On a hyperbolic conservation law of electron transport in solid materials for electron probe microanalysis

The prediction of X-ray intensities based on the distribution of electrons throughout solid materials is essential to solve the inverse problem of quantifying the composition of materials in electron probe microanalysis (EPMA) [3]. We present a hyperbolic conservation law for electron transport in solid materials and investigate its validity under conditions typical for EPMA experiments. The conservation law is based on the time-stationary Boltzmann equation for binary electron-atom scattering. We model the energy loss of the electrons with a continuous slowing-down approximation. A first order moment approximation with respect to the angular variable is discussed. We propose to use a minimum entropy closure to derive a system of hyperbolic conservation laws, known as the M1 model [11]. A finite volume scheme for the numerical solution of the resulting equations is presented. Important numerical aspects of the scheme are discussed, such as bounds for the finite propagation speeds, as well as difficulties arising fromspatial discontinuities in thematerial coefficients and the scaling of the characteristic velocities with the stopping power of the electrons.We compare the accuracy and performance of the numerical solution of the hyperbolic conservation law to Monte Carlo simulations. The results indicate a reasonable accuracy of the proposed method and showthat compared to the MonteCarlo simulation the finite volume scheme is computationally less expensive.     

Numerical treatment in resonant regime for shallow water equations with discontinuous topography

This paper deals with numerical treatments for the shallow water equations with discontinuous topography when the initial data belong to both supersonic region and subsonic region. This kind of data are present in both engineering and rivers, but they are not always well-treated in existing schemes. Our goal is to improve the well-balanced scheme constructed earlier in our work by introducing a computing corrector into the construction of the scheme. First, a further study in the construction of the well-balanced scheme reveals that the errors could make the approximate states near the critical surface that ought to be in one side of the critical surface fall into the other side. This qualitative change, though small, may cause much larger errors following stationary hydraulic jumps formed from these approximate states due to the jump of the bottom. Then, we introduce a corrector in the computing algorithm that selects the equilibrium states in the construction of the well-balanced scheme such that the approximate stationary hydraulic jumps always remain in the right region. Numerical tests show that the well-balanced method using an underlying numerical flux such as Lax–Friedrichs flux, FORCE, GFORCE, or Roe fluxes can approximate very well the exact solution even when the initial data are on both supercritical region and subcritical region.     

Fixed-point fast sweeping weighted essentially non-oscillatory method for multi-commodity continuum traffic equilibrium assignment problem

This work presents a fixed-point fast sweeping weighted essentially non-oscillatory method for the multi-commodity continuum traffic equilibrium assignment problem with elastic travel demand. The commuters’ origins (i.e. home locations) are continuously dispersed over the whole city with several highly compact central business districts. The traffic flows from origins to the same central business district are considered as one commodity. The continuum traffic equilibrium assignment model is formulated as a static conservation law equation coupled with an Eikonal equation for each commodity. To solve the model, a pseudo-time-marching approach and a third order finite volume weighted essentially non-oscillatory scheme with Lax–Friedrichs flux splitting are adopted to solve the conservation law equation, coupled with a third order fast sweeping numerical method for the Eikonal equation on rectangular grids. A fixed-point fast sweeping method that utilizes Gauss–Seidel iterations and alternating sweeping strategy is designed to improve the convergence for steady state computations of the problem. A numerical example is given to show the feasibility of the model and the effectiveness of the solution algorithm.     

Simulation of a compositional model for multiphase flow in porous media

In this article, we consider the simulation of a compositional model for three‐dimensional, three‐phase, multicomponent flow in a porous medium. This model consists of Darcy's law for volumetric flow velocities, mass conservation for hydrocarbon components, thermodynamic equilibrium for mass interchange between phases, and an equation of state for saturations. A discretization scheme based on the block‐centered finite difference method for pressures and compositions is developed. Numerical results are reported for the benchmark problem of the third comparative solution project (CSP) organized by the society of petroleum engineers (SPE). © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

A complete flux scheme for the stationary advection-diffusion-reaction equation

Expressions for the numerical flux of a conservation law of advection-diffusion-reaction type are derived from a local solution of the entire conservation law, including the source term. The resulting complete flux scheme is given for one-dimensional (in Cartesian and spherical coordinates) and two-dimensional model equations. Combined with a finite volume method, the numerical scheme is second order accurate, uniformly in the Peclet numbers. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

不可压饱和多孔弹性杆动力响应的多辛方法

研究了不可压饱和多孔弹性杆的一维动力响应问题.基于多孔介质理论,在流相和固相微观不可压、固相骨架小变形的假定下,建立了不可压流体饱和多孔弹性杆一维轴向动力响应的数学模型.利用Hamilton空间体系的多辛理论,构造了不可压饱和多孔弹性杆轴向振动方程的多辛形式及其多种局部守恒律.采用中点Box离散方法得到轴向振动方程的多辛离散格式和局部能量守恒律以及局部动量守恒律的离散格式;数值模拟了不可压饱和多孔弹性杆的轴向振动过程,记录了每一时间步的局部能量数值误差和局部动量数值误差.结果表明,已构造的多辛离散格式具有很高的精确性和较长时间的数值稳定性,这为解决饱和多孔介质的动力响应问题提供了新的途径.     

Secure communications based on the synchronization of the hyperchaotic Chen and the unified chaotic systems

This paper deals with the adaptive synchronization of two identical hyperchaotic master and slave systems. The master system and the slave system each consists of two subsystems: a hyperchaotic Chen subsystem and a unified chaotic subsystem. The asymptotic convergence of the errors between the states of the master system and the states of the slave system is proven using Lyapunov theory. Simulation results are presented to illustrate the ability of the control law to synchronize the master and slave systems. Moreover, the proposed control scheme is applied to encrypt and decrypt discrete signals such as digital images where computer simulation results are provided to show that the proposed control law works well.     

One-sided stability and convergence of the Nessyahu–Tadmor scheme

Non-oscillatory schemes are widely used in numerical approximations of nonlinear conservation laws. The Nessyahu–Tadmor (NT) scheme is an example of a second order scheme that is both robust and simple. In this paper, we prove a new stability property of the NT scheme based on the standard minmod reconstruction in the case of a scalar strictly convex conservation law. This property is similar to the One-sided Lipschitz condition for first order schemes. Using this new stability, we derive the convergence of the NT scheme to the exact entropy solution without imposing any nonhomogeneous limitations on the method. We also derive an error estimate for monotone initial data.     

A flux-limiter method for dam-break flows over erodible sediment beds

Finite volume methods for dam-break flows over erodible sediment beds require a monotone numerical flux. In the present study we present a new flux-limiter scheme based on the Lax–Wendroff method coupled with a non-homogeneous Riemann solver and a flux limiter function. The non-homogeneous Riemann solver consists of a predictor stage for the discretization of gradient terms and a corrector stage for the treatment of source terms. The proposed method satisfy the conservation property such that the discretization of the flux gradients and the source terms are well-balanced in the numerical solution of suspended sediment models. The flux-limiter method provides accurate results avoiding numerical oscillations and numerical dissipation in the approximated solutions. Several standard test examples are considered to verify the performance and the accuracy of the proposed method.     

Multisymplectic Fourier pseudo-spectral integrators for Klein-Gordon-Schrödinger equations

A multisymplectic Fourier pseudo-spectral scheme,which exactly preserves the discrete multisymplectic conservation law,is presented to solve the Klein-Gordon-Schrdinger equations.The scheme is of spectral accuracy in space and of second order in time.The scheme preserves the discrete multisymplectic conservation law and the charge conservation law.Moreover,the residuals of some other conservation laws are derived for the geometric numerical integrator.Extensive numerical simulations illustrate the numerical behavior of the multisymplectic scheme,and demonstrate the correctness of the theoretical analysis.     

Implicit kinetic schemes for scalar conservation laws

Based on kinetic formulation for scalar conservation laws, we present implicit kinetic schemes. For time stepping these schemes require resolution of linear systems of algebraic equations. The scheme is conservative at steady states. We prove that if time marching procedure converges to some steady state solution, then the implicit kinetic scheme converges to some entropy steady state solution. We give sufficient condition of the convergence of time marching procedure. For scalar conservation laws with a stiff source term we construct a stiff numerical scheme with discontinuous artificial viscosity coefficients that ensure the scheme to be equilibrium conserving. We couple the developed implicit approach with the stiff space discretization, thus providing improved stability and equilibrium conservation property in the resulting scheme. Numerical results demonstrate high computational capabilities (stability for large CFL numbers, fast convergence, accuracy) of the developed implicit approach. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 26–43, 2002     

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89-112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge-Kutta methods, Appl. Numer. Math. 58 (2008) 1675-1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed.A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.     

Decoupled energy‐law preserving numerical schemes for the Cahn–Hilliard–Darcy system

We study two novel decoupled energy‐law preserving and mass‐conservative numerical schemes for solving the Cahn‐Hilliard‐Darcy system which models two‐phase flow in porous medium or in a Hele–Shaw cell. In the first scheme, the velocity in the Cahn–Hilliard equation is treated explicitly so that the Darcy equation is completely decoupled from the Cahn–Hilliard equation. In the second scheme, an intermediate velocity is used in the Cahn–Hilliard equation which allows for the decoupling. We show that the first scheme preserves a discrete energy law with a time‐step constraint, while the second scheme satisfies an energy law without any constraint and is unconditionally stable. Ample numerical experiments are performed to gauge the efficiency and robustness of our scheme. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 936–954, 2016     

Analyse d'un schéma décalé pour l'approximation de systèmes hyperboliques non conservatifs

We consider a model system made of two nonlinear equations which are non conservative. A conservation law can be obtained from these equations through linear operations only, which don't modify the shock waves. A numerical scheme based on a different mesh adapted to each variable is proposed. By choosing a shifted mesh, we have un explicit Riemann solver and we can derive a finite volume scheme. We prove a priori estimates in L^∞ norm and Total Variation for the system, which lead to a strong convergence in L¹ norm towards a solution satisfying the associated conservation law.     

A higher-order moment method of the lattice Boltzmann model for the conservation law equation

In this paper, we proposed a higher-order moment method in the lattice Boltzmann model for the conservation law equation. In contrast to the lattice Bhatnagar–Gross–Krook (BGK) model, the higher-order moment method has a wide flexibility to select equilibrium distribution function. This method is based on so-called a series of partial differential equations obtained by using multi-scale technique and Chapman–Enskog expansion. According to Hirt’s heuristic stability theory, the stability of the scheme can be controlled by modulating some special moments to design the third-order dispersion term and the fourth-order dissipation term. As results, the conservation law equation is recovered with higher-order truncation error. The numerical examples show the higher-order moment method can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the conservation law equation.     

Analysis of Nonlinear Resonance in Conservation Laws with Point Sources and Well-Balanced Scheme

The purpose of this paper is to study the wave behavior of the hyperbolic conservation law with concatenation of point sources: for i ∈ I some finite index set, and where δ( ) is the Dirac measure. Special features of this problem are the discontinuities that appear along the t -axis at the point sources x = x _{i +1/2}. Resonance occurs when the speed of the nonlinear wave is close to zero. In addition to classical shock waves, the equation exhibits overcompressive and undercompressive waves. The Riemann problem "with resonance" is solved, and we show global existence via the Glimm scheme. Analytical understanding is used to design a well-balanced numerical scheme, of the Godunov type, which preserves the balance between the sources terms and the fluxes terms. Some numerical tests are reported.     

Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms

We focus in this study on the convergence of a class of relaxation numerical schemes for hyperbolic scalar conservation laws including stiff source terms. Following Jin and Xin, we use as approximation of the scalar conservation law, a semi-linear hyperbolic system with a second stiff source term. This allows us to avoid the use of a Riemann solver in the construction of the numerical schemes. The convergence of the approximate solution toward a weak solution is established in the cases of first and second order accurate MUSCL relaxed methods.

     

Well-balanced unstaggered central schemes for one and two-dimensional shallow water equation systems

We propose a new well-balanced unstaggered central finite volume scheme for hyperbolic balance laws with geometrical source terms. In particular we construct a new one and two-dimensional finite volume method for the numerical solution of shallow water equations on flat/variable bottom topographies. The proposed scheme evolves a non-oscillatory numerical solution on a single grid, avoids the time consuming process of solving Riemann problems arising at the cell interfaces, and is second-order accurate both in space and time. Furthermore, the numerical scheme follows a well-balanced discretization that first discretizes the geometrical source term according to the discretization of the flux terms, and then mimics the surface gradient method and discretizes the water height according to the discretization of the water level. The resulting scheme exactly satisfies the C-property at the discrete level. The proposed scheme is then applied and classical one and two-dimensional shallow water equation problems with flat or variable bottom topographies are successfully solved. The obtained numerical results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potential and efficiency of the proposed method.     

Parabolic approximations of diffusive–dispersive equations

We consider a lower-order approximation for a third-order diffusive–dispersive conservation law with nonlinear flux. It consists of a system of two second-order parabolic equations; a coupling parameter is also added. If the flux has an inflection point it is well-known, on the one hand, that the diffusive–dispersive law admits traveling-wave solutions whose end states are also connected by undercompressive shock waves of the underlying hyperbolic conservation law. On the other hand, if the diffusive–dispersive regularization vanishes, the solutions of the corresponding initial-value problem converge to a weak solution of the hyperbolic conservation law. We show that both of these properties also hold for the lower-order approximation. Furthermore, when the coupling parameter tends to infinity, we prove that solutions of initial value problems for the approximation converge to a weak solution of the diffusive–dispersive law. The proofs rely on new a priori energy estimates for higher-order derivatives and the technique of compensated compactness.     

Second order scheme for scalar conservation laws with discontinuous flux

Burger, Karlsen, Torres and Towers in [9] proposed a flux TVD (FTVD) second order scheme with Engquist–Osher flux, by using a new nonlocal limiter algorithm for scalar conservation laws with discontinuous flux modeling clarifier thickener units. In this work we show that their idea can be used to construct FTVD second order scheme for general fluxes like Godunov, Engquist–Osher, Lax–Friedrich, … satisfying (A, B)-interface entropy condition for a scalar conservation law with discontinuous flux with proper modification at the interface. Also corresponding convergence analysis is shown. We show further from numerical experiments that solutions obtained from these schemes are comparable with the second order schemes obtained from the minimod limiter.