An algebraic approach to discrete mechanics

Using basic ideas from algebraic geometry, we extend the methods of Lagrangian and symplectic mechanics to treat a large class of discrete mechanical systems, that is, systems such as cellular automata in which time proceeds in integer steps and the configuration space is discrete. In particular, we derive an analog of the Euler-Lagrange equation from a variational principle, and prove an analog of Noether's theorem. We also construct a symplectic structure on the analog of the phase space, and prove that it is preserved by time evolution.     

Strongly differentiable solutions of the discrete coagulation-fragmentation equation

We examine an infinite system of ordinary differential equations that models the binary coagulation and multiple fragmentation of clusters. In contrast to previous investigations, our analysis does not involve finite-dimensional truncations of the system. Instead, we treat the problem as an infinite-dimensional differential equation, posed in an appropriate Banach space, and apply perturbation results from the theory of strongly continuous semigroups of operators. The existence and uniqueness of physically meaningful solutions are established for uniformly bounded coagulation rates but with no growth restrictions imposed on the fragmentation rates.     

Riccati-coupled similarity shock wave solutions for multispeed discrete Boltzmann models

We study nonstandard shock wave similarity solutions for three multispeed discrete Boltzmann models: (1) the square 8_i, model with speeds 1 and 2 with thex axis along one median, (2) the Cabannes cubic 14_i model with speeds 1 and 3 and thex axis perpendicular to one face, and (3) another 14_i, model with speeds 1 and 2. These models have five independent densities and two nonlinear Riccati-coupled equations. The standard similarity shock waves, solutions of scalar Riccati equations, are monotonic and the same behavior holds for the conservative macroscopic quantities. First, we determine exact similarity shock-wave solutions of coupled Riccati equations and we observe nonmonotonic behavior for one density and a smaller effect for one conservative macroscopic quantity when we allow a violation of the microreversibility. Second, we obtain new results on the Whitham weak shock wave propagation. Third, we solve numerically the corresponding dynamical system, with microreversibility satisfied or not, and we also observe the analogous nonmonotonic behavior.     

Exact 1+1 dimensional solutions of the discrete velocity Broadwell-Boltzmann model

For the 6-velocity Broadwell model with four independent densities, exact 1+1 dimensional solutions (space x, time t) are found. They are sums of two similarity waves with components either real or complex conjugate. The first case provides 1+1 dimensional shock-waves. In the second case we build periodic solutions in the space which either are propagating or nonpropagating in time. The same method has been successfully applied to other discrete models.     

Multiplication of solutions for linear overdetermined systems of partial differential equations

A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE’s contains the Cauchy–Riemann equations and the cofactor pair systems, included as special cases. The multiplication provides a method for generating, in a pure algebraic way, large classes of non-trivial solutions that can be constructed by forming convergent power series of trivial solutions.     

A discrete velocity model of a gas: Global in time solutions of the BBGKY hierarchy

In the present paper we study the evolution of a system of hard disks moving in the plane with a finite number of velocities as in the framework of a discrete velocity model of the Enskog equation, proposed in previous papers. Starting from the BBGKY hierarchy of such a system we give existence and uniqueness results for the initial value problem in suitable Banach spaces. In particular, the main result presented is the global in time weak solution to the BBGKY hierarchy for local equilibrium initial data, in the thermodynamic limit.     

Orthogonal polynomial approach to discrete Lax pairs for initial boundary-value problems of the QD algorithm

Using orthogonal polynomial theory, we construct the Lax pair for the quotient-difference algorithm in the natural Rutishauser variables. We start by considering the family of orthogonal polynomials corresponding to a given linear form. Shifts on the linear form give rise to adjacent families. A compatible set of linear problems is made up from two relations connecting adjacent and original polynomials. Lax pairs for several initial boundary-value problems are derived and we recover the discrete-time Toda chain equations of Hirota and of Suris. This approach allows us to derive a Bäcklund transform that relates these two different discrete-time Toda systems. We also show that they yield the same bilinear equation up to a gauge transformation. The singularity confinement property is discussed as well.     

Metastable fluid flow described via a discrete-velocity coagulation-fragmentation model

A discrete-velocity Boltzmann model is introduced. It is based on two principles: (i) clusters of particles move in ³ with seven fixed momenta; (ii) clusters may gain or lose particles according to the rules of Becker-Döring cluster equations. The model provides a kinetic representation of evaporation and condensation. The model is used to obtain macroscopic fluid equations which are valid into the metastable fluid regime, , where is any positive number, is the inelastic Knudsen number, and _s is the saturation density.     

Geometric discrete analogues of tangent bundles and constrained Lagrangian systems

Discretizing variational principles, as opposed to discretizing differential equations, leads to discrete-time analogues of mechanics, and, systematically, to geometric numerical integrators. The phase space of such variational discretizations is often the set of configuration pairs, analogously corresponding to initial and terminal points of a tangent vector. We develop alternative discrete analogues of tangent bundles, by extending tangent vectors to finite curve segments, one curve segment for each tangent vector. Towards flexible, high order numerical integrators, we use these discrete tangent bundles as phase spaces for discretizations of the variational principles of Lagrangian systems, up to the generality of nonholonomic mechanical systems with nonlinear constraints. We obtain a self-contained and transparent development, where regularity, equations of motion, symmetry and momentum, and structure preservation, all have natural expressions.     

Large-time behavior of the Broadwell model of a discrete velocity gas

We study the behavior of solutions of the one-dimensional Broadwell model of a discrete velocity gas. The particles have velocity ±1 or 0; the total mass is assumed finite. We show that at large time the interaction is negligible and the solution tends to a free state in which all the mass travels outward at speed 1. The limiting behavior is stable with respect to the initial state. No smallness assumptions are made.Partially supported by N.S.F. Grant No. NSF-DMS-84-08393     

A new contribution to nonlinear stability of a discrete velocity model

For a class of discrete velocity models of kinetic theory we prove exponential nonlinear conditional stability of the constant basic state in the slab [0, 1].     

The geometry of peaked solitons and billiard solutions of a class of integrable PDE's

The purpose of this Letter is to investigate the geometry of new classes of soliton-like solutions for integrable nonlinear equations. One example is the class of peakons introduced by Camassa and Holm [10] for a shallow water equation. We put this equation in the framework of complex integrable Hamiltonian systems on Riemann surfaces and draw some consequences from this setting. Amongst these consequences, one obtains new solutions such as quasiperiodic solutions,n-solitons, solitons with quasiperiodic background, billiard, andn-peakon solutions and complex angle representations for them. Also, explicit formulas for phase shifts of interacting soliton solutions are obtained using the method of asymptotic reduction of the corresponding angle representations. The method we use for the shallow water equation also leads to a link between one of the members of the Dym hierarchy and geodesic flow onN-dimensional quadrics. Other topics, planned for a forthcoming paper, are outlined.Research supported in part by DOE CHAMMP and HPCC programs.Research partially supported by the Department of Energy, the Office of Naval Research and the Fields Institute for Research in the Mathematical Sciences.     

A fast iterative model for discrete velocity calculations on triangular grids

A fast synthetic type iterative model is proposed to speed up the slow convergence of discrete velocity algorithms for solving linear kinetic equations on triangular lattices. The efficiency of the scheme is verified both theoretically by a discrete Fourier stability analysis and computationally by solving a rarefied gas flow problem. The stability analysis of the discrete kinetic equations yields the spectral radius of the typical and the proposed iterative algorithms and reveal the drastically improved performance of the latter one for any grid resolution. This is the first time that stability analysis of the full discrete kinetic equations related to rarefied gas theory is formulated, providing the detailed dependency of the iteration scheme on the discretization parameters in the phase space. The corresponding characteristics of the model deduced by solving numerically the rarefied gas flow through a duct with triangular cross section are in complete agreement with the theoretical findings. The proposed approach may open a way for fast computation of rarefied gas flows on complex geometries in the whole range of gas rarefaction including the hydrodynamic regime.     

Classical Yang-Mills-Dirac equations: Qualitative analysis of some solutions with a noncompact symmetry group

We consider classical Yang-Mills and Yang-Mills-Dirac equations on Minkowski space, with gauge group SU(2), and look for solutions invariant (up to a gauge transformation) under SO(3)×SO₀(1, 1) and SO₀(2, 1)×SO(2), respectively. In each case, we analyze the qualitative features of the solutions, in particular the asymptotic behavior as the solution approaches its singularities. The method is based on standard theorems from the theory of nonlinear ordinary differential equations.     

Lagrangian submanifolds and the Euler-Lagrange equations in higher-order mechanics

A direct construction of the Euler-Lagrange equations in higher-order mechanics as a submanifold of a higher-order tangent bundle is given, starting from the Lagrangian submanifold defined by the Lagrangian function. This construction uses higher-order tangent bundle geometry, derives the Euler-Lagrange equations as the constraint equations of a submanifold, and makes no assumptions about the regularity of the Lagrangian.     

On the oscillons in the signum-Gordon model

New periodic solutions of signum-Gordon equation are presented. We first find solutions φ₀(x, t) defined for (x, t) ∈ ? × [0, T ] and satisfying the condition φ₀(x, 0) = φ₀(x, T ) = 0. Then these solutions are extended to the whole spacetime by using (2.4).     

Microlocal analysis and phase-space decompositions

A general class of phase-space decompositions f(x)=a(x,p) dp of functions f defined in ⁿ is presented. In the latter, a(x,p) depends on values of the Fourier transform F of f in a region around p whose width tends to zero as |x| increases and it decays exponentially, for each p, in all directions in x-space outside the microsupport at p of F, with a rate of exponential fall-off linked to analyticity properties of F in local tubes (in complex space) around p. A possible application in quantum-field theory is mentioned.     

Monte Carlo solution of the Boltzmann equation via a discrete velocity model

A new discrete velocity scheme for solving the Boltzmann equation is described. Directly solving the Boltzmann equation is computationally expensive because, in addition to working in physical space, the nonlinear collision integral must also be evaluated in a velocity space. Collisions between each point in velocity space with all other points in velocity space must be considered in order to compute the collision integral most accurately, but this is expensive. The computational costs in the present method are reduced by randomly sampling a set of collision partners for each point in velocity space analogous to the Direct Simulation Monte Carlo (DSMC) method. The present method has been applied to a traveling 1D shock wave. The jump conditions across the shock wave match the Rankine–Hugoniot jump conditions. The internal shock wave structure was compared to DSMC solutions, and good agreement was found for Mach numbers ranging from 1.2 to 10. Since a coarse velocity discretization is required for efficient calculation, the effects of different velocity grid resolutions are examined. Additionally, the new scheme’s performance is compared to DSMC and it was found that upstream of the shock wave the new scheme performed nearly an order of magnitude faster than DSMC for the same upstream noise. The noise levels are comparable for the same computational effort downstream of the shock wave.