topologies and the need for higher-order reconstruction is demonstrated. Numerical experiments for several cases including incompressible viscous flows establish the IDeC(k) reconstruction as a generic, fast, robust and higher-order accurate algorithm on arbitrary polygonal meshes.     

Enriched multi-point flux approximation for general grids

It is well known that the two-point flux approximation, a numerical scheme used in most commercial reservoir simulators, has O(1) error when grids are not K-orthogonal. In the last decade, the multi-point flux approximations have been developed as a remedy. However, non-physical oscillations can appear when the anisotropy is really strong. We found out the oscillations are closely related to the poor approximation of pressure gradient in the flux computation.In this paper, we propose the control volume enriched multi-point flux approximation (EMPFA) for general diffusion problems on polygonal and polyhedral meshes. Non-physical oscillations are not observed for realistic and strongly anisotropic heterogeneous material properties described by a full tensor. Exact linear solutions are recovered for grids with non-planar interfaces, and a first and second order convergence are achieved for the flux and scalar unknowns, respectively.     

Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes

We consider a non-linear finite volume (FV) scheme for stationary diffusion equation. We prove that the scheme is monotone, i.e. it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients. The scheme is extended to regular star-shaped polygonal meshes and isotropic heterogeneous coefficients.     

Fission dynamics of excited nuclei within the liquid-drop model

We evaluate the temperature T_scis at the scission point and the saddle-to-scission time τ_scis for the fission of heated nuclei. We use classical Lagrange-like equations of motion within the liquid-drop model. The nuclear surface is parameterized by a two-parameter family of the Lawrence shapes. Conservative forces are defined through the free energy of the nucleus at finite temperatures. We use the friction tensor that is derived from the Navier-Stokes momentum-flux tensor and which takes into account the boundary conditions at the nuclear surface. The scission line is determined from the instability condition of the nuclear shape with respect to variations of the neck radius. A numerical solution to the dynamical equations is obtained for the ²³⁶U nucleus. The viscosity coefficient μ is deduced from a comparison of experimental data on the kinetic energy of fission fragments with the computed one. It is found that μ obtained by using our approach deviates significantly from μ of the standard hydrodynamic model.     

Unified treatment of complete orthonormal sets for wave functions, and Slater orbitals of particles with arbitrary spin in coordinate, momentum and four-dimensional spaces

The new analytical relations of complete orthonormal sets for the tensor wave functions and the tensor Slater orbitals of particles with arbitrary spin in coordinate, momentum and four-dimensional spaces are derived using the properties of tensor spherical harmonics and complete orthonormal scalar basis sets of ψα-exponential type orbitals, ?α-momentum space orbitals and zα-hyperspherical harmonics introduced by the author for particles with spin s=0, where the . All of the tensor wave functions obtained are complete without the inclusion of the continuum and, therefore, their group of transformations is the four-dimensional rotation group O(4). The analytical formulas in coordinate space are also derived for the overlap integrals over tensor Slater orbitals with the same screening constant. We notice that the new idea presented in this work is the combination of tensor spherical harmonics of rank s with complete orthonormal scalar sets for radial parts of ψα-, ?α- and zα-orbitals, where .     

A finite volume method for approximating 3D diffusion operators on general meshes

A finite volume method is presented for discretizing 3D diffusion operators with variable full tensor coefficients. This method handles anisotropic, non-symmetric or discontinuous variable tensor coefficients while distorted, non-matching or non-convex n-faced polyhedron meshes can be used. For meshes of polyhedra whose faces have not more than four edges, the associated matrix is positive definite (and symmetric if the diffusion tensor is symmetric). A second-order (resp. first-order) accuracy is numerically observed for the solution (resp. gradient of the solution).     

A tensor artificial viscosity using a finite element approach

We derive a tensor artificial viscosity suitable for use in a 2D or 3D unstructured arbitrary Lagrangian–Eulerian (ALE) hydrodynamics code. This work is similar in nature to that of Campbell and Shashkov [1]; however, our approach is based on a finite element discretization that is fundamentally different from the mimetic finite difference framework. The finite element point of view leads to novel insights as well as improved numerical results. We begin with a generalized tensor version of the Von Neumann–Richtmyer artificial viscosity, then convert it to a variational formulation and apply a Galerkin discretization process using high order Gaussian quadrature to obtain a generalized nodal force term and corresponding zonal heating (or shock entropy) term. This technique is modular and is therefore suitable for coupling to a traditional staggered grid discretization of the momentum and energy conservation laws; however, we motivate the use of such finite element approaches for discretizing each term in the Euler equations. We review the key properties that any artificial viscosity must possess and use these to formulate specific constraints on the total artificial viscosity force term as well as the artificial viscosity coefficient. We also show, that under certain simplifying assumptions, the two-dimensional scheme from [1] can be viewed as an under-integrated version of our finite element method. This equivalence holds on general distorted quadrilateral grids. Finally, we present computational results on some standard shock hydro test problems, as well as some more challenging problems, indicating the advantages of the new approach with respect to symmetry preservation for shock wave propagation over general grids.     

An improved monotone finite volume scheme for diffusion equation on polygonal meshes

We construct a new nonlinear monotone finite volume scheme for diffusion equation on polygonal meshes. The new scheme uses the cell-edge unknowns instead of cell-vertex unknowns as the auxiliary unknowns in order to improve the accuracy of monotone scheme. Our scheme is locally conservative and has only cell-centered unknowns. Numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes. Specially, numerical results show that the new scheme is robust, and more accurate than the existing monotone scheme on some kinds of meshes.     

Permeability tensor versus permittivity one in theory of nonreciprocal optical components

We discuss symmetry properties of the combined systems: electromagnetic field (TE and TM modes) + medium (nondiagonal permeability tensor [μ] and nondiagonal permittivity one [?]) in 2D photonic crystals with transverse magnetization. We show that the origin of orthogonality of the pairs (TE mode + [?]) and (TM mode + [μ]) is different parities of the corresponding fields with respect to the horizontal plane of symmetry of the crystal. As a result of this symmetry, a common use only of the permittivity tensor [?] in the electromagnetic theory of nonreciprocal optical components, ignoring the permeability tensor [μ], leads to loss of a significant part of possible solutions related to TE modes.     

The finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes

We construct a new nonlinear finite volume scheme for diffusion equation on polygonal meshes and prove that the scheme satisfies the discrete extremum principle. Our scheme is locally conservative and has only cell-centered unknowns. Numerical results are presented to show how our scheme works for preserving discrete extremum principle and positivity on various distorted meshes.     

Cosmological perturbations: a new gauge-invariant approach

A new gauge-invariant approach for describing cosmological perturbations is developed. It is based on a physically motivated splitting of the stress-energy tensor of the perturbation into two parts—the bare perturbation and the complementary perturbation associated with stresses in the background gravitational field induced by the introduction of the bare perturbation. The complementary perturbation of the stress-energy tensor is explicitly singled out and taken to the left side of the perturbed Einstein equations so that the bare stress-energy tensor is the sole source for the perturbation of the metric tensor and both sides of these equations are gauge invariant with respect to infinitesimal coordinate transformations. For simplicity we analyze the perturbations of the spatially-flat Friedmann–Lemaı̂tre–Robertson–Walker (FLRW) dust model. A cosmological gauge can be chosen such that the equations for the perturbations of the metric tensor are completely decoupled for the h₀₀, h_0i, and h_ij metric components and explicitly solvable in terms of retarded integrals.     

Generator-coordinate study of 3He+3He elastic scattering with a semirealistic nucleon-nucleon potential

The elastic scattering of ³He by ³He is studied in the framework of the generator coordinate method, using a new semirealistic nucleon-nucleon potential. All components of this potential have a soft core in the sense that they can be used in calculations where short-range correlations between nucleons are not taken into account. With this potential, a good agreement between calculated and measured cross sections and polarizations is obtained. The tensor component of the potential is found to be essential to explain the J-dependence of the empirical phase shifts, although the calculated polarizations are insensitive to the strength of the tensor potential.     

Relationships in infrared intensity theories,in particular the Mayants-Averbukh theory

A brief survey is given of the Mayants-Averbukh infrared intensity theory in relation to the more well-known but equivalent polar tensor theory. In addition, the appearances of the symmetry invariant parameter matrices D_n⁰ of the Mayants-Averbukh theory were derived and tabulated for various symmetries about the midpoint of bond n. The use of these matrices and a single bond coordinate system will offer a convenient alternative to the Mayants-Averbukh treatment of a central, symmetric bond. The rotational mode equations of the Mayants-Averbukh and polar tensor theories have been investigated to elucidate the constrainsts which they impose on infrared intensity theories based on the bond dipole moment model and the atomic point charge model. It was found that the valence-optical theory is in full conformity with the rotational modes only if all electrooptical parameters $\text{?μ}{}_{\mathrm{n}}\text{?γ}{}_{\mathrm{i}}$ are neglected, where γ_i is the ith internal angular coordinate. The constraints imposed on the equilibrium charge-charge flux theory correspond to neglect of all charge flux parameters. The generalized valence-optical theory was found to be incompatible with the rotational mode equations of the Mayants-Averbukh theory. However, its basic dipole moment equation was found useful for suggesting a unique interpretation of a set of d parameters (elements of D_n⁰) in terms of bond dipole moment components.     

Vaidya spacetime in the diagonal coordinates

We have analyzed the transformation from initial coordinates (v, r) of the Vaidya metric with light coordinate v to the most physical diagonal coordinates (t, r). An exact solution has been obtained for the corresponding metric tensor in the case of a linear dependence of the mass function of the Vaidya metric on light coordinate v. In the diagonal coordinates, a narrow region (with a width proportional to the mass growth rate of a black hole) has been detected near the visibility horizon of the Vaidya accreting black hole, in which the metric differs qualitatively from the Schwarzschild metric and cannot be represented as a small perturbation. It has been shown that, in this case, a single set of diagonal coordinates (t, r) is insufficient to cover the entire range of initial coordinates (v, r) outside the visibility horizon; at least three sets of diagonal coordinates are required, the domains of which are separated by singular surfaces on which the metric components have singularities (either g ₀₀ = 0 or g ₀₀ = ∞). The energy–momentum tensor diverges on these surfaces; however, the tidal forces turn out to be finite, which follows from an analysis of the deviation equations for geodesics. Therefore, these singular surfaces are exclusively coordinate singularities that can be referred to as false fire-walls because there are no physical singularities on them. We have also considered the transformation from the initial coordinates to other diagonal coordinates (η, y), in which the solution is obtained in explicit form, and there is no energy–momentum tensor divergence.     

New numerical method for solving time-harmonic Maxwell equations with strong singularity

In this paper we introduce a notion of R_ν-generalized solution to time-harmonic Maxwell equations with strong singularity in a 2D nonconvex polygonal domain. We develop a new weighted edge FEM. Results of numerical experiments prove the efficiency of this method.     

Calculation of the viscosity of SU(2) gluodynamics with the lattice simulation

The viscosity of SU(2) gluodynamics within the simulation of the lattice quantum chromodynamics at a temperature of T/T _c = 1.2 has been calculated with the Kubo formula relating the viscosity to the spectral function of the correlation function of the energy-momentum tensor. The correlation function of the energy-momentum tensor has been calculated using the numerical simulation of the lattice SU(2) gluodynamics on supercomputers.     

Shear viscosity of superfluid He-A1 at low temperatures

The shear viscosity tensor of the A₁-phase of superfluid ³He is calculated at low temperatures and melting pressure, by using the Boltzmann equation approach. The two normal and superfluid components take part in elements of the shear viscosity tensor differently. The interaction between normal and Bogoliubov quasiparticles in the collision integrals is considered in the binary, decay, and coalescence processes. We show that the elements of the shear viscosities η_xy, η_xz, and η_zz are proportional to (T/T_c)⁻². The constant of proportionality is in nearly good agreement with the experimental results of Roobol et al.