On weakly nonlinear waves in media exhibiting mixed nonlinearity

We obtain the transport equations governing small amplitude high frequency disturbances, that include both quadratic and cubic nonlinearities inherent in hyperbolic systems of conservation laws. The coefficients of the nonlinear terms in the transport equation are obtained in terms of the Glimm interaction coefficients. For symmetric and isotropic systems the mean curvature of the wave front, which appears as the coefficient of the linear term in the transport equation, is shown to be related to the derivative of the ray tube area along the bicharacteristics; the amplitude of the disturbance is shown to become unbounded in the neighborhood of the point where the ray tube collapses. We also obtain a formula, akin to the one obtained by R. Rosales (1991), for the energy dissipated across shocks.     

Singularities of multivalued solutions of nonlinear differential equations,and nonlinear phenomena

The purpose of this paper is to use the geometrical theory of nonlinear partial differential equations and the theory of singularities of maps in order to obtain the general scheme for constructing shock waves from multivalued solutions, given by smooth integral manifolds. This scheme is illustrated by some examples from gas dynamics, mechanics, acoustics and thermodynamics.     

On the shock dynamics of weak converging shock waves in solid materials

Ricerche di Matematica - We presented a geometrical shock dynamics model to predict the behavior of weak converging shock waves in solid materials. Taking into consideration...     

On evolutionary ray-projection dynamics

We introduce the ray-projection dynamics in evolutionary game theory by employing a ray projection of the relative fitness (vector) function, i.e., a projection unto the unit simplex along a ray through the origin. Ray-projection dynamics are weakly compatible in the terminology of Friedman (Econometrica 59:637–666, 1991), each of their interior fixed points is an equilibrium and each interior equilibrium is one of its fixed points. Furthermore, every interior evolutionarily stable strategy is an asymptotically stable fixed point, and every strict equilibrium is an evolutionarily stable state and an evolutionarily stable equilibrium. We also employ the ray-projection on a set of functions related to the relative fitness function and show that several well-known evolutionary dynamics can be obtained in this manner.     

On shock dynamics

This is in continuation of our paper On the propagation of a multi-dimensional shock of arbitrary strength’ published earlier in this journal (Srinivasan and Prasad [9]). We had shown in our paper that Whitham’s shock dynamics, based on intuitive arguments, cannot be relied on for flows other than those involving weak shocks and that too with uniform flow behind the shock. Whitham [12] refers to this as misinterpretation of his approximation and claims that his theory is not only correct but also provides a natural closure of the open system of the equations of Maslov [3]. The main aim of this note is to refute Whitham’s claim with the help of an example and a numerical integration of a problem in gasdynamics.     

Stoßwellenausbreitung in Rohrverzweigungen

Summary The interaction of a shock wave with the junction of a straight duct having a side branch is considered. The paper checks the validity of three hypotheses. The first and second one calculate the shock waves after the junctions only by a geometrical consideration. A comparison with experimental values shows only an unsatisfactory result for weak shock waves. The third theory assumes that the pressure drop across the junction, in the quasi-steady flow which takes place after the incident shock wave, is the same as in steady flow tests. The pressure drops were measured for different junctions and the results of the calculation were compared with experimental values from a shock tube, both for circular and rectangular channels. A good agreement was obtained.     

Variable elasticity of substituition in a discrete time Solow-Swan growth model with differential saving

We study the dynamics shown by the discrete time neoclassical one-sector growth model with differential savings as in Bohm and Kaas [4] while assuming VES production function in the form given by Revankar [24]. It is shown that the model can exhibit unbounded endogenous growth despite the absence of exogenous technical change and the presence of non-reproducible factors if the elasticity of substitution is greater than one. We then consider parameters range related to non-trivial dynamics (i.e. the elasticity of substitution in less than one and shareholders save more than workers) and we focus on local and global bifurcations causing the transition to more and more complex asymptotic dynamics. In particular, as our map is non-differentiable in a subset of the states space, we show that border collision bifurcations occur. Several numerical simulations support the analysis.     

具弱色散误差的单调差分格式

本文基于[1]、[2]的理论分析,分别给出了计算(一维、二维)非线性双曲方程组的两类具弱色散误差的单调格式。§2针对一维气动力学方程组,通过数值结果。就激波的逼近问题,与已有的格式进行了比较。     

Use of geometry in finite element thermal radiation combined with ray tracing

In heat transfer for space applications, the exchanges of energy by radiation play a significant role. In this paper, we present a method which combines the geometrical definition of the model with a finite element mesh. The geometrical representation is advantageous for the radiative component of the thermal problem while the finite element mesh is more adapted to the conductive part. Our method naturally combines these two representations of the model. The geometrical primitives are decomposed into cells. The finite element mesh is then projected onto these cells. This results in a ray tracing acceleration technique. Moreover, the ray tracing can be performed on the exact geometry, which is necessary if specular reflectors are present in the model. We explain how the geometrical method can be used with a finite element formulation in order to solve thermal situation including conduction and radiation. We illustrate the method with the model of a satellite.     

An investigation of the internal structure of shock profiles for shock capturing schemes

The theoretical understanding of discrete shock transitions obtained by shock capturing schemes is very incomplete. Previous experimental studies indicate that discrete shock transitions obtained by shock capturing schemes can be modeled by continuous functions, so called continuum shock profiles. However, the previous papers have focused on linear methods. We have experimentally studied the trajectories of discrete shock profiles in phase space for a range of different high resolution shock capturing schemes, including Riemann solver based flux limiter methods, high resolution central schemes and ENO type methods. In some cases, no continuum profiles exists. However, in these cases the point values in the shock transitions remain bounded and appear to converge toward a stable limit cycle. The possibility of such behavior was anticipated in Bultelle, Grassin and Serre, 1998, but no specific examples, or other evidence, of this behavior have previously been given. In other cases, our results indicate that continuum shock profiles exist, but are very complicated. We also study phase space orbits with regard to post shock oscillations.     

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.     

Multiplicative nonholonomic/Newton-like algorithm

We construct new algorithms, which use the fourth order cumulant of stochastic variables for the cost function. The multiplicative updating rule here constructed is natural from the homogeneous nature of the Lie group and has numerous merits for the rigorous treatment of the dynamics. As one consequence, the second order convergence is shown. For the cost function, functions invariant under the componentwise scaling are chosen. By identifying points which can be transformed to each other by the scaling, we assume that the dynamics is in a coset space. In our method, a point can move toward any direction in this coset. Thus, no prewhitening is required.     

Shock Wave Tracking for Hyperbolic Systems Exhibiting Local Linear Degeneracy

Extending the results of our previous study [2], we now investigate the propagation of interior shocks corresponding to the signaling problem of small-amplitude, high-frequency type. We derive a formula for the shock front and show that the previously constructed asymptotic solution is valid on both sides of this front. This solution is further distinguished to a higher order in which the effects of material inhomogeneity are accounted for. Moreover, if λ = λ( u , x) represents the eigenvalue under consideration, we show that the single-wave-mode boundary disturbance of [2] can lead only to a λ-shock. We also derive an entropy condition for the shock wave. As an application of our theory, the fluid-filled hyperelastic tube problem of [7] is further examined and an example calculation made in which we show that a compressive shock wave is generated at the shock-initiation point. This demonstration is effected as a particular example of the solution to a general bifurcation problem.     

二相平面渗流问题的不变流管解法及其理论分析

本文总结和改进了工程上广泛应用的求解二相平面渗流问题的不变流管近似方法.对其核心部份即给定压差时一维变截面流管中的二相驱替问题作了全面的考察,证明了解的存在唯一性,给出了准确解、数值解及其收敛性和稳定性分析.     

Computability and complexity of ray tracing

The ray-tracing problem is, given an optical system and the position and direction of an initial light ray, to decide if the light ray reaches some given final position. For many years ray tracing has been used for designing and analyzing optical systems. Ray tracing is now used extensively in computer graphics to render scenes with complex curved objects under global illumination. We show that ray-tracing problems in some three-dimensional simple optical systems (purely geometrical optics) are undecidable. These systems may consist of either reflective objects that are represented by rational quadratic equations, or refractive objects that are represented by rational linear equations. Some problems in more restricted models are shown to be PSPACE-hard or sometimes in PSPACE. A preliminary version of this paper appeared as “The Computability and Complexity of Optical Beam Tracing” in theProceedings of the 31st Annual Symposium on Foundations of Computer Science, October 1990, Vol. I, pp. 106–114. The research of J. H. Reif and A. Yoshida was supported in part by Air Force Contract No. AFOSR-87-0386, DARPA/ARO Contract No. DAAL03-88-K-0195, DARPA/ISTO Contract No. N00014-88-K-0458, and NASA subcontract 550-63 of primecontract NASS-30428. J.D. Tygar's research was supported in part by the Defense Advanced Research Projects Agency under Contract No. F33615-87-C-1449 and by a National Science Foundation Presidential Young Investigator Award under Contract No. CCR-8858087.     

Acoustic Wave Propagation in an Underwater Sound Channel 1. Qualitative Theory

We give a qualitative study of wave propagation in an inhomogeneousmedium principally by geometrical optics and ray theory. Theinhomogeneity is represented by a sound-speed profile whichis dependent upon one coordinate, namely the depth; and we discussthe general characteristics of wave propagation which resultfrom a source placed on the sound channel axis. We show thatour mathematical model of the sound-speed in the ocean actuallypredicts some of the behaviour of the observed physical phenomenain the underwater sound channel. Using ray theoretic techniqueswe investigate the implications of our profile on the followingcharacteristics of SOFAR propagation: (i) the sound energy travellingfurther away from the axis takes less time to travel from sourceto receiver than sound energy travelling closer to the axis,(ii) the focusing of sound energy in the sound channel at certainranges, (iii) the overall ray picture in the sound channel.     

Geometrical Optics and Models of Computer Memory Fragmentation

We analyze a two-dimensional difference equation that arose in a stochastic model of memory fragmentation in computers. We use the ray method of geometrical optics, and other singular perturbation methods, to solve this equation asymptotically. The asymptotic limit corresponds to one of heavy system usages. We interpret the results from the ray method probabilistically.     

On the propagation of a multi-dimensional shock of arbitrary strength

In this paper the kinematics of a curved shock of arbitrary strength has been discussed using the theory of generalised functions. This is the extension of Moslov’s work where he has considered isentropic flow even across the shock. The condition for a nontrivial jump in the flow variables gives the shock manifold equation (sme). An equation for the rate of change of shock strength along the shock rays (defined as the characteristics of the sme) has been obtained. This exact result is then compared with the approximate result of shock dynamics derived by Whitham. The comparison shows that the approximate equations of shock dynamics deviate considerably from the exact equations derived here. In the last section we have derived the conservation form of our shock dynamic equations. These conservation forms would be very useful in numerical computations as it would allow us to derive difference schemes for which it would not be necessary to fit the shock-shock explicitly.