Broadwell Equation Research Articles

Vanishing mean free path limit for interacting shock waves of Broadwell equation

In this paper, we study the vanishing mean free path limit for interacting shock waves of Broadwell equation. Given any entropy solution of Broadwell model system which consists of two different families of shocks interacting at some positive time, we show that such solution is the vanishing mean free path limit of a family of smooth global solution for the Broadwell model system.

Shock wave structure in a lattice gas

The motion and structure of shock and expansion waves in a simple particle system, a lattice gas and cellular automaton, are determined in an exact computation. Shock wave solutions, also exact, of a continuum description, a model Boltzmann equation, are compared with the lattice results. The comparison demonstrates that, as proved by Caprino et al. [“A derivation of the Broadwell equation,” Commun. Math. Phys. 135, 443 (1991)] only when the lattice processes are stochastic is the model Boltzmann description accurate. In the strongest shock wave, the velocity distribution function is the bimodal function proposed by Mott-Smith.

Fluid-Dynamic Limit for the Centered Rarefaction Wave of the Broadwell Equation

We study the asymptotic equivalence of the 1-d Broadwell model of the nonlinear Boltzmann equation to its corresponding Euler equation of compressible gas dynamics in the limit of small mean free path. We consider the case where the initial data are allowed to have jump discontinuities such that the corresponding solutions to the Euler equation contain centered rarefaction waves. In particular, Riemann data connected by rarefaction curves are included. We show that, as long as the initial data are a small perturbation of a non-vacuum constant state, the solution for the Broadwell equation exists globally in time and converges, in the small mean free path limit, to the solution of the corresponding Euler equation uniformly except for an initial layer whose width is essentially the order of the mean free path.

Two dimensional half-space problems for the Broadwell discrete velocity model

Stationary boundary value problems for the Broadwell model in a half-space and in a half-infinite channel are considered. By means of the analogy between the stationary boundary value problems for the Broadwell equations and the initial-boundary value problem of Carleman's system, solutions are found for various situations. Uniqueness and non-uniqueness of solutions is discussed as well. The non-uniqueness problem in the channel leads to the investigation of the initial value problem for Carleman's equation with partly negative initial densities. Some new results for this problem are given.

Kinetic and viscous boundary layers for broadwell equations

In this paper, we investigate the boundary layer behavior of solutions to the one dimensional Broadwell model of the nonlinear Boltzmann equation for small mean free path. We consider the analogue ...

Kinetic limits of the HPP cellular automaton

We study the Boltzmann-Grad limit in various versions of the two-dimensional HPP cellular automaton. In the completely deterministic case we prove convergence to an evolution that is not of kinetic type, a well-known phenomenon after Uchyiama's paper on the Broadwell gas, whereas the limiting equation becomes of kinetic type in the model with random collisions. The main part of the paper concerns the case where the collisions are deterministic and the randomness comes from inserting, between any two successive HPP updatings,ɛ-ν stirring updatings, ν<1 being any fixed positive number andɛ a parameter which tends to 0. The initial measure is a product measure with average occupation numbers of the order ofɛ (low-density limit) and varying on distances of the order of ɛ−1. The limit asɛ → 0 of the system evolved for times of the order ofɛ-1-ν corresponds to the Boltzmann-Grad limit. We prove propagation of chaos and that the renormalized average occupation numbers (i.e., divided byɛ) converge to the solution of the Broadwell equation. Convergence is proven at all times for which the solution of the Broadwell equation is bounded.

THE INITIAL BOUNDARY VALUE PROBLEM FOR THE FOUR-VELOCITY PLANE BROADWELL MODEL

The initial boundary value problem for the four-velocity Broadwell equations on a square with specular reflection on the boundary is considered. Uniqueness and existence of solutions on finite time interval is established. The length of the time interval depends on the initial data. The proof of existence uses the procedure known in numerical analysis as the fractional step method. It shows that the method is convergent justifying its usage in computations.

The fluid‐dynamic limit of the Broadwell model of the nonlinear Boltzmann equation in the presence of shocks

AbstractThis paper studies the asymptotic equivalence of the Broadwell model of the nonlinear Boltzmann equation to its corresponding Euler equation of compressible gas dynamics in the limit of small mean free path ε. It is shown that the fluid dynamical approximation is valid even if there are shocks in the fluid flow, although there are thin shock layers in which the convergence does not hold. More precisely, by assuming that the fluid solution is piecewise smooth with a finite number of noninteracting shocks and suitably small oscillations, we can show that there exist solutions to the Broadwell equations such that the Broadwell solutions converge to the fluid dynamical solutions away from the shocks at a rate of order (ε) as the mean free path ε goes to zero. For the proof, we first construct a formal solution for the Broadwell equation by matching the truncated Hilbert expansion and shock layer expansion. Then the existence of Broadwell solutions and its convergence to the fluid dynamic solution is reduced to the stability analysis for the approximate solution. We use an energy method which makes full use of the inner structure of time dependent shock profiles for the Broadwell equations.

Decay theorems for the Broadwell equations

In this note we establish boundedness and decay theorems for finitely supported solutions to the Broadwell equations with four velocity components. There is a vast literature on these and similar equations (for details see the survey of Platkowski & Illner and the references contained therein). The particular variant of them we examine seems new; though nonlinear, they are homogeneous of degree one

A derivation of the Broadwell equation

We consider a stochastic system of particles in a two dimensional lattice and prove that, under a suitable limit (i.e.N→∞, e→0,Ne2→const, whereN is the number of particles and e is the mesh of the lattice) the one-particle distribution function converges to a solution of the two-dimensional Broadwell equation for all times for which the solution (of this equation) exists. Propagation of chaos is also proven.

Stability of shock waves for the Broadwell equations

For the Broadwell model of the nonlinear Boltzmann equation, there are shock profile solutions, i.e. smooth traveling waves that connect two equilibrium states. For weak shock waves, we prove asymptotic (in time) stability with respect to small perturbations of the initial data. Following the work of Liu [7] on shock wave stability for viscous conservation laws, the method consists of analyzing the solution as the sum of a shock wave, a diffusive wave, a linear hyperbolic wave and an error term. The diffusive and linear hyperbolic waves are approximate solutions of the fluid dynamic equations corresponding to the Broadwell model. The error term is estimated using a variation of the energy estimates of Kawashima and Matsumura [6] and the characteristic energy method of Liu [7].