基于Euler方程有限差分方法驻波晃动模拟

研究在二维水槽带非线性自由面边界条件的Euler方程的数值解,数值模拟了驻波的波高．将不规则的物理区域变换为一个固定的正方形计算区域,在计算区域使用交错网格技术的目的是准确捕捉流场瞬间的波高值,应用由Bang-fuh Chen建立的时间无关有限差分方法求解不可压缩无粘Euler方程的数值解．通过数值结果表明,数值解很好地吻合分析解和以前出版的文献结果．从数值解可以看出,非线性现象和拍的现象非常明显,同时数值模拟了带初始驻波的水平激励和垂直激励运动,具有很好的数值效果．     

Standing Wave Solutions in Nonhomogeneous Delayed Synaptically Coupled Neuronal Networks

The authors establish the existence and stability of standing wave solutions of a nonlinear singularly perturbed systemof integral differential equations and a nonlinear scalar integral differential equation. It will be shown that there exist six standing wave solutions (u(x,t),w(x,t))=(U(x),W(x)) to the nonlinear singularly perturbed system of integral differential equations. Similarly, there exist six standing wave solutions u(x,t)=U(x) to the nonlinear scalar integral differential equation. The main idea to establish the stability is to construct Evans functions corresponding to several associated eigenvalue problems.     

Uniform boundary stabilization of a wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping

We consider a wave equation with nonlinear acoustic boundary conditions. This is a nonlinearly coupled system of hyperbolic equations modeling an acoustic/structure interaction, with an additional boundary damping term to induce both existence of solutions as well as stability. Using the methods of Lasiecka and Tataru for a wave equation with nonlinear boundary damping, we demonstrate well-posedness and uniform decay rates for solutions in the finite energy space, with the results depending on the relationship between (i) the mass of the structure, (ii) the nonlinear coupling term, and (iii) the size of the nonlinear damping. We also show that solutions (in the linear case) depend continuously on the mass of the structure as it tends to zero, which provides rigorous justification for studying the case where the mass is equal to zero.     

Modeling weakly nonlinear acoustic wave propagation

Three weakly nonlinear models of lossless, compressible fluidflow—a straightforward weakly nonlinear equation (WNE),the inviscid Kuznetsov equation (IKE) and the Lighthill–Westerveltequation (LWE)—are derived from first principles and theirrelationship to each other is established. Through a numericalstudy of the blow-up of acceleration waves, the weakly nonlinearequations are compared to the ‘exact’ Euler equations,and the ranges of applicability of the approximate models areassessed. By reformulating these equations as hyperbolic systemsof conservation laws, we are able to employ a Godunov-type finite-differencescheme to obtain numerical solutions of the approximate modelsfor times beyond the instant of blow-up (that is, shock formation),for both density and velocity boundary conditions. Our studyreveals that the straightforward WNE gives the best results,followed by the IKE, with the LWE's performance being the poorestoverall.     

The fluid dynamic limit of the nonlinear boltzmann equation

Solutions of the nonlinear Boltzmann equation are constructed up to the first appearance of shocks in the corresponding fluid dynamics. This construction assumes the knowledge of solutions of the Euler equations for compressible gas flow. The Boltzmann solution is found as a truncated Hilbert expansion with a remainder, and the remainder term solves a weakly nonlinear equation which is solved by iteration. The solutions found have special initial values. They should serve as “outer expansions” to which initial layers, boundary layers and shock layers can be matched.     

A Canonical System of lntegrodifferential Equations Arising in Resonant Nonlinear Acoustics

In general, weakly nonlinear high frequency almost periodic wave trains for systems of hyperbolic conservation laws interact and resonate to leading order. In earlier work the first two authors and J. Hunter developed simplified asymptotic equations describing this resonant interaction. In the important special case of compressible fluid flow in one or several space dimensions, these simplified asymptotic equations are essentially two inviscid Burgers equations for the nonlinear sound waves, coupled by convolution with a known kernel given by the sum of the initial vortex strength and the derivative of the initial entropy. Here we develop some of the remarkable new properties of the solutions of this system for resonant acoustics. These new features include substantial almost periodic exchange of energy between the nonlinear sound waves, the existence of smooth periodic wave trains, and the role of such smooth wave patterns in eliminating or suppressing the strong temporal decay of sawtooth profile solutions of the decoupled inviscid Burgers equations. Our approach combines detailed numerical modeling to elucidate the new phenomena together with rigorous analysis to obtain exact solutions as well as other elementary properties of the solutions of this system.     

Zero-viscosity-capillarity limit to the planar rarefaction wave for the 2D compressible Navier–Stokes–Korteweg equations

We shall consider the two-dimensional (2D) isentropic Navier–Stokes–Korteweg equations which are used to model compressible fluids with internal capillarity. Formally, the 2D isentropic Navier–Stokes–Korteweg equations converge, as the viscosity and the capillarity vanish, to the corresponding 2D inviscid Euler equations, and we do justify this for the case that the corresponding 2D inviscid Euler equations admit a planar rarefaction wave solution. More precisely, it is proved that there exists a family of smooth solutions for the 2D isentropic compressible Navier–Stokes–Korteweg equations converging to the planar rarefaction wave solution with arbitrary strength for the 2D Euler equations. A uniform convergence rate is obtained in terms of the viscosity coefficient and the capillarity away from the initial time. The key ingredients of our proof are the re-scaling technique and energy estimate, in which we also introduce the hyperbolic wave to recover the physical viscosities and capillarity of the inviscid rarefaction wave profile.     

A Numerical Study of the Exact Evolution Equations for Surface Waves in Water of Finite Depth

We describe a pseudo-spectral numerical method to solve the systems of one-dimensional evolution equations for free surface waves in a homogeneous layer of an ideal fluid. We use the method to solve a system of one-dimensional integro-differential equations, first proposed by Ovsjannikov and later derived by Dyachenko, Zakharov, and Kuznetsov, to simulate the exact evolution of nonlinear free surface waves governed by the two-dimensional Euler equations. These equations are written in the transformed plane where the free surface is mapped onto a flat surface and do not require the common assumption that the waves have small amplitude used in deriving the weakly nonlinear Korteweg–de Vries and Boussinesq long-wave equations. We compare the solution of the exact reduced equations with these weakly nonlinear long-wave models and with the nonlinear long-wave equations of Su and Gardner that do not assume the waves have small amplitude. The Su and Gardner solutions are in remarkably close agreement with the exact Euler solutions for large amplitude solitary wave interactions while the interactions of low-amplitude solitary waves of all four models agree. The simulations demonstrate that our method is an efficient and accurate approach to integrate all of these equations and conserves the mass, momentum, and energy of the Euler equations over very long simulations.     

直接边界积分方法计算水波的稳定性

1.引 言 数值模拟流体自由界面运动一直是研究水波的主要方法.海浪攀爬海岸的研究是水动力学中的一个很经典且很具有挑战性的课题,因为在水面附近的方程是高度非线性的.对于二维情形下有一致倾斜度的海岸上的水波,Carrier＆Greeenspan[7]建立了基于浅水模型的非线性理论,Tuck＆Hwang引入变量代换,把最初的非线性方程转变成更容易分析的线性方程.这种直接对单一流体用浅水方程计算自由界面的办法仍然被广泛应用.Zhang,Wu,Hou[23]给出了这个问题的Euler-Langrange混合格式,Li ＆ Zhang[13]借助这个格式,并引入人工边界条件对海浪攀爬海岸问题进行了整体的数值模拟.     

Some Explicit Resonating Waves in Weakly Nonlinear Gas Dynamics

Equations governing leading order wave amplitudes of resonating almost periodic wave trains in weakly nonlinear acoustics have been obtained by Majda and Rosales [Stud. Appl. Math. 71:149–179 (1984)]. These equations consist of a pair of Burgers equations coupled through an integral term with a known kernel. Numerical experiments reported by Majda, Rosales, and Schonbek have suggested the existence of smooth solutions of this system whose components consist of traveling waves moving in opposite directions. For the simplest cosine kernel, explicit formulae are given here for such resonating wave solutions. There is a wave of maximum amplitude with a “peak.” For more general kernels, small amplitude resonating waves are constructed via bifurcation.     

Weak Continuity and Compactness for Nonlinear Partial Differential Equations

This paper presents several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a significant role. The compactness and convergence of vanishing viscosity solutions for nonlinear hyperbolic conservation laws are first analyzed, including the inviscid limit from the Navier-Stokes equations to the Euler equations for homentropic flow, the vanishing viscosity method to construct the global spherically symmetric solutions to the multidimensional compressible Euler equations, and the sonic-subsonic limit of solutions of the full Euler equations for multi-dimensional steady compressible fluids. Then the weak continuity and rigidity of the Gauss-Codazzi-Ricci system and corresponding isometric embeddings in differential geometry are revealed. Further references are also provided for some recent developments on the weak continuity and compactness for nonlinear partial differential equations.     

Ground states for Schrödinger-type equations with nonlocal nonlinearity

The problem of the existence of stable solitary wave solutions for nonlinear Schrödinger-type equations with a generalized cubic nonlinearity is considered. These types of equations have recently arisen in the context of optical communications as averaging approximations to nonlinear dispersive equations with widely separated time scales. In this paper, it is shown that under general conditions on the kernel of the nonlocal term, stable standing wave solutions exist for these equations.     

High-order accurate dissipative weighted compact nonlinear schemes

Based on the method deriving dissipative compact linear schemes (DCS), novel high-order dissipative weighted compact nonlinear schemes (DWCNS) are developed. By Fourier analysis,the dissipative and dispersive features of DWCNS are discussed. In view of the modified wave number, the DWCNS are equivalent to the fifth-order upwind biased explicit schemes in smooth regions and the interpolations at cell-edges dominate the accuracy of DWCNS. Boundary and near boundary schemes are developed and the asymptotic stabilities of DWCNS on both uniform and stretching grids are analyzed. The multi-dimensional implementations for Euler and Navier-Stokes equations are discussed. Several numerical inviscid and viscous results are given which show the good performances of the DWCNS for discontinuities capturing, high accuracy for boundary layer resolutions, good convergent rates (the root-mean-square of residuals approaching machine zero for solutions with strong shocks) and especially the damping effect on the spudous oscillations which were found in the solutions obtained by TVD and ENO schemes.     

Quadrature formulas for periodic functions and their application to the boundary element method

Two-dimensional and axisymmetric boundary value problems for the Laplace equation in a domain bounded by a closed smooth contour are considered. The problems are reduced to integral equations with a periodic singular kernel, where the period is equal to the length of the contour. Taking into account the periodicity property, high-order accurate quadrature formulas are applied to the integral operator. As a result, the integral equations are reduced to a system of linear algebraic equations. This substantially simplifies the numerical schemes for solving boundary value problems and considerably improves the accuracy of approximation of the integral operator. The boundaries are specified by analytic functions, and the remainder of the quadrature formulas decreases faster than any power of the integration step size. The examples include the two-dimensional potential inviscid circulation flow past a single blade or a grid of blades; the axisymmetric flow past a torus; and free-surface flow problems, such as wave breakdown, standing waves, and the development of Rayleigh-Taylor instability.     

Modulation theory solution for nonlinearly resonant,fifth‐order Korteweg–de Vries,nonclassical, traveling dispersive shock waves

A new class of resonant dispersive shock waves was recently identified as solutions of the Kawahara equation— a Korteweg–de Vries (KdV) type nonlinear wave equation with third‐ and fifth‐order spatial derivatives— in the regime of nonconvex, linear dispersion. Linear resonance resulting from the third‐ and fifth‐order terms in the Kawahara equation was identified as the key ingredient for nonclassical dispersive shock wave solutions. Here, nonlinear wave (Whitham) modulation theory is used to construct approximate nonclassical traveling dispersive shock wave (TDSW) solutions of the fifth‐ order KdV equation without the third derivative term, hence without any linear resonance. A self‐similar, simple wave modulation solution of the fifth order, weakly nonlinear KdV–Whitham equations is obtained that matches a constant to a heteroclinic traveling wave via a partial dispersive shock wave so that the TDSW is interpreted as a nonlinear resonance. The modulation solution is compared with full numerical solutions, exhibiting excellent agreement. The TDSW is shown to be modulationally stable in the presence of sufficiently small third‐order dispersion. The Kawahara–Whitham modulation equations transition from hyperbolic to elliptic type for sufficiently large third‐order dispersion, which provides a possible route for the TDSW to exhibit modulational instability.     

弱粘性流体中垂直激励表面波的阻尼效应

在竖直振动的圆柱形容器中,将Navier-Stokes方程线性化,利用两时间尺度奇异摄动展开法研究了弱粘性流体的单一自由面驻波运动．整个流场被分为外部势流区和内部边界层区两部分,对两部分区域分别求解,得到包含阻尼项和外驱动影响的线性振幅方程．利用稳定性分析,得到形成稳定表面波的条件,给出了临界曲线．此外,还获得了阻尼系数的解析表达式．最后,将线性阻尼加到理想流体条件下所得到的色散关系中对其进行修正,理论结果证明修正后的驱动频率更加接近实验的结果．通过计算发现,当驱动的频率较低时,流体的粘性对表面波模式选择有重要影响,而表面张力的影响不明显;但当驱动频率较高时,流体的表面张力起主要作用,而流体的粘性影响甚小．     

强阻尼波方程解的整体存在性和一致衰减性

这篇文章研究一类带非线性源项和非线性边界阻尼项的强阻尼波 方程强解和弱解的整体存在性和唯一性,进而也讨论解的一致衰减.     

The inviscid limit for an inflow problem of compressible viscous gas in presence of both shocks and boundary layers

In this paper, we study the inviscid limit problem for the Navier-Stokes equations of one-dimensional compressible viscous gas on half plane. We prove that if the solution of the inviscid Euler system on half plane is piecewise smooth with a single shock satisfying the entropy condition, then there exist solutions to Navier-Stokes equations which converge to the inviscid solution away from the shock discontinuity and the boundary at an optimal rate of ε¹ as the viscosity ε tends to zero.     

Boundary stabilization of a 3-dimensional structural acoustic model

The main result of this paper provides uniform decay rates obtained for the energy function associated with a three-dimensional structural acoustic model described by coupled system consisting of the wave equation and plate equation with the coupling on the interface between the acoustic chamber and the wall. The uniform stabilization is achieved by introducing a nonlinear dissipation acting via boundary forces applied at the edge of the plate and viscous or boundary damping applied to the wave equation. The results obtained in this paper extend, to the non-analytic, hyperbolic-like setting, the results obtained previously in the literature for acoustic problems modeled by structurally damped plates (governed by analytic semigroups). As a bypass product, we also obtain optimal uniform decay rates for the Euler Bernoulli plate equations with nonlinear boundary dissipation acting via shear forces only and without (i) any geometric conditions imposed on the domain ,(ii) any growth conditions at the origin imposed on the nonlinear function. This is in contrast with the results obtained previously in the literature ([22] and references therein).     

A FLUID-PARTICLE MODEL WITH ELECTRIC FIELDS NEAR A LOCAL MAXWELLIAN WITH RAREFACTION WAVE

The paper is concerned with time-asymptotic behavior of solution near a local Maxwellian with rarefaction wave to a fluid-particle model described by the Vlasov-Fokker-Planck equation coupled with the compressible and inviscid fluid by Euler-Poisson equations through the relaxation drag frictions, Vlasov forces between the macroscopic and microscopic momentums and the electrostatic potential forces. Precisely, based on a new micro-macro decomposition around the local Maxwellian to the kinetic part of the fluid-particle coupled system, which was first developed in [16], we show the time-asymptotically nonlinear stability of rarefaction wave to the one-dimensional compressible inviscid Euler equations coupled with both the Vlasov-Fokker-Planck equation and Poisson equation.