Interior and boundary difference equations for hyperbolic differential equations

Interior and boundary difference equations are derived for several hyperbolic partial differential equations by means of an integral method. The method is applied to a simple transport equation, to waves in a compressible, isentropic fluid, and to surface waves in shallow water. Boundary conditions treated are (a) a perfectly reflecting boundary, (b) an open boundary with outgoing waves and a specified incoming wave, and (c) a partially reflecting boundary. For open boundaries, the major assumption for the algorithms to be valid is that outgoing waves can be defined, an assumption equivalent to the most general statement of Sommerfeld's radiation condition. The difference equations obtained are conservative, second-order accurate, two time-level, explicit, and stable (for one-dimensional, time-dependent problems) for cΔt/Δx ? 1 where c is the wave speed, Δt is the temporal grid size, and Δx is the spatial grid size. Numerical calculations demonstrate the excellent accuracy of the procedure.     

Numerical simulation of oblique and multidirectional wave propagation and breaking on steep slope based on FEM model of Boussinesq equations

Creating a representative numerical simulation of the propagation and breaking of waves along slopes is an important problem in engineering design. Most studies on wave breaking have focused on the propagation of normal incident waves on gentle slopes. In practice, however, waves on steep slopes are obliquely incident or multidirectional irregular waves. In this paper, the eddy viscosity term is introduced to the momentum equation of the improved Boussinesq equations to model wave dissipation caused by breaking and friction, and a numerical model based on an unstructured finite element method (FEM) is established based on the governing equations. It is applied to simulate wave propagation on a steep slope of 1:5. Parallel physical experiments are conducted for comparative analysis that considered a large number of cases, including those featuring of normal and oblique incident regular and irregular waves, and multidirectional waves. The heights of the incident wave increase for different periods to represent different kinds of waves breaking. Based on examination, the effectiveness and accuracy of the numerical model is verified through a comprehensive comparison between the numerical and the experimental results, including in terms of variation in wave height, wave spectrum, and nonlinear parameters. Satisfactory agreement between the numerical and experimental values shows that the proposed model is effective in representing the breaking of oblique incident regular waves, irregular waves, and multidirectional incident irregular waves. However, the initial threshold of the breaking parameter η_t^(I) takes different values for oblique and multidirectional waves. This needs to be paid attention when the breaking of waves is simulated using the Boussinesq equations.     

Existence and stability of travelling waves in fixed-bed reactors

The model equations of the catalytic fixed-bed reactor often possess solutions in the form of travelling wave fronts similar to the well-known case of Fisher's equation. The mathematical investigation of these waves requires searching for solutions of singular boundary value problems in the phase plane or in the three-dimensional phase space. In this paper necesary and sufficient conditions are derived which are to be satisfied by the model parameters and the propagation velocity of the wave front if wave solutions exist. Moreover, sufficient conditions for the asymptotic stability of these solutions are proved where the perturbations are supposed to belong to a certain weighted L²-space. Finally, the connection between the initial distribution of the state variable and the velocity of the wave is discussed.     

Wave generation in a computation domain

One of the possible methods to deal with the reflecting waves at the incident boundary in numerical modeling is to generate waves in the computation domain and absorb the outgoing waves at the incident boundary. A source function is introduced into the momentum equation of Boussinesq equations for generating wave in a computation domain in this paper. Typical numerical examples are given for the verification of the proposed method. Numerical examination for the wave diffraction through a breakwater gap shows that the proposed method is especially useful for multidirectional waves.     

Exact boundary controllability of nodal profile for 1-D quasilinear wave equations

Based on the theory of semi-global C ² solution for 1-D quasilinear wave equations, the local exact boundary controllability of nodal profile for 1-D quasilinear wave equations is obtained by a constructive method, and the corresponding global exact boundary controllability of nodal profile is also obtained under certain additional hypotheses.     

The Boundary Value Problem for the Nonlinear Shallow Water Equations

We propose an approximate analytical solution of the boundary value problem (BVP) for the nonlinear shallow waters equations. Our work, based on the Carrier and Greenspan [ 1 ] hodograph transformation, focuses on the propagation of nonlinear nonbreaking waves over a uniformly plane beach. Available results are briefly discussed with specific emphasis on the comparison between the Initial Value Problem and the BVP; the latter more completely representing the physical phenomenon of wave propagation on a beach. The solution of the BVP is achieved through a perturbation approach solely using the assumption of small waves incoming at the seaward boundary of the domain. The most significant results, i.e., the shoreline position estimation, the actual wave height and velocity at the seaward boundary, the reflected wave height and velocity at the seaward boundary are given for three specific input waves and compared with available solutions.     

Internal Kelvin waves in a two-layer liquid model

The subinertial internal Kelvin wave solutions of a linearized system of the ocean dynamics equations for a semi-infinite two-layer f-plane model basin of constant depth bordering a straight, vertical coast are imposed. A rigid lid surface condition and no-slip wall boundary condition are imposed. Some trapped wave equations are presented and approximate solutions using an asymptotic method are constructed. In the absence of bottom friction, the solution consists of a frictionally modified Kelvin wave and a vertical viscous boundary layer. With a no-slip bottom boundary condition, the solution consists of a modified Kelvin wave, two vertical viscous boundary layers, and a large cross-section scale component. The numerical solutions for Kelvin waves are obtained for model parameters that take account of a joint effect of lateral viscosity, bottom friction, and friction between the layers.     

Kelvin–Helmholtz instability with a free surface

The well-posedness of the hydrostatic equations is linked to long wave stability criteria for parallel shear flows. We revisit the Kelvin--Helmholtz instability with a free surface. In the wall-bounded case, the flow is unstable to all wave lengths. Short wave instabilities are localized and independent of boundary conditions. On the other hand, long waves are shown to be stable if the upper boundary is a free surface and gravity is sufficiently small. We also consider smooth velocity profiles of the base flow rather than a velocity jump. We show that stability of long waves for small gravity generally holds for monotone profiles U(y). On the other hand, this need not be the case if U is not monotone.     

Operator Matrices as Generators of Cosine Operator Functions

We introduce an abstract setting that allows to discuss wave equations with time-dependent boundary conditions by means of operator matrices. We show that such problems are well-posed if and only if certain perturbations of the same problems with homogeneous, time-independent boundary conditions are well-posed. As applications we discuss two wave equations in L^p(0, 1) and in L²(Ω) equipped with dynamical and acoustic-like boundary conditions, respectively.     

Exact boundary observability for nonautonomous quasilinear wave equations

By means of a direct and constructive method based on the theory of semiglobal C² solution, the local exact boundary observability is shown for nonautonomous 1-D quasilinear wave equations. The essential difference between nonautonomous wave equations and autonomous ones is also revealed.     

General Theory of Acoustic Wave Propagation in Liquids and Gases

We study the propagation of small-amplitude acoustic waves in liquids and gases and use the hydrodynamic equations to obtain an exact dispersion equation. This equation in dimensionless variables contains only two material constants p and q. We solve the dispersion equation, obtaining an exact solution that holds for all values of the parameters and all frequencies up to hypersonic, and thus analytically establish exactly how the speed of sound c, the wave vector k, and the damping factor x depend on the frequency ω and the dimensionless material constants p and q. Studying the behavior of the solution in the sonic and ultrasonic frequency bands for ω < 10⁷ sec^-1 results in an expression for the damping factor, which differs from the Kirchhoff formula. The speed of sound c and the wave vector k are shown to have finite nonzero values for all hypersonic frequencies. At the same time, there exists a certain maximum frequency value, ω_max ≈ 10¹¹-10¹² sec^-1, at which the damping factor x is zero. This frequency determines the boundary of the applicability domain for the hydrodynamic equations. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 340–352, February, 2006.     

Asymptotic Behavior of Massless Dirac Waves in Schwarzschild Geometry

In this paper, we show that massless Dirac waves in the Schwarzschild geometry decay to zero at a rate t ^?2λ , where λ = 1, 2, . . . is the angular momentum. Our technique is to use Chandrasekhar’s separation of variables whereby the Dirac equations split into two sets of wave equations. For the first set, we show that the wave decays as t ^?2λ . For the second set, in general, the solutions tend to some explicit profile at the rate t ^?2λ . The decay rate of solutions of Dirac equations is achieved by showing that the coefficient of the explicit profile is exactly zero. The key ingredients in the proof of the decay rate of solutions for the first set of wave equations are an energy estimate used to show the absence of bound states and zero energy resonance and the analysis of the spectral representation of the solutions. The proof of asymptotic behavior for the solutions of the second set of wave equations relies on careful analysis of the Green’s functions for time independent Schrödinger equations associated with these wave equations.     

General exact solutions for linear and nonlinear waves in a Thirring model

In the present paper, we construct exact solutions to a system of partial differential equations iu_x + v + u | v | ² = 0, iv_t + u + v | u | ² = 0 related to the Thirring model. First, we introduce a transform of variables, which puts the governing equations into a more useful form. Because of symmetries inherent in the governing equations, we are able to successively obtain solutions for the phase of each nonlinear wave in terms of the amplitudes of both waves. The exact solutions can be described as belonging to two classes, namely, those that are essentially linear waves and those which are nonlinear waves. The linear wave solutions correspond to waves propagating with constant amplitude, whereas the nonlinear waves evolve in space and time with variable amplitudes. In the traveling wave case, these nonlinear waves can take the form of solitons, or solitary waves, given appropriate initial conditions. Once the general solution method is outlined, we focus on a number of more specific examples in order to show the variety of physical solutions possible. We find that radiation naturally emerges in the solution method: if we assume one of u or v with zero background, the second wave will naturally include both a solitary wave and radiation terms. The solution method is rather elegant and can be applied to related partial differential systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Exact boundary controllability of nodal profile for quasilinear wave equations in a planar tree‐like network of strings

Based on the theory of semi‐global piecewise C² solutions to 1D quasilinear wave equations, the local exact boundary controllability of nodal profile for quasilinear wave equations in a planar tree‐like network of strings with general topology is obtained by a constructive method. The principles of providing nodal profiles and of choosing and transferring boundary controls are presented, respectively. Copyright © 2013 John Wiley & Sons, Ltd.     

A Kind of Discrete Non-Reflecting Boundary Conditions for Varieties of Wave Equations

Abstract In this paper, a new kind of discrete non-reflecting boundary conditions is developed.It can be usedfor a variety of wave equations such as the acoustic wave equation, the isotropic and anisotropic elastic waveequations and the equations for wave propagation in multi-phase media and so on.In this kind of boundaryconditions,the composition of all artifical reflected waves,but not the individual reflected ones,is consideredand eliminated.Thus, it has a uniform formula for different wave equations.The velocity C_A of the composedreflected wave is determined in the way to make the reflection coefficients minimal,the value of which depends onequations.In this psper,the construction of the boundary conditions illustrated and C_A is found,numericalresults are presented to illustrate the effectiveness of the boundary conditions.     

Global solvability for the Kirchhoff equations in exterior domains of dimension three

We consider the initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole space of dimension three, and show that these problems admit time-global solutions, provided the norms of the initial data in the usual Sobolev spaces of appropriate order are sufficiently small. We obtain uniform estimates of the L¹(R) norms with respect to time variable at each point in the domain, of solutions of initial (boundary) value problem for the linear wave equations. We then show that the estimates above yield the unique global solvability for the Kirchhoff equations.     

Well-posedness of the poincar�� problem in a cylindrical domain for the higher-dimensional wave equation

It is known that waves (acoustic waves, radio waves, elastic waves, and electric waves) in cylindrical tubes are described by the wave equation. In the theory of hyperbolic-type partial differential equations, boundary-value problems with data on the whole boundary serve as examples of ill-posedness of the posed problems. In this work, it is shown that the Poincar´e problem in a cylindrical domain for the higher-dimensional wave equation is uniquely solvable. A uniqueness criterion for a regular solution is also obtained.     

Exact boundary observability for 1‐D quasilinear wave equations

By means of a direct and constructive method based on the theory of semi‐global C² solution, the local exact boundary observability and an implicit duality between the exact boundary controllability and the exact boundary observability are shown for 1‐D quasilinear wave equations with various boundary conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

Integral equations for the Fourier-transformed boundary values for the transmission problems for right-angled wedges and octants

A 4×4-system of integral equations for the Fourier transformed boundary values of the normal derivatives of the wave functions defined in the four quadrants of R ²-space is derived. This system results from the scalar transmission problem with continuous passage of the boundary values of the total wave-fields and of the weighted normal derivatives corresponding to the case of magnetically polarized fields. Several equivalent systems of integral equations are deduced then which show that Banach's fixed point principle may be applied at least for slightly differing media in the four quadrants. The method, which is equivalent to a compatibility condition for holomorphic functions, may be generalized to the case of the scalar transmission problem for octants in R ³-space. There a 12 × 12-system of integral equations for the Fourier transformed normal derivatives on the quarter-plane faces is established.     

The blow-up curve of solutions of mixed problems for semilinear wave equations with exponential nonlinearities in one space dimension, II

In this paper, we consider mixed problems with a timelike boundary derivative (or a Dirichlet) condition for semilinear wave equations with exponential nonlinearities in a quarter plane. The case when the boundary vector field is tangent to the characteristic which leaves the domain in the future is also considered. We show that solutions either are global or blow up on a C¹ curve which is spacelike except at the point where it meets the boundary; at that point, it is tangent to the characteristic which leaves the domain in the future.