High order finite difference methods with subcell resolution for advection equations with stiff source terms

A new high order finite-difference method utilizing the idea of Harten ENO subcell resolution method is proposed for chemical reactive flows and combustion. In reaction problems, when the reaction time scale is very small, e.g., orders of magnitude smaller than the fluid dynamics time scales, the governing equations will become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present proposed method is a modified fractional step method which solves the convection step and reaction step separately. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with the computed flow variables in the shock region modified by the Harten subcell resolution idea. For numerical experiments, a fifth-order finite-difference WENO scheme and its anti-diffusion WENO variant are considered. A wide range of 1D and 2D scalar and Euler system test cases are investigated. Studies indicate that for the considered test cases, the new method maintains high order accuracy in space for smooth flows, and for stiff source terms with discontinuities, it can capture the correct propagation speed of discontinuities in very coarse meshes with reasonable CFL numbers.     

以Maxwell-Boltzmann分布函数为基础的流矢量分裂方法

将以微观气体分子运动论为基础的流矢量分裂法和二时间步的算法相结合,用于计算无粘理想气体流动.方程中的流矢量按局部平衡的Maxwell-Boltzmann分布函数分解.3个一维的算例给出了激波、接触间断和稀疏波的计算结果,并与精确解做了对比.     

Effect of solid dust particles on the propagation of shock wave in planar and non-planar gasdynamics

The aim of the present study is to analyze the propagation of shock wave along the characteristic path in planar and non-planar unsteady compressible ideal gas flow in presence of small solid dust particles. The analytical solution of the governing quasilinear hyperbolic system is computed in the characteristic plane and it is found that this analytical linear solution in this plane can exhibit non-linear phenomenon in the physical plane. The effect of the dust particles on the evolutionary behavior of the propagating shock wave in ideal gas flow is discussed. The transport equations leading to the evolution of shock wave is determined which introduces the conditions of shock formation. The growth and decay of compressive waves and expansive waves, respectively, in planar and non-planar ideal gas dynamics influenced by the presence of small solid dust particles, is discussed.     

Three-dimensional finite-difference lattice Boltzmann model and its application to inviscid compressible flows with shock waves

In this paper, a three-dimensional (3D) finite-difference lattice Boltzmann model for simulating compressible flows with shock waves is developed in the framework of the double-distribution-function approach. In the model, a density distribution function is adopted to model the flow field, while a total energy distribution function is adopted to model the temperature field. The discrete equilibrium density and total energy distribution functions are derived from the Hermite expansions of the continuous equilibrium distribution functions. The discrete velocity set is obtained by choosing the abscissae of a suitable Gauss–Hermite quadrature with sufficient accuracy. In order to capture the shock waves in compressible flows and improve the numerical accuracy and stability, an implicit–explicit finite-difference numerical technique based on the total variation diminishing flux limitation is introduced to solve the discrete kinetic equations. The model is tested by numerical simulations of some typical compressible flows with shock waves ranging from 1D to 3D. The numerical results are found to be in good agreement with the analytical solutions and/or other numerical results reported in the literature.     

The Self-Similarity Index of the Convergence of Strong Cylindrical Shock Waves in a Gas with a Uniform Density

A model of the convergence of cylindrical shock waves (SWs) in a gas with a uniform density has been considered. The partial differential equations of this model have been reduced to ordinary differential equations, from which the law of convergence of such shock waves and the dependence α = f(γ, γ_eff) of their self-similarity index α on the heat-capacity ratio in front of the shock wave (γ) and behind the shock wave front (γ_eff) of the gas have been found. This dependence for cylindrical shock waves has been shown to agree with the experimental data within the measurement error.     

Free-energy transition in a gas of noninteracting nonlinear wave particles

We investigate the dynamics of a gas of noninteracting particlelike soliton waves, demonstrating that phase transitions originate from their collective behavior. This is predicted by solving exactly the nonlinear equations and by employing methods of the statistical mechanics of chaos. In particular, we show that a suitable free energy undergoes a metamorphosis as the input excitation is increased, thereby developing a first-order phase transition whose measurable manifestation is the formation of shock waves. This demonstrates that even the simplest phase-space dynamics, involving independent (uncoupled) degrees of freedom, can sustain critical phenomena.     

A method for tractable dynamical studies of single and double shock compression

A new multiscale simulation method is formulated for the study of shocked materials. The method combines molecular dynamics and the Euler equations for compressible flow. Treatment of the difficult problem of the spontaneous formation of multiple shock waves due to material instabilities is enabled with this approach. The method allows the molecular dynamics simulation of the system under dynamical shock conditions for orders of magnitude longer time periods than is possible using the popular nonequilibrium molecular dynamics approach. An example calculation is given for a model potential for silicon in which a computational speedup of 10(5) is demonstrated. Results of these simulations are consistent with the recent experimental observation of an anomalously large elastic precursor on the nanosecond time scale.     

Gas dynamics in strong gravitational fields

The steady and axially symmetric flow of a perfect fluid is studied in the context of general relativistic gas dynamics. It is assumed that the flow occurs in the background field of a rotating black hole (or any compact object). The hydrodynamic equations are referred to a locally nonrotating frame and their characteristics are found. The equations describing oblique shock waves are also obtained.     

Identification of discontinuities in plasma plume evolution

The ejection of material during laser ablation gives rise to the development of discontinuities in the ambient gas. Several of these discontinuities are observed and characterized, including externally and internally propagating shock waves, contact surface, and the ionization front. Qualitative experimental observations and analysis of these discontinuities are presented. Results from shadowgraphy enabled determination of an irradiance threshold between two different ablation mechanisms, and determination of several stages of plasma plume evolution. Consideration of the refractive index as a dynamic sum of the contributions from gas and electrons led to separate identification of ionization front from the contact surface. Furthermore, ionization front was observed to lead the shock wave at the earlier stage of the ablation.     

Numerical model of two-dimensional heterogeneous combustion in porous media under natural convection or forced filtration

A novel mathematical model and original numerical method for investigating the two-dimensional waves of heterogeneous combustion in porous media are proposed and described in detail. The mathematical model is constructed within the framework of the model of interacting interpenetrating continua and includes equations of state, continuity, momentum conservation and energy for solid and gas phases. Combustion, considered in the paper, is due to the exothermic reaction between fuel in the porous solid medium and oxidiser contained in the gas flowing through the porous object. The original numerical method is based on a combination of explicit and implicit finite-difference schemes. A distinctive feature of the proposed model is that the gas velocity at the open boundaries (inlet and outlet) of the porous object is unknown and has to be found from the solution of the problem, i.e. the flow rate of the gas regulates itself. This approach allows processes to be modelled not only under forced filtration, but also under free convection, when there is no forced gas input in porous objects, which is typical for many natural or anthropogenic disasters (burning of peatlands, coal dumps, landfills, grain elevators). Some two-dimensional time-dependent problems of heterogeneous combustion in porous objects have been solved using the proposed numerical method. It is shown that two-dimensional waves of heterogeneous combustion in porous media can propagate in two modes with different characteristics, as in the case of one-dimensional combustion, but the combustion front can move in a complex manner, and gas dynamics within the porous objects can be complicated. When natural convection takes place, self-sustaining combustion waves can go through the all parts of the object regardless of where an ignition zone was located, so the all combustible material in each part of the object is burned out, in contrast to forced filtration.     

铁-铍介质界面上激波折射现象的数值研究

用数值方法研究铁-铍介质界面上的激波折射现象.运用激波极曲线理论分析不同强度的激波从正规折射过渡到非正规折射的临界角变化.运用一个具有二阶精度和波传播性质的激波捕捉法,数值求解激波折射运动的流体力学方程组.对正规折射,数值结果与激波极曲线理论一致;对非正规折射,不同强度的激波大都存在前驱的折射激波,并且入射激波的强度不同、入射角度不同,激波折射的图像也不同.     

Acoustic wave propagation simulation in a poroelastic medium saturated by two immiscible fluids using a staggered finite-difference with a time partition method

Based on the three-phase theory proposed by Santos, acoustic wave propagation in a poroelastic medium saturated by two immiscible fluids was simulated using a staggered high-order finite-difference algorithm with a time partition method, which is firstly applied to such a three-phase medium. The partition method was used to solve the stiffness problem of the differential equations in the three-phase theory. Considering the effects of capillary pressure, reference pressure and coupling drag of two fluids in pores, three compressional waves and one shear wave predicted by Santos have been correctly simulated. Influences of the parameters, porosity, permeability and gas saturation on the velocities and amplitude of three compressional waves were discussed in detail. Also, a perfectly matched layer (PML) absorbing boundary condition was firstly implemented in the three-phase equations with a staggered-grid high-order finite-difference. Comparisons between the proposed PML method and a commonly used damping method were made to validate the efficiency of the proposed boundary absorption scheme. It was shown that the PML works more efficiently than the damping method in this complex medium. Additionally, the three-phase theory is reduced to the Biot’s theory when there is only one fluid left in the pores, which is shown in Appendix. This reduction makes clear that three-phase equation systems are identical to the typical Biot’s equations if the fluid saturation for either of the two fluids in the pores approaches to zero. Supported by the Key Program of the National Natural Science Foundation of China (Grant No. 10534040) and the National Natural Science Foundation of China (Grant No. 10674148)     

A central Rankine–Hugoniot solver for hyperbolic conservation laws

A numerical method in which the Rankine–Hugoniot condition is enforced at the discrete level is developed. The simple format of central discretization in a finite volume method is used together with the jump condition to develop a simple and yet accurate numerical method free of Riemann solvers and complicated flux splittings. The steady discontinuities are captured accurately by this numerical method. The basic idea is to fix the coefficient of numerical dissipation based on the Rankine–Hugoniot (jump) condition. Several numerical examples for scalar and vector hyperbolic conservation laws representing the inviscid Burgers equation, the Euler equations of gas dynamics, shallow water equations and ideal MHD equations in one and two dimensions are presented which demonstrate the efficiency and accuracy of this numerical method in capturing the flow features.     

Multi-domain Fourier-continuation/WENO hybrid solver for conservation laws

We introduce a multi-domain Fourier-continuation/WENO hybrid method (FC–WENO) that enables high-order and non-oscillatory solution of systems of nonlinear conservation laws, and which enjoys essentially dispersionless, spectral character away from discontinuities, as well as mild CFL constraints (comparable to those of finite difference methods). The hybrid scheme employs the expensive, shock-capturing WENO method in small regions containing discontinuities and the efficient FC method in the rest of the computational domain, yielding a highly effective overall scheme for applications with a mix of discontinuities and complex smooth structures. The smooth and discontinuous solution regions are distinguished using the multi-resolution procedure of Harten [J. Comput. Phys. 115 (1994) 319–338]. We consider WENO schemes of formal orders five and nine and a FC method of order five. The accuracy, stability and efficiency of the new hybrid method for conservation laws is investigated for problems with both smooth and non-smooth solutions. In the latter case, we solve the Euler equations for gas dynamics for the standard test case of a Mach three shock wave interacting with an entropy wave, as well as a shock wave (with Mach 1.25, three or six) interacting with a very small entropy wave and evaluate the efficiency of the hybrid FC–WENO method as compared to a purely WENO-based approach as well as alternative hybrid based techniques. We demonstrate considerable computational advantages of the new FC-based method, suggesting a potential of an order of magnitude acceleration over alternatives when extended to fully three-dimensional problems.     

Bending loss analyses of photonic crystal fibers based on the finite-difference time-domain method

This is a report on an effective simulation method for the bending loss analyses of photonic crystal fibers. This method is based on the two-dimensional finite-difference time-domain algorithm and a conformal transformation of the refractive index profile. We observed the temporal dynamics of light waves in a bent fiber in a simulation and obtained the bending loss as a function of bend radius and optical wavelength for the commercial photonic crystal fibers. The accuracy of this method was verified by good agreement between the simulation and experimental data.     

Shock profile solutions of the Boltzmann equation

Shock waves in gas dynamics can be described by the Euler Navier-Stokes, or Boltzmann equations. We prove the existence of shock profile solutions of the Boltzmann equation for shocks which are weak. The shock is written as a truncated expansion in powers of the shock strength, the first two terms of which come exactly from the Taylor tanh (x) profile for the Navier-Stokes solution. The full solution is found by a projection method like the Lyapunov-Schmidt method as a bifurcation from the constant state in which the bifurcation parameter is the difference between the speed of soundc ₀ and the shock speeds.Research supported in part by the National Science Foundation, the Army Research Office, the Air Force of Scientific Research, the Office of Naval Research, and the Department of Energy     

Current discontinuities on superconducting cosmic strings

The propagation of current perturbations on superconducting cosmic strings is considered. The conditions for the existence of discontinuities similar to shock waves have been found. The formulas relating the string parameters and the discontinuity propagation speed are derived. The current growth law in a shock wave is deduced. The propagation speeds of shock waves with arbitrary amplitudes are calculated. The reason why there are no shock waves in the case of time-like currents (in the “electric” regime) is explained; this is attributable to the shock wave instability with respect to perturbations of the string world sheet.     

Decay of initial nonlinear Alfvén wave discontinuities in a collisionless plasma

A system of Whitham equations for the derivative nonlinear Schrödinger equation in the Riemann form was used to analyze possible types of decay of discontinuities which accompany the overturning of the simplest quasilongitudinal nonlinear Alfvén wave. A whole class of structures is found which are formed when the discontinuities decay into two simple collisionless shock waves without forming a plateau between them. Different types of decay of the initial discontinuities are considered for cases when the overturning of the Alfvén wave is modulationally stable.Checheno—Ingushk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 110–117, July, 1994.     

Description of Finite-amplitude Standing Acoustic Waves Using Convection-diffusion Equations

This paper deals with problems of finite-amplitude standing waves in acoustical resonators of variable cross-section. Set of two one-dimensional partial differential equations in the third approximation, formulated in conservative form, is derived from the fundamental equations of gas dynamics. The model equations which takes into account external driving force, gas dynamic nonlinearities and thermoviscous dissipation are solved numerically in time domain using a central scheme developed for convection-diffusion equations integration. In this paper numerical results for closed air-filled acoustic resonators are presented.     

Obliquely propagating ion-acoustic solitary and shock waves in magnetized quantum degenerate multi-ions plasma in the presence of trapped electrons

The properties of obliquely propagating ion-acoustic waves have been investigated in multi-ions magnetized plasma comprising of inertial, positively and negatively charged ion fluids, trapped electrons, and negatively charged stationary heavy ions. The propagation of the waves is oblique to the ambient magnetic field which is along the z-direction. Only fast type of modes exists in the linear regime. The reductive perturbation method was adopted to derive the Korteweg– de Vries (KdV) and Burger equations, as well as the solitary and shock wave solutions of the evolved equations, have been used to analyze the properties of the small but finite amplitude waves. The effects of the constituent plasma parameters, namely, the trapping effect of electrons, the electron degenerate temperature and the viscosity coefficient on the dynamics of the small amplitude solitary and shock waves have been examined. The influence of the magnetic field and the obliquity parameter on the propagation characteristics of ion-acoustic waves are discussed.