Delta shock waves with Dirac delta function in both components for systems of conservation laws

We study a class of non-strictly and weakly hyperbolic systems of conservation laws which contain the equations of geometrical optics as a prototype. The Riemann problems are constructively solved. The Riemann solutions include two kinds of interesting structures. One involves a cavitation where both state variables tend to zero forming a singularity, the other is a delta shock wave in which both state variables contain Dirac delta function simultaneously. The generalized Rankine–Hugoniot relation and entropy condition are proposed to solve the delta shock wave. Moreover, with the limiting viscosity approach, we show all of the existence, uniqueness and stability of solution involving the delta shock wave. The generalized Rankine–Hugoniot relation is also confirmed. Then our theory is successfully applied to two typical systems including the geometric optics equations. Finally, we present the numerical results coinciding with the theoretical analysis.     

The singular solutions to a nonsymmetric system of Keyfitz-Kranzer type with initial data of Riemann type

The solutions to the Riemann problem for a nonsymmetric system of Keyfitz-Kranzer type are constructed explicitly when the initial data are located in the quarter phase plane. In particular, some singular hyperbolic waves are discovered when one of the Riemann initial data is located on the boundary of the quarter phase plane, such as the delta shock wave and some composite waves in which the contact discontinuity coincides with the shock wave or the wave back of rarefaction wave. The double Riemann problem for this system with three piecewise constant states is also considered when the delta shock wave is involved. Furthermore, the global solutions to the double Riemann problem are constructed through studying the interaction between the delta shock wave and the other elementary waves by using the method of characteristics. Some interesting nonlinear phenomena are discovered during the process of constructing solutions; for example, a delta shock wave is decomposed into a delta contact discontinuity and a shock wave.     

Riemann problem for a geometrical optics system

The paper solves analytically the Riemann problem for a nonstrictly hyperbolic system of conservation laws arising in geometrical optics,in which the flux contains the nonconvex function possessing an infinite number of inflection points.Firstly,the generalized Rankine–Hugoniot relations and entropy condition of delta shock waves and left(right)-contact delta shock waves are proposed and clarified.Secondly,with the help of the convex hull,seven kinds of structures of Riemann solutions are obtained.The solutions fall into three broad categories with a series of geometric structures involving simultaneously contact discontinuities,vacuums and delta shock waves.Finally,numerical experiments confirm the theoretical analysis.     

The limits of Riemann solutions to the simplified pressureless Euler system with flux approximation

The formation of vacuum state and delta shock wave in the solutions to the Riemann problem for the simplified pressureless Euler system is considered under the linear approximations of flux functions. The method is to perturb the non‐strictly hyperbolic system into a nearby strictly hyperbolic system by introducing appropriately the linear approximations of flux functions. The solutions to the Riemann problem for the approximated system can be constructed explicitly and then the formation of vacuum state and delta shock wave can be observed by taking the perturbation parameter tend to zero in the solutions.     

Approximate inertial manifolds of Burgers equation approached by nonlinear Galerkin’s procedure and its application

The approximate inertial manifolds (AIMs) of Burgers equation is approached by nonlinear Galerkin methods, and it can be used to capture and study the shock wave numerically in a reduced system with low dimension. Following inertial manifolds, the asymptotic behavior of Burgers equation, an infinite dimensional dissipative dynamic systems, will evolve to a compact set known as a global attractor, which is finite-dimensional, and the nonlinear phenomena are included and captured in such global attractor. In the application, nonlinear Galerkin methods is introduced to approach such inertial manifolds. By this method, the solution of the original system is projected onto the complete space spanned by the eigenfunctions or the modes of the linear operator of Burgers equation, and nonlinear Galerkin method splits the infinite-dimensional phase space into two complementary subspaces: a finite-dimensional one and its infinite-dimensional complement. Then, the post-processed Galerkin’s procedure is used to approximate the solution of the reduced system, with the introduction of the interaction between lower and higher modes. Additionally, some numerical examples are presented to make a comparison between the traditional Galerkin method and nonlinear Galerkin method, in particular, some sharp jumping phenomena, which are related to the shock wave, have been captured by the numerical method presented. As the conclusion, it can be drawn that it is possible to completely describe the dynamics on the attractor of a nonlinear partial differential equation (PDE) with a finite-dimensional dynamical system, and the study can provide a numerical method for the analysis of the nonlinear continuous dynamic systems and complicated nonlinear phenomena in finite-dimensional dynamic system, whose nonlinear dynamics has been developed completely compared with infinite-dimensional dynamic system.     

Delta and singular delta locus for one‐dimensional systems of conservation laws

This work gives a condition for existence of singular and delta shock wave solutions to Riemann problem for 2×2 systems of conservation laws. For a fixed left‐hand side value of Riemann data, the condition obtained in the paper describes a set of possible right‐hand side values. The procedure is similar to the standard one of finding the Hugoniot locus. Fluxes of the considered systems are globally Lipschitz with respect to one of the dependent variables. The association in a Colombeau‐type algebra is used as a solution concept. Copyright © 2004 John Wiley &Sons, Ltd.     

Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws

We prove that the Riemann solutions are stable for a nonstrictly hyperbolic system of conservation laws under local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the interactions of delta shock waves with shock waves and rarefaction waves. During the interaction process of the delta shock wave with the rarefaction wave, a new kind of nonclassical wave, namely a delta contact discontinuity, is discovered here, which is a Dirac delta function supported on a contact discontinuity and has already appeared in the interaction process for the magnetohydrodynamics equations [M. Nedeljkov and M. Oberguggenberger, Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl. 344 (2008) 1143-1157]. Moreover, the global structures and large time asymptotic behaviors of the solutions are constructed and analyzed case by case.     

Response of multiwall carbon nanotubes to impact loading

Dynamic behaviors of multiwall carbon nanotubes (MWCNTs) with finite length are investigated using an analytical method. Multiple elastic shells and linearized model of van der Waals forces are used for development a comprehensive continuum dynamic model of MWCNTs. By applying Laplace transform, analytical solution for thin and thick MWCNTs under dynamic loading are obtained. Dynamic responses of 3-, 9-, and 11-layer MWCNTs under external pressure shock are examined and accuracy of results are verified by comparison the results with those obtained by numerical methods. Both displacement and stress analysis are performed for layers of MWCNTs and frequencies of oscillations are obtained. Also, effects of axial wave created by external pressure shock are studied in MWCNTs with two-dimensional analyses. Dynamic responses of MWCNTs with initial axial displacement are also proposed and the propagation of the axial wave through the length of tubes is illustrated. Furthermore, wave propagation velocity is found by analysis of time history diagram.     

Shadow wave solution for the generalized Langmuir isotherm in chromatography

The shadow wave solution of the Riemann problem for the chromatography system under the mixed competitive-cooperative generalized Langmuir isotherm is constructed. It is shown that this shadow wave solution is weakly unique in the sense that all the entropy shadow wave solutions have the same distributional limit which is exactly the corresponding delta shock wave solution of the Riemann problem. Furthermore, the generalized Riemann problem with the delta type initial data is also considered and the existence and weak uniqueness of a solution are obtained in the framework of shadow wave solution.     

Global structure of Riemann solutions to a system of two-dimensional hyperbolic conservation laws

The Riemann problem for a two-dimensional nonstrictly hyperbolic system of conservation laws is considered. Without the restriction that each jump of the initial data projects one planar elementary wave, ten topologically distinct solutions are obtained by applying the method of generalized characteristic analysis. Some of these solutions involve the nonclassical waves, i.e., the delta shock wave and the delta contact discontinuity, for which we explicitly give the expressions of their strengths, locations and propagation speeds. Moreover, we demonstrate that the nature of our solutions is identical with that of solutions to the corresponding one-dimensional Cauchy problem, which provides a verification that our construction produces the correct unique global solutions.     

Linearization of the Riemann problem for a triangular system of conservation laws and delta shock wave formation process

Using the weak asymptotic method, we approximate a triangular system of conservation laws arising from the so‐called generalized pressureless gas dynamics by a diagonal linear system. Then, we apply the usual method of characteristics to find approximate solution to the original system. As a consequence, we shall see how the delta shock wave naturally arises along the characteristics. Also, we propose a procedure that could be applied to more general systems of conservation laws. Copyright © 2009 John Wiley & Sons, Ltd.     

3D dynamic Green’s functions in a multilayered poroelastic half-space

The complete 3D dynamic Green’s functions in the multilayered poroelastic media are presented in this study. A method of potentials in cylindrical coordinate system is applied first to decouple the Biot’s wave equations into four scalar Helmholtz equations, and then, general solutions to 3D wave propagation problems are obtained. After that, a three vector base and the propagator matrix method are introduced to treat 3D wave propagation problems in the stratified poroelastic half-space disturbed by buried sources. It is known that the original propagator algorithm has the loss-of-precision problem when the waves become evanescent. At present, an orthogonalization procedure is inserted into the matrix propagation loop to avoid the numerical difficulty of the original propagator algorithm. At last, the validity of the present approach for accurate and efficient calculating 3D dynamic Green’s functions of a multilayered poroelastic half-space is confirmed by comparing the numerical results with the known exact analytical solutions of a uniform poroelastic half-space.     

Transonic shock and rarefaction wave interactions of two-dimensional Riemann problems for the self-similar nonlinear wave system

This paper addresses the self-similar transonic irrotational flow in gas dynamics in two space dimensions.We consider a configuration that the incident shock becomes a transonic shock as it enters the sonic circle, interacts with the rarefaction wave downstream, and then becomes sonic. The rarefaction wave further downstream becomes sonic (degenerate) creating an unknown boundary for the governing system. We present the Riemann data for this configuration, provide the characteristic decomposition of the system, and formulate the boundary value problem for this configuration. The numerical results are presented, and a method to establish the existence result is briefly discussed.     

Interactions of delta shock waves for a class of nonstrictly hyperbolic system of conservation laws

In this paper, we study the perturbed Riemann problem for a class of nonstrictly hyperbolic system of conservation laws, and focuse on the interactions of delta shock waves with the shock waves and the rarefaction waves. The global solutions are constructed completely with the method of splitting delta function. In solutions, we find a new kind of nonclassical wave, which is called delta contact discontinuity with Dirac delta function in both components. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. Moreover, by letting perturbed parameter $\varepsilon$ tend to zero, we analyze the stability of Riemann solutions.     

Time-domain finite element method to generalized diffusion-elasticity problems with the concentration-dependent elastic constants and the diffusivity

The investigations of mechanical-diffusion coupling are of great importance for the micro-electromechanical devices under non-uniform concentration environment, especially with the development of energy storage technology for a rapid charging system. In recent years there have been many experimental and theoretical studies show that the elastic constants and the diffusivity depend on the concentration of diffusing substances. In view of this, present work aims to study generalized diffusion-elasticity problems considering the concentration-dependent elastic constants and the diffusivity by time-domain finite element method. By using principle of virtual work, the obtained nonlinear finite element equations are solved directly in time domain to minimize precision losses in the application of integrated transformation method, and then the nonlinear solutions can be obtained. As numerical examples, the developed method is used to investigate the transient response of a thick circular plate subjected to the shock loading of the concentration. The results demonstrate that the developed method can faithfully predict the deformation of structure and most importantly the diffusive wave feature in both one-/two-dimensional solids whilst it is commonly difficult to model, especially for two-dimensional case, by using transform method. Parametric studies are performed to evaluate and discuss the effects of concentration-dependent elastic constants and diffusivity on the structural dynamic responses.     

A new averaging scheme for the riemann problem in pure water

The numerical investigation of shock phenomena in gas or liquid media where enthalpy is the preferred thermodynamic variable poses special problems. When an expression for internal energy is available, the usual procedure is to employ a splitting scheme to remove source terms from the Euler equations, then upwind-biased shock capturing algorithms are built around the Riemann problem for the conservative system which remains. However, when the governing equations arc formulated in terms of total enthalpy, treatment of a pressure time derivative as a source term leads to a Riemann problem for a system where one equation is not a conservation law. The present research establishes that successful upwind-biased shock capturing schemes can be based upon the pseudo-conservative system. A new averaging scheme for solving the associated Riemann problem is developed. The method is applied to numerical simulations of shock wave propagation in pure water.     

Vanishing pressure limits of Riemann solutions to the isentropic relativistic Euler system for Chaplygin gas

The Riemann solutions to the isentropic relativistic Euler system for Chaplygin gas with a small parameter are considered. Unlike the polytropic or barotropic gas cases, we find that firstly, as the parameter decreases to a certain critical number, the two-shock solution converges to a delta shock wave solution of the same system. Moreover, as the parameter goes to zero, that is, the pressure vanishes, the solution is nothing but the delta shock wave solution to the zero-pressure relativistic Euler system. Meanwhile, the two-rarefaction wave solution tends to the vacuum solution to the zero-pressure relativistic system, and the solution containing one rarefaction wave and one shock wave tends to the contact discontinuity solution to the zero-pressure relativistic system as pressure vanishes.     

Shadow waves for pressureless gas balance laws

The pressureless gas dynamic model with body force as a source is considered. The problem is solved using the procedure used for finding delta shock type solutions to a special conservation laws known as Shadow Waves. If the body force is interpreted as the acceleration constant multiplied by the density, the solution obtained in this paper looks physically reasonable since the velocities of waves are changed accordingly with the acceleration.     

一维Chaplygin气体欧拉方程组中波的相互作用

研究一维Chaplygin气体欧拉方程组中波的相互作用.方程组的波包含接触间断和在密度变量以及内能变量上同时具有狄拉克函数的狄拉克激波.根据这些波的不同组合,问题被分成了7种情形.通过详细地构造每种情形的整体解,获得了各种波相互作用的完整结果.特别地,对于一类初值,两个接触间断相互作用后,产生了一个狄拉克激波;然而,对于另外一类初值,一个狄拉克激波与一个接触间断相互作用后,狄拉克激波消失.这些都是波相互作用中非常特别的现象.     

Riemann problem and elementary wave interactions in isentropic magnetogasdynamics

In this paper, we consider the Riemann problem for a quasilinear hyperbolic system of equations governing the one dimensional unsteady simple wave flow of an isentropic, inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field. This class of equations includes, as a special case, the equations of isentropic gasdynamics. We study the shock and rarefaction waves and their properties, and discuss the geometry of shock curves using the Riemann invariant coordinates. Under certain conditions, we show the existence and uniqueness of the solution to the Riemann problem for arbitrary initial data, and then discuss the vacuum state in isentropic magnetogasdynamics. Finally, we discuss numerical results for different initial data, and discuss all possible interactions of elementary waves. It is noticed that although the magnetogasdynamic system is more complex than the corresponding gasdynamic system, all the parallel results remain identical. However, unlike the ordinary gasdynamic case, the solution inside rarefaction waves in magnetogasdynamics cannot be obtained directly and explicitly; indeed, it requires an extra iteration procedure. It is also observed that the presence of a magnetic field makes both the shock and rarefaction stronger compared to what they would have been in the absence of a magnetic field.