The influence of numerical diffusion on the solution of the radiative transfer equations

We investigate the explicit numerical solution strategies of multi-dimensional radiative transfer equations which are commonly used, e.g., to determine the radiation emerging from astrophysical objects surrounded by absorbing and scattering matter. For explicit grid solvers, we identify numerical diffusion as a severe source of error in first-order discretization schemes, underestimated in former work about radiative transfer. Using the simple example of a beam propagating through vacuum, we illustrate the influence of the diffusion on the solution and discuss various techniques to reduce it. In view of the large required storage for implicit solvers, we propose to use second-order explicit grid techniques to solve 3D radiative transfer problems.     

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Thomas-Fermi model can be applied to describe state of electrons for mixtures. A method to solve the mixture Thomas-Fermi equation is proposed. With the proper initial test solution and step length, this method searches the solution in a way that reduces the steps by half, which can get solutions with various densities and temperatures fast.     

On numerical solution of the shallow water equations with chemical reactions on icosahedral geodesic grid

The shallow water equations coupled to the set of reaction–advection–diffusion equations are discretized on a geodesic icosahedral mesh using the finite volume technique. The method of solution of this coupled system is based on the principle of semi-discretization. The algorithm is mass conserving and stable for advection with the Courant numbers up to 2.7. The important part of the methodology is the optimization of the node positions in the icosahedral grid. It is shown that a slight adjustment of the mesh is instrumental in improving the accuracy of the numerical approximation. The convergence of the approximation of the differential operators is evaluated and compared to the data published in the literature. Numerical tests performed with the shallow water solver include two advection experiments, steady and unsteady zonal balanced flow, mountain flow, and the Rossby wave. The mountain flow and the Rossby wave cases are used to test the transport properties of the method in the case of both passive and reactive scalar fields. The investigation of essential numerical characteristics of the method is concluded by the simulation of an unstable zonal jet. The numerical simulation is performed using the set of shallow water equations without dissipation as well as with the viscosity term added to the momentum equation. Results show that the behavior of the model is consistent with both the literature published on the subject and the general empirical evidence.     

On the phenomenological theory of diffusion

A phenomenological theory of diffusion is given along the lines of the treatment of De Groot and Mazur¹). Simple relations between the diffusion matrices of the various kinds of composition variables are established. In linear approximation all kinds of composition variables obey Fick's second law. The special role of the mean volume velocity is discussed and attention is given to Onsager coefficients and mobilities (friction coefficients).     

High-order explicit-implicit numerical methods for nonlinear anomalous diffusion equations

In this paper, the high-order finite difference/element methods for the nonlinear anomalous diffusion equations of subdiffusion and superdiffusion are developed, where the high-order finite difference methods are used to approximate the time-fractional derivatives and the finite element methods are used in the spatial domain. The stability and error estimates are proved for both cases of superdiffusion and subdiffusion. Numerical examples are provided to confirm the theoretical analysis.     

The Huygens subgridding for the numerical solution of the Maxwell equations

The Huygens subgridding (HSG) is a subgridding technique developed for the numerical solution of the Maxwell equations. It relies on the theoretical equivalence of any physical volume with two or more fictitious volumes connected by equivalent currents. The application of this concept to the finite-difference time-domain (FDTD) method has been previously published in the one dimensional and two dimensional cases. In this paper the HSG is extended to the general three dimensional case, the exchange of the electromagnetic energy between the two FDTD grids is investigated theoretically, and some modifications to the HSG algorithm are presented with the objective of simplifying its implementation.     

Equivalence of integral equations in the molecular theory of fluids

It is proven that, under physically reasonable conditions, the correlation functions satisfying the BBGKY equations for an infinite system are also solutions of the Mayer-Montroll and Kirkwood-Salsburg equations. The relation between these correlation functions and the probability distributions for finding a fixed number of particles in a given finite region of an infinite system is investigated. The Gibbsian nature of these probability distributions is shown to depend on the range of the intermolecular forces.     

New approach for numerical solution of configuration-space Faddeev equations

A new computational scheme for solving the bound state configuration-space Faddeev equations is applied. The scheme is based on the spline-approximation and the adiabatic limit of Faddeev equations. An ordering of variables being in agreement with the limit was chosen. As a result the matrix of the eigenvalue problem has a sparse block structure. Calculations of the bound states of µpp, µdd, µtt mesic molecules and ¯pdd, ¯ptt antiprotonic ones, were performed. To check the method, calculations of the binding energies for such systems as the positronium ion Ps_–,³H and³He were carried out. The results are compared with the best results of other authors.     

A numerical method for the solution of the bidomain equations in cardiac tissue

A numerical scheme for efficient integration of the bidomain model of action potential propagation in cardiac tissue is presented. The scheme is a mixed implicit-explicit scheme with no stability time step restrictions and requires that only linear systems of equations be solved at each time step. The method is faster than a fully explicit scheme and there is no increase in algorithmic complexity to use this method instead of a fully explicit method. The speedup factor depends on the timestep size, which can be set solely on the basis of the demands for accuracy. (c) 1998 American Institute of Physics.     

On the solution of constrained hartree-fock-bogolyubov equations

A method is described to solve constrained HFB-equations without introduction of a Lagrangian multiplier. To demonstrate the versatility of the method rotational bands in¹⁶⁴Er and¹⁶⁸Yb are calculated.     

On Zamolodchikov's solution of the tetrahedron equations

The tetrahedron equations arise in field theory as the condition for theS-matrix in 2+1-dimensions to be factorizable, and in statistical mechanics as the condition that the transfer matrices of three-dimensional models commute. Zamolodchikov has proposed what appear (from numerical evidence and special cases) to be non-trivial particular solutions of these equations, but has not fully verified them. Here it is proved that they are indeed solutions.     

Generalization of Sturm-Liouville theory to a system of ordinary differential equations with Dirac type spectrum

The Sturm-Liouville theory is generalized to Dirac-equation-like systems of ordinary differential equations. It is shown how the comparison theorem and conversion to integral equations can be generalized.Dedicated to Walter Thirring on his 60^th birthday     

On the solution of the number-projected Hartree-Fock-Bogolyubov equations

The numerical solution of the recently formulated number-projected Hartree-Fock-Bogolyubov (HFB) equations is studied in an exactly solvable cranked-deformed shell-model Hamiltonian. It is found that the solution of these number-projected equations involves similar numerical effort as that of bare HFB. We consider that this is significant progress in the mean-field studies of quantum many-body systems. The results of the projected calculations are shown to be in almost complete agreement with the exact solutions of the model Hamiltonian. The phase transition obtained in the HFB theory as a function of the rotational frequency is shown to be smeared out with the projection.     

On the numerical solution of the evolution equation in QCD

We discuss in some detail the algorithm for the numerical solution of QCD evolution equations in the next-to-leading order. Relative merits and shortcomings of this method are presented and compared with those of other available approaches. Effective control of the accuracy of obtained results is stressed.     

麦克斯韦积分方程的一种数值解法

直接从麦克斯韦积分方程出发,采用FIT算法,模拟电磁场的分布情况。     

On a partial solution of the diffusion equation

The process is considered of establishing the equilibrium spatial distribution of the concentration of particles in a one-dimensional bounded space region, subjected to a constant force normal to impermeable region boundaries. This process is described by the solution of the third boundary-value problem with homogeneous boundary conditions for the two-dimensional parabolic equation. It is shown that the found solution to the seemingly well-known problem of mathematical physics, but being of great importance in applications, cannot be obtained using theGreen’s function of this problem, known in the literature.     

On the classes of integral equations nonderivable by graph theory means: An improved molecular theory for classical liquids in equilibrium

Starting from the definition of molecular distribution functions a new hierarchy of integral equations for the equilibrium classical liquids is derived. An outline is given of why this hierarchy cannot be derived by graph theory means. In the theory, the sum of elementary diagrams is given explicitly as a functional of the higher order correlation functions. Analysing this functional the most general truncation scheme within the framework of the first order theories is presented. The pairwise additivity of the intermolecular potential is not assumed.     

On a certain solution of the Einstein-Maxwell equations

A connection is established between a certain exterior solution of the Einstein-Maxwell equations and an interior cylindrically symmetric stationary solution found earlier by one of the authors.