A Vertex-Centered and Positivity-Preserving Finite Volume Scheme for Two-Dimensional Three-Temperature Radiation Diffusion Equations on General Polygonal Meshes

Two-dimensional three-temperature (2-D 3-T) radiation diffusion equations are widely used to approximately describe the evolution of radiation energy within a multimaterial system and explain the exchange of energy among electrons, ions and photons. In this paper, we suggest a new positivity-preserving finite volume scheme for 2-D 3-T radiation diffusion equations on general polygonal meshes. The vertex unknowns are treated as primary ones for which the finite volume equations are constructed. The edge-midpoint and cell-centered unknowns are used as auxiliary ones and interpolated by the primary unknowns, which makes the final scheme a pure vertex-centered one. By comparison, most existing positivity-preserving finite volume schemes are cell-centered and based on the convex decomposition of the co-normal. Here, the co-normal decomposition is not convex in general, leading to a fixed stencil of the flux approximation and avoiding a certain search algorithm on complex grids. Moreover, the new scheme effectively alleviates the numerical heat-barrier issue suffered by most existing cell-centered or hybrid schemes in solving strongly nonlinear radiation diffusion equations. Numerical experiments demonstrate the second-order accuracy and the positivity of the solution on various distorted grids. For the problem without analytic solution, the contours of the numerical solutions obtained by our scheme on distorted meshes accord with those on smooth quadrilateral meshes.     

二维三温热传导方程的并行自适应多重网格算法求解

3-T heat conduct equation including electron, ion and photon (radiation) temperatures can be used to approximately describe the energy broadcast across multimedia for radial flow dynamics and discover the energy swapping among photon,electron and ion. Owing to the strong nonlinear diffusion coefficients and energy swapping coefficients and strong discontinuous coefficients across media interfaces,this equation is difficult to be solved with high numerical resolution. Based on the parallel adaptive multigrid software framework UG on 2-D unstructured grid, this paper successfully solved such equation with high resolution by combining the finite volume implicit discretization scheme and parallel adaptive multigrid algorithm, and gained much significant results.     

Eigenvalues of kinematical conservation laws (KCL) based 3-D weakly nonlinear ray theory (WNLRT)

System of kinematical conservation laws (KCL) govern evolution of a curve in a plane or a surface in space, even if the curve or the surface has singularities on it. In our recent publication [K.R. Arun, P. Prasad, 3-D kinematical conservation laws (KCL): evolution of a surface in R³-in particular propagation of a nonlinear wavefront, Wave Motion 46 (2009) 293-311] we have developed a mathematical theory to study the successive positions and geometry of a 3-D weakly nonlinear wavefront by adding an energy transport equation to KCL. The 7 × 7 system of equations of this KCL based 3-D weakly nonlinear ray theory (WNLRT) is quite complex and explicit expressions for its two nonzero eigenvalues could not be obtained before. In this short note, we use two different methods: (i) the equivalence of KCL and ray equations and (ii) the transformation of surface coordinates, to derive the same exact expressions for these eigenvalues. The explicit expressions for nonzero eigenvalues are important also for checking stability of any numerical scheme to solve 3-D WNLRT.     

Remarks on the Regularity to 3-D Ideal Magnetohydrodynamic Equations

In this paper we are interested in the sufficient conditions which guarantee the regularityof solutions of 3-D ideal magnetohydrodynamic equations in the arbitrary time interval [0,T].Fivesufficient conditions are given.Our results are motivated by two main ideas:one is to control theaccumulation of vorticity alone;the other is to generalize the corresponding geometric conditions of3-D Euler equations to 3-D ideal magnetohydrodynamic equations.     

解线性抛物方程的一类新格式

解线性抛物方程的一类新格式孙志忠（中国科学院计算中心）ＡＮＥＷＣＬＡＳＳＯＦＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＳＦＯＲＬＩＮＥＡＲＰＡＲＡＢＯＬＩＣＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳ￥ＳｕｎＺｈｉ－ｚｈｏｎｇ（ＣｏｍｐｕｔｉｎｇＣｅｎｔｅｒ，Ａ...     

Approximate global convergence and quasireversibility for a coefficient inverse problem with backscattering data

A numerical method possessing the approximate global convergence property is developed for a 3-D coefficient inverse problem for hyperbolic partial differential equations with backscattering data resulting from a single measurement. An important part of this technique is the quasireversibility method. An approximate global convergence theorem is proved. Results of two numerical experiments are presented. Bibliography: 46 titles. Illustrations: 2 figures.     

Pointwise structure of a radiation hydrodynamic model in one-dimension

In this paper, we study the nonlinear stability and the pointwise structure around a constant equilibrium for a radiation hydrodynamic model in 1-dimension, in which the behavior of the fluid is described by a full Euler equation with certain radiation effect. It is well-known that the classical solutions of the Euler equation in 1-D may blow up in finite time for general initial data. The global existence of the solution in this paper means that the radiation effect stabilizes the system and prevents the formation of singularity when the initial data is small. To study the precise effect of the radiation in this model, we also treat the pointwise estimates of the solution for the original nonlinear problem by combining the Green's function for the linearized radiation hydrodynamic equations with the Duhamel's principle. The result in this paper shows that the pointwise structure of this model is similar to that of full Navier-Stokes equations in 1-D.     

非定常 N-S 方程的非线性 Galerkin 算法及其误差估计

本文给出了二维非定常N-S方程的三种数值格式,其中空间变量用谱非线性Galerkin算法进行离散,时间变量用有限差分离散,并研究了这些格式数值解的逼近精度．最后,给出了部分数值计算结果．     

非定常沿岸波流的有限单元分析(Ⅰ)──数学模型

本文提出了一个预测由风和波流相互作用所产生的非定常近海波流的数学模型,其基础方程组中除了传统的连续性方程、动量方程、能量方程外,还引进了色散方程及折射方程。同时,在基础方程组中本文还分析了横向剪应力、风的应力、辐射应力及海底摩擦应力的作用,因而期望它能较全面和较精确地反映波流相互作用的规律。在(Ⅱ)中将给出二步显含有限单元法及几个计算例子。     

High-order ADI method for stream-function vorticity equations

Many recent works have demonstrated the efficiency of high-order compact (HOC) difference schemes on the stream-function and vorticity formulation of 2-D incompressible Navier-Stokes equations. HOC discretizations induce cross spatial derivatives which are treated explicitly in most ADI schemes. Recently, Karaa and Zhang proposed a fourth-order ADImethod for solving convection-diffusion problems efficiently. In this work, we extend this method to the solution of incompressible Navier-Stokes equations. The driven flow in a square cavity is used as a model problem and numerical results are compared with other results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic modelling and input-energy comparison for the elevator system

The elevator system driven by a permanent magnet synchronous motor (PMSM) is studied in this paper. The mathematical model of the elevator system includes the electrical and mechanical equations, and the dimensionless forms are derived for the purpose of practicable upward and downward movement. In this paper, the trapezoidal, cycloidal, five-degree (5-D) and seven-degree (7-D) polynomial and industry trajectories are designed and compared numerically in various motion and the absolute input energies. From numerical simulations, it is found that the trapezoidal trajectory consumes the minimum energy; the 7-D polynomial trajectory consumes the maximum one. The less end-point constraints are required, the less energy is consumed. Finally, the proposed sliding mode controller (SMC) is employed to demonstrate the robustness and well tracking control performance numerically.     

Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion

We prove the global existence and the decay estimates of small smooth solution for the 2-D MHD equations without magnetic diffusion. This confirms the numerical observation that the energy of the MHD equations is dissipated at a rate independent of the ohmic resistivity.     

Modelling dispersion in laminar and turbulent flows in an open channel based on centre manifolds using 1D-IRBFN method

Centre manifold method is an accurate approach for analytically constructing an advection–diffusion equation (and even more accurate equations involving higher-order derivatives) for the depth-averaged concentration of substances in channels. This paper presents a direct numerical verification of this method with examples of the dispersion in laminar and turbulent flows in an open channel with a smooth bottom. The one-dimensional integrated radial basis function network (1D-IRBFN) method is used as a numerical approach to obtain a numerical solution for the original two-dimensional (2-D) advection–diffusion equation. The 2-D solution is depth-averaged and compared with the solution of the 1-D equation derived using the centre manifolds. The numerical results show that the 2-D and 1-D solutions are in good agreement both for the laminar flow and turbulent flow. The maximum depth-averaged concentrations for the 1-D and 2-D models gradually converge to each other, with their velocities becoming practically equal. The obtained numerical results also demonstrate that the longitudinal diffusion can be neglected compared to the advection.     

Haar wavelets method for solving Poisson equations with jump conditions in irregular domain

In this paper, Haar wavelets method is used to solve Poisson equations in the presence of interfaces where the solution itself may be discontinuous. The interfaces have jump conditions which need to be enforced. It is critical for the approximation of the boundaries of the irregular domain. An irregular domain can be treated by embedding the domain into a rectangular domain and Poisson equation is solved by using Haar wavelets method on the rectangle. Firstly, we demonstrate this method in the case of 1-D region, then we consider the solution of the Poisson equations in the case of 2-D region. The efficiency of the method is demonstrated by some numerical examples.     

Solving generalized complex-symmetric eigenvalue problems arising from resonant MEMS simulations with PETSc

In the operation of high frequency resonators in micro electromechanical systems(MEMS), there is a strong need to be able to accurately determine the energy loss rates or alternatively the quality of resonance. The resonance quality is directly related to a designers ability to assemble a high fidelity system response for signal filtering, for example. This has implications on robustness and quality of electronic communication and also strongly influences overall rates of power consumption in such devices – i.e. battery life. An index of the quality of resonance Q is inversely proportional to the amount of energy loss existing in the system, and can be defined in terms of the complex-valued eigenvalues ω of the system as, . In this presentation, we examine methods for accurately simulating energy loss using the direct modeling of acoustic radiation via perfectly matched layer (PML) technology for 3-D structures, which results in complex-symmetric mass and stiffness matrices. Thus it is crucial that one is able to accurately compute eigenvalues of a complex-symmetric system. 3-D modeling increases the number of degrees of freedom and complexity in obtaining the solution, resulting in a requirement for more computing power and memory which can only be supplied by parallel computing methods. The simulations are conducted through the MEMS simulation software HiQLab in combination with the parallel numerical libraries PETSc and Trilinos. A modified smoothed aggregation algebraic multigrid method for solving a linear system of equations is combined with a Jacobi-Davidson method to solve for desired eigenvalues. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

THE VANISHING PRESSURE LIMIT OF SOLUTIONS TO THE SIMPLIFIED EULER EQUATIONS FOR ISENTROPIC FLUIDS

In this paper,the Riemann problem of the 1-D reduced model for the 2-D Euler equations is considered and the Riemann solutions are obtained.It is proved that,as the pressure vanishes,they converge to two kinds of Riemann solutions to the 1D reduced model for the 2-D transport equations:one contains δ-shocks,the other contains vacuum.     

A higher-order Godunov method for modeling finite deformation in elastic-plastic solids

In this paper we develop a first-order system of conservation laws for finite deformation in solids, describe its characteristic structure, and use this analysis to develop a second-order numerical method for problems involving finite deformation and plasticity. The equations of mass, momentum, and energy conservation in Lagrangian and Eulerian frames of reference are combined with kinetic equations of state for the stress and with caloric equations of state for the internal energy, as well as with auxiliary equations representing equality of mixed partial derivatives of the deformation gradient. Particular attention is paid to the influence of a curl constraint on the deformation gradient, so that the characteristic speeds transform properly between the two frames of reference. Next, we consider models in rate-form for isotropic elastic-plastic materials with work-hardening, and examine the circumstances under which these models lead to hyperbolic systems for the equations of motion. In spite of the fact that these models violate thermodynamic principles in such a way that the acoustic tensor becomes nonsymmetric, we still find that the characteristic speeds are always real for elastic behavior, and essentially always real for plastic response. These results allow us to construct a second-order Godunov method for the computation of three-dimensional displacement in a one-dimensional material viewed in the Lagrangian frame of reference. We also describe a technique for the approximate solution of Riemann problems in order to determine numerical fluxes in this algorithm. Finally, we present numerical examples of the results of the algorithm.     

On the existence of the weak solution with local energy inequality to the 3-D inhomogeneous incompressible Navier–Stokes equations

In this paper, we show there exists a weak solution to the 3-D inhomogeneous incompressible Navier–Stokes equations satisfying in addition the local energy inequality. The same result in the homogeneous incompressible case was described in [1, Theorem 3.2].     

A Conservative Space-time Mesh Refinement Method for the 1-D Wave Equation. Part II: Analysis

Summary. We introduced in [2] a new method for space-time refinement for the 1-D wave equation. This method is based on the conservation of a discrete energy through two different discretization grids which guarantees the stability of the scheme. In this second part, we analyse the accuracy of this scheme in a detailed way by means of a plane wave analysis and numerical experiments that permit us to point out spurious numerical phenomena and explain how to control them. Mathematics Subject Classification (2000):65N12