Explicit-implicit difference scheme for the joint solution of the radiative transfer and energy equations by the splitting method

High-order accurate explicit and implicit conservative predictor-corrector schemes are presented for the radiative transfer and energy equations in the multigroup kinetic approximation solved together by applying the splitting method with respect to physical processes and spatial variables. The original system of integrodifferential equations is split into two subsystems: one of partial differential equations without sources and one of ordinary differential equations (ODE) with sources. The general solution of the ODE system and the energy equation is written in quadratures based on total energy conservation in a cell. A feature of the schemes is that a new approximation is used for the numerical fluxes through the cell interfaces. The fluxes are found along characteristics with the interaction between radiation and matter taken into account. For smooth solutions, the schemes approximating the transfer equations on spatially uniform grids are second-order accurate in time and space. As an example, numerical results for Fleck’s test problems are presented that confirm the increased accuracy and efficiency of the method.     

大气环流闭合方程组的总能量守恒格式及全球预报误差的严格估计

本文讨论大气环流闭合方程组,由于同时考虑了热传导效应,内摩擦效应及表达动能向内能转化的耗散项,因此符合总能量守恒律,文中对这一方程组建立了加权平均守恒型差分格式,并证明当选择最优参数时,它满足离散形式的总能量守恒律,通常的二次守恒格式是其次优的情况,文中还综合应用了Jessen不等式,Hardy不等式等等,从而严格证明了在一定条件下,存在t₀>0,当t     

The completely conservative difference schemes for the nonlinear Landau-Fokker-Planck equation

Conservativity and complete conservativity of finite difference schemes are considered in connection with the nonlinear kinetic Landau-Fokker-Planck equation. The characteristic feature of this equation is the presence of several conservation laws. Finite difference schemes, preserving density and energy are constructed for the equation in one- and two-dimensional velocity spaces. Some general methods of constructing such schemes are formulated. The constructed difference schemes allow us to carry out the numerical solution of the relaxation problem in a large time interval without error accumulation. An illustrative example is given.     

A kinetic energy preserving DG scheme applied to mechanical FSI

In the context of mechanical fluid-structure interaction (FSI) comprising moving or deforming structures, fluid discretizations need to cope with time-dependent fluid domains and resulting grid deformations in addition to the general challenges regarding e.g. boundary layers and turbulent phenomena. Recent approaches in the simulation of compressible turbulent flow are based on so-called split forms of conservation laws to guarantee the preservation of secondary physical quantities such as kinetic energy. For the simulation of turbulent flows, this often leads to a better representation of the kinetic energy spectrum. Initially, kinetic energy preserving(KEP) DG schemes have been constructed on Gauss-Legendre-Lobatto(GLL) nodes containing the interval end points, however, KEP DG schemes based on the classical Gauss-Legendre(GL) nodes are potentially more accurate and may be also more efficient than its GLL variant for certain applications. In this work, the KEP-DG schemes both on GL and GLL nodes are applied to the classical moving piston test case via an ALE formulation on moving fluid grids showing a more accurate frequency representation of the structure displacement in case of GLL nodes. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mathematical model of a kinetic boundary layer

The two-dimensional (plane) problem of a hypersonic kinetic boundary layer developing on a thin body in the case of a homogeneous polyatomic gas flow with no dissociation or electron excitation is considered assuming that energy exchange between translational and internal molecular degrees of freedom is easy. (The approximation of a hypersonic kinetic boundary layer arises from the kinetic theory of gases and, within the thin-layer model, takes into account the strong nonequilibrium of the hypersonic flow with respect to translational and internal degrees of freedom of the gas particles.) A method is proposed for constructing the solution of the given kinetic problem in terms of a given solution of an equivalent well-studied classical Navier-Stokes hypersonic boundary layer problem (which is traditionally formulated on the basis of the Navier-Stokes equations).     

朝向产热或吸热伸展平面的MHD驻点流动

分析了微极流体朝向加热伸展平面的磁流体动力学(MHD)驻点流动,考虑了粘性耗散和内部产热/吸热对流动的影响.讨论了指定表面温度(PST)和指定热通量(PHF)两种情况,采用同伦分析方法(HAM)求解边界层流动和能量方程.通过图表的显示,研究了感兴趣物理量的变化.注意到高伸展参数时解的存在与外加应用磁场密切相关.     

Qualitatively stability of nonstandard 2-stage explicit Runge–Kutta methods of order two

When one solves differential equations, modeling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. Nonstandard finite differences (NSFDs) schemes can improve the accuracy and reduce computational costs of traditional finite difference schemes. In addition NSFDs produce numerical solutions which also exhibit essential properties of solution. In this paper, a class of nonstandard 2-stage Runge–Kutta methods of order two (we call it nonstandard RK2) is considered. The preservation of some qualitative properties by this class of methods are discussed. In order to illustrate our results, we provide some numerical examples.     

Kinetic Magnetohydrodynamics of Neutron Matter

Equations for the magnetohydrodynamics of neutron matter are derived within a microscopic approach based on the Landau theory of a Fermi liquid. Along with the strong short-distance nuclear interactions, the equations account for the weak long-distance magnetic interactions. Applications of the derived magnetohydrodynamic equations to the theory of shock waves in neutron matter are discussed.     

Conservative scheme for the thermodiffusion Stefan problem

We suggest further development of the principle of conservation for problems with moving boundaries. Using the problem of phase transitions in binary compounds as an example, we demonstrate a technique for constructing divergence and nondivergence finite difference schemes guaranteeing that the energy and mass conservation laws hold in the discrete model. In a class of front tracking methods, we prove the equivalence of the approach based on the use of moving grids with that based on a dynamic change of variables which permits one to solve the problem on a fixed grid.     

Monotone Finite Difference Schemes for Nonlinear Systems with Mixed Quasi-Monotonicity

Monotone finite difference schemes are proposed for nonlinear systems with mixed quasi-monotonicity. Two monotone iteration processes for the corresponding discrete problems are presented, which converge monotonically to the quasi-solutions of the discrete problems. The limits are the exact solutions under some conditions. A monotone finite difference scheme on uniform mesh with the accuracy of fourth order is constructed. The numerical results coincide with theoretical analysis.     

Unsteady Magnetohydrodynamic Flows

It is easy enough to deduce from the exact solution for a basicunsteady magnetohydrodynamic channel flow that there are twotime scales present. The possible existence of two time scalesenables one to formulate a method of constructing approximatesolutions for several flow problems. The approximation schemeemployed here has the advantage over the more usual methodsof boundary layer analysis in that it yields for each physicalquantity a single representation which is valid for all times.Moreover, it is shown that the error involved in using thisapproximate solution is suitably small.     

Completely conservative difference schemes for dynamic problems of linear elasticity and viscoelasticity

On the basis of a mixed statement (velocity-strain), we complete the development of a general theory of completely conservative adjoint-coordinated difference schemes for dynamic problems of linear elasticity and viscoelasticity. In particular, our explicitly solvable discrete models permit controlling the total energy imbalance and have the same parallelization degree as the conventional explicit schemes.     

Heat and mass transfer in MHD non-Darcian flow of a micropolar fluid over a stretching sheet embedded in a porous media with non-uniform heat source and thermal radiation

A mathematical analysis has been carried out to study magnetohydrodynamic boundary layer flow, heat and mass transfer characteristic on steady two-dimensional flow of a micropolar fluid over a stretching sheet embedded in a non-Darcian porous medium with uniform magnetic field. Momentum boundary layer equation takes into account of transverse magnetic field whereas energy equation takes into account of Ohmic dissipation due to transverse magnetic field, thermal radiation and non-uniform source effects. An analysis has been performed for heating process namely the prescribed wall heat flux (PHF case). The governing system of partial differential equations is first transformed into a system of non-linear ordinary differential equations using similarity transformation. The transformed equations are non-linear coupled differential equations which are then linearized by quasi-linearization method and solved very efficiently by finite-difference method. Favorable comparisons with previously published work on various special cases of the problem are obtained. The effects of various physical parameters on velocity, temperature, concentration distributions are presented graphically and in tabular form.     

Equilibrium of a Degenerate Gas of Nucleons and Electrons in a Strong Magnetic Field

A model of a degenerate ideal gas of nucleons and electrons in a superstrong magnetic field is used to describe the state of matter in the central region of a strongly magnetized neutron star. The influence of a constant uniform superstrong magnetic field on the equilibrium conditions and the equation of state for the degenerate gas of neutrons, protons, and electrons is investigated in the framework of this model. The contribution determined by the interaction of the anomalous magnetic moments of the fermions with the magnetic field is taken into account. The influence of the superstrong magnetic field on the process of gravitational collapse of a magnetized neutron star is discussed under the assumption that the central region of the star consists mostly of degenerate neutrons. We show that if the densities of electrons, protons, and neutrons are relatively low depending on the field strength, the fermion gases in a superstrong uniform magnetic field become totally polarized with respect to the spin. We discuss the possibility of spontaneous magnetization occurring in a system of degenerate neutrons where the exchange interaction effects are taken into account.     

Hybrid LES-RANS Modeling of Complex Turbulent Flows

The paper describes a state-of-the-art hybrid LES-URANS method for the simulation of complex internal and external turbulent flows. Relying on a unified LES-URANS approach with a soft interface the methodology is designed for wall-bounded non-equilibrium flows. The unsteady Reynolds-averaged Navier-Stokes (URANS) mode within the hybrid approach is taken into account by an explicit algebraic Reynolds stress model (EARSM), which guarantees an appropriate representation of the anisotropic near-wall turbulence. All non-closed terms in the transport equation of the turbulent kinetic energy are modeled by enhanced formulations using the EARSM (production and diffusion term) and the splitting of the dissipation rate into a homogeneous and an inhomogeneous contribution. The former is expressed analytically by a Taylor series expansion of the homogeneous lateral Taylor microscale in the vicinity of the wall guaranteeing the correct asymptotic behavior. The latter is incorporated into the diffusion term. The interface location between the large-eddy simulation (LES) mode and the URANS mode is determined automatically on-the-fly based on the modeled turbulent kinetic energy and the distance to the wall. For transitional (external) flows an additional dynamic transition criterion is applied which determines the laminar and the turbulent flow regimes and is combined with the existing interface criterion. An internal flow over a periodic arrangement of hills and an external flow past a SD7003 airfoil with a laminar separation bubble are taken into account for a detailed evaluation of the method. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New Conservative Schemes for Regularized Long Wave Equation

In this paper, two finite difference schemes are presented for initial-boundary value problems of Regularized Long-Wave(RLW) equation. They all have the advantages that there are discrete energies which are conserved. Convergence and stability of difference solutions with order O(h2 τ2) are proved in the energy norm. Numerical experiment results demonstrate the effectiveness of the proposed schemes.     

On splitting‐based mass and total energy conserving arbitrary order shallow‐water schemes

A new method for numerical solution to the shallow‐water equations is suggested. The method allows constructing a family of finite difference schemes of different approximation order that conserve the mass and the total energy. Our approach is based on the method of splitting, and unlike others it permits to derive conservative numerical schemes after discretizing all the partial derivatives, both spatial and temporal. The schemes thus appear to be fully discrete, both in time and in space. Besides, due to a simple structure of the matrices appeared therewith, the method provides essential benefits in the computational cost of solution and is easy‐to‐implement in the Cartesian and spherical geometries. Numerical results confirm functionality and efficiency of the developed method. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Formulation and Numerical Implementation of Ferroelectrics at Large Strains

In recent years, an increasing interest has been shown in functional materials such as ferroelectric polymers. For such materials, viscous effects and electric polarizations cause hysteresis phenomena accompanied with possibly large remanent strains and rotations. Ferroelectric polymers have many attractive characteristics. They are light, inexpensive, fracture tolerant, and pliable. Furthermore, they can be manufactured into almost any conceivable shape and their properties can be tailored to suit a broad range of requirements. In this work, continuous and discrete variational formulations are exploited for the treatment of the non-linear dissipative response of ferroelectric polymers under electrical loading. The point of departure is a general internal variable formulation that determines the hysteretic response of this class of materials in terms of an energy storage and a rate-dependent dissipation function. The ferroelectric constitutive assumptions, which account for specific problems arising in the geometric nonlinear setting, are discussed. With regard to the choice of the internal variables, a critical factor is the kinematic assumptions. Here, we propose the multiplicative decomposition of the local deformation gradient into reversible and remanent parts, where the latter is characterized by a metric tensor. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Conservative arbitrary order finite difference schemes for shallow-water flows

The classical nonlinear shallow-water model (SWM) of an ideal fluid is considered. For the model, a new method for the construction of mass and total energy conserving finite difference schemes is suggested. In fact, it produces an infinite family of finite difference schemes, which are either linear or nonlinear depending on the choice of certain parameters. The developed schemes can be applied in a variety of domains on the plane and on the sphere. The method essentially involves splitting of the model operator by geometric coordinates and by physical processes, which provides substantial benefits in the computational cost of solution. Besides, in case of the whole sphere it allows applying the same algorithms as in a doubly periodic domain on the plane and constructing finite difference schemes of arbitrary approximation order in space. Results of numerical experiments illustrate the skillfulness of the schemes in describing the shallow-water dynamics.     

THE UNCONDITIONAL STABLE DIFFERENCE METHODS WITH INTRINSIC PARALLELISM FOR TWO DIMENSIONAL SEMILINEAR PARABOLIC SYSTEMS

In this paper we are going to discuss the difference schemes with intrinsic parallelism for the boundary value problem of the two dimesional semilinear parabolic systems.The unconditional stability of the general finite difference schemes with intrinsic parallelism is justified in the sense of the continuous dependence of the discrete vector solution of the difference schemes on the discrete data of the original problems in the discrete W2^(2,1) norms.Then the uniqueness of the discrete vector solution of this difference scheme follows as the consequence of the stability.