Discrete Evolution Equations Research Articles

Fast Summation by Interval Clustering for an Evolution Equation with Memory

We solve a fractional diffusion equation using a piecewise-constant, discontinuous Galerkin method in time combined with a continuous, piecewise-linear finite element method in space. If there are $N$ time levels and $M$ spatial degrees of freedom, then a direct implementation of this method requires $O(N^2M)$ operations and $O(NM)$ active memory locations, owing to the presence of a memory term: at each time step, the discrete evolution equation involves a sum over all previous time levels. We show how the computational cost can be reduced to $O(MN\log N)$ operations and $O(M\log N)$ active memory locations.

A theorem allowing the derivation of deterministic evolution equations from stochastic evolution equations. III The Markovian–non-Markovian mix

The proof of a theorem that allows one to construct deterministic evolution equations from a set, with two subsets, containing two types of discrete stochastic evolution equation is developed. One subset evolves Markovianly and the other non-Markovianly. As an illustrative example, the deterministic evolution equations of quantum electrodynamics are derived from two sets of Markovian and non-Markovian stochastic evolution equations, of different type, after an average over realization, using the theorem. This example shows that deterministic differential equations that contain both first-order and second-order time derivatives can be derived after a Taylor series expansion of the dynamical variables. It is shown that the derivation of such deterministic differential equations can be done by solving a set of linear equations. Two explicit examples, the first containing updating rules that depend on one previous time step and the second containing updating rules that depend on two previous time steps, are given in detail in order to show step by step the linear transformations that allow one to obtain the deterministic differential equations.

Smoothed particle hydrodynamics simulation of shear-induced powder migration in injection moulding

We present the application of the smoothed particle hydrodynamics (SPH) discretization scheme to Phillips' model for shear-induced particle migration in concentrated suspensions. This model provides an evolution equation for the scalar mean volume fraction of idealized spherical solid particles of equal diameter which is discretized by the SPH formalism. In order to obtain a discrete evolution equation with exact conservation properties we treat in fact the occupied volume of the solid particles as the degree of freedom for the fluid particles. We present simulation results in two- and three-dimensional channel flow. The two-dimensional results serve as a verification by a comparison to analytic solutions. The three-dimensional results are used for a comparison with experimental measurements obtained from computer tomography of injection moulded ceramic microparts. We observe the best agreement of measurements with snapshots of the transient simulation for a ratio D(c)/D(η)=0.1 of the two model parameters.

Non-commutative integrability, paths and quasi-determinants

In previous work, we showed that the solution of certain systems of discrete integrable equations, notably Q and T-systems, is given in terms of partition functions of positively weighted paths, thereby proving the positive Laurent phenomenon of Fomin and Zelevinsky for these cases. This method of solution is amenable to generalization to non-commutative weighted paths. Under certain circumstances, these describe solutions of discrete evolution equations in non-commutative variables: Examples are the corresponding quantum cluster algebras (Berenstein and Zelevinsky (2005) [3]), the Kontsevich evolution (Di Francesco and Kedem (2010) [10]) and the T-systems themselves (Di Francesco and Kedem (2009) [8]). In this paper, we formulate certain non-commutative integrable evolutions by considering paths with non-commutative weights, together with an evolution of the weights that reduces to cluster algebra mutations in the commutative limit. The general weights are expressed as Laurent monomials of quasi-determinants of path partition functions, allowing for a non-commutative version of the positive Laurent phenomenon. We apply this construction to the known systems, and obtain Laurent positivity results for their solutions in terms of initial data.

Perturbation theory, stability, boundedness and asymptotic behaviour for second order evolution equation in discrete time

This work deals with the existence, uniqueness and stability of solutions for semilinear discrete second order evolution equations in Banach spaces by using recent characterization of well-posedness for second order evolution equations in terms of R-boundedness and l p -multipliers. We also give results on boundedness and asymptotic behaviour for semilinear discrete evolution equations.

Lattice Boltzmann method for Lennard-Jones fluids based on the gradient theory of interfaces

In the present study we propose a lattice Boltzmann equation (LBE) model derived from density gradient expansions of the discrete BBGKY evolution equations. The model is based on the mechanical approach of the gradient theory of interfaces. The basic input is the radial distribution function, which is related exclusively to the molecular interaction potential, rather than semiempirical equations of state used in previous LBE models. This function can be provided from independent molecular simulations or from approximate theories. Evidently the accuracy of the interaction potential, and thus the radial distribution function, reflects on the accuracy of the thermodynamic properties and consistency of the derived LBE model. We have applied the proposed model to obtain equilibrium bulk and interfacial properties of a Lennard-Jones fluid at different temperatures, T, close to critical, T(c). The results demonstrate that the LBE model is in excellent agreement with gradient theory as well as with independent literature results based on different molecular simulation approaches. Hence the proposed LBE model can recover accurately bulk and interfacial thermodynamics for a Lennard Jones fluid at T/T(c)>0.9.

Well-posedness of second order evolution equation on discrete time

We characterize the well-posedness for second order discrete evolution equations in unconditional martingale difference spaces by means of Fourier multipliers and R-boundedness properties of the resolvent operator which defines the equation. Applications to semilinear problems are given.

Initial-boundary-value problems for discrete linear evolution equations

We present a transform method for solving initial-boundary-value problems (IBVPs) for linear semidiscrete (differential-difference) and fully discrete (difference-difference) evolution equations. The method is the discrete analogue of the one recently proposed by A. S. Fokas to solve IBVPs for evolution linear partial differential equations. We show that any discrete linear evolution equation can be written as the compatibility condition of a discrete Lax pair, namely, an overdetermined linear system of equations containing a spectral parameter. As in the continuum case, the method employs the simultaneous spectral analysis of both parts of the Lax pair, the symmetries of the evolution equation and a relation, called the global algebraic relation, that couples all known and unknown boundary values. The method applies for differential-difference equations in one lattice variable as well as for multi-dimensional and fully discrete evolution equations. We demonstrate the method by discussing explicitly several examples.

Algorithms for coupled problems that preserve symmetries and the laws of thermodynamics: Part I: Monolithic integrators and their application to finite strain thermoelasticity

We propose discretizations for a general class of nonlinear, coupled, thermomechanical problems of evolution. The most salient feature of the new methods is that they rigorously preserve the two laws of thermodynamics as well as the symmetries of the systems they model. To formulate such methods we exploit the geometric structure afforded by the GENERIC formalism of non-equilibrium thermodynamics and we follow a systematic methodology that results in discrete evolution equations which mimic the GENERIC structure. As an illustration, a complete discretization of finite strain thermoelasticity is presented, using finite elements in space and a monolithic integrator in time. Simulations are provided which demonstrate the conservation features of the algorithm as well as its remarkable robustness.

Repeated Quantum Interactions and Unitary Random Walks

Among the discrete evolution equations describing a quantum system ℋ S undergoing repeated quantum interactions with a chain of exterior systems, we study and characterize those which are directed by classical random variables in ℝ N . The characterization we obtain is entirely algebraical in terms of the unitary operator driving the elementary interaction. We show that the solutions of these equations are then random walks on the group U(ℋ0) of unitary operators on ℋ0.

Some nonlinear dynamic inequalities on time scales and applications

Some nonlinear dynamic inequalities on a time scale T are formulated in this paper. Some sufficient conditions for global existence and an estimate of the rate of decay of solutions are obtained. The results not only unify the results of differential and difference inequalities but can be applied on different types of time scales. These inequalities are of interest in a study of continuous and discrete dynamical systems and nonlinear evolution equations as well as in oscillation theory of dynamic equations on time scales and can be applied to the study of global existence of nonlinear PDE. Some applications illustrating the main results are given.

Application of the Exp-function method for nonlinear differential-difference equations

In this paper, we established abundant travelling wave solutions for some nonlinear differential-difference equations. It is shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful new method for discrete nonlinear evolution equations in mathematical physics.

Low Density Limit and the Quantum Langevin Equation for the Heat Bath

We consider a repeated quantum interaction model describing a small system [Formula: see text] in interaction with each of the identical copies of the chain ⊗ℕ* ℂn+1, modeling a heat bath, one after another during the same short time intervals [0, h]. We suppose that the repeated quantum interaction Hamiltonian is split into two parts: a free part and an interaction part with time scale of order h. After giving the GNS representation, we establish the connection between the time scale h and the classical low density limit. We introduce a chemical potential µ related to the time h as follows: h2 = eβµ. We further prove that the solution of the associated discrete evolution equation converges strongly, when h tends to 0, to the unitary solution of a quantum Langevin equation directed by the Poisson processes.

Discretization of nonlinear evolution equations over associative function algebras

A general approach is proposed for discretizing nonlinear dynamical systems and field theories on suitable functional spaces, defined over a regular lattice of points, in such a way that both their symmetry and integrability properties are preserved. A class of discrete KdV equations is introduced. Also, new hierarchies of discrete evolution equations of Gelfand–Dickey type are defined.

The long wave limiting of the discrete nonlinear evolution equations

We show here that rational, positon, negaton, breather solutions of some discrete nonlinear evolution equations are presented via long wave limiting method. The discrete nonlinear evolution equations concerned are 1D Toda lattice, differential-difference KdV, differential-difference analogue KdV equations.

Discrete stochastic evolution rules and continuum evolution equations

The continuum evolution equations are derived from updating rules for three classes of stochastic models. The first class corresponds to models whose stochastic continuum equations are of the Langevin type obtained after carrying out a “local average” known as coarse-graining. The second class consists of a hierarchy of continuum equations for the correlations of the dynamical variables obtained after making an average over realizations. This average generates a hierarchy of deterministic partial differential equations except when the dynamical variables do not depend on the values of the neighboring dynamical variables, in which case a hierarchy of ordinary differential equations is obtained. The third class of evolution equations for the correlations of the dynamical variable constitutes another hierarchy after calculating an average over both realizations and all the sites of the lattice. This double average generates a hierarchy of deterministic ordinary differential equations. The second and third classes of equations are truncated using a mean field ( m , n ) -closure approximation in order to obtain a finite set of equations. Illustrative examples of every class are given.

Modulation instability in nonlinear coupled resonator optical waveguides and photonic crystal waveguides

Modulation instability (MI) in a coupled resonator optical waveguide (CROW) and photonic-crystal waveguide (PCW) with nonlinear Kerr media was studied by using the tight-binding theory. By considering the coupling between the defects, we obtained a discrete nonlinear evolution equation and termed it the extended discrete nonlinear Schrödinger (EDNLS) equation. By solving this equation for CROWs and PCWs, we obtained the MI region and the MI gains, G(p,q), for different wavevectors of the incident plane wave (p) and perturbation (q) analytically. In CROWs, the MI region, in which solitons can be formed, can only occur for pa being located either before or after pi/2, where a is the separation of the cavities. The location of the MI region is determined by the number of the separation rods between defects and the sign of the Kerr coefficient. However, in the PCWs, pa in the MI region can exceed the pi/2. For those wavevectors close to pi/2, the MI profile, G(q), can possess two gain maxima at fixed pa. It is quite different from the results of the nonlinear CROWs and optical fibers. By numerically solving the EDNLS equation using the 4th order Runge-Kutta method to observe exponential growth of small perturbation in the MI region, we found it is consistent with our analytic solutions.

New exact solutions for the generalized Kawahara and modified Kawahara equation using the modified extended direct algebraic method

In this paper, we consider the nonlinear equations such as the generalized Kawahara equation and Modified Kawahara equation. By applied the modified extended direct algebraic (MEDA) method, the traveling wave solutions for these equations are presented. New exact traveling solutions are explicitly obtained with the help of symbolic computation, provides a very effective and powerful (mathematical tools) for discrete nonlinear evolution equations in mathematical physics. The obtained solutions include compactons, solutions, solitary patterns and periodic solutions.

Noether symmetry and Mei symmetry of discrete holonomic system in phase space

The Noether symmetry, the Mei symmetry and the conserved quantities of discrete holonomic system in phase space are studied in this paper. Using the difference discrete variational approach, the difference discrete variational principle of discrete holonomic system in phase space is derived. The discrete canonical equations and energy evolution equation are established. The criterion of Noether symmetry and Mei symmetry of the system are given. The discrete Noether and Mei conserved quantities and the conditions for their existence are obtained. An example is discussed to show the applications of the results.

Initial-boundary-value problems for discrete evolution equations: discrete linear Schrödinger and integrable discrete nonlinear Schrödinger equations

We present a method to solve initial-boundary-value problems for linear and integrable nonlinear differential-difference evolution equations. The method is the discrete version of the one developed by A S Fokas to solve initial-boundary-value problems for linear and integrable nonlinear partial differential equations via an extension of the inverse scattering transform. The method takes advantage of the Lax pair formulation for both linear and nonlinear equations, and is based on the simultaneous spectral analysis of both parts of the Lax pair. A key role is also played by the global algebraic relation that couples all known and unknown boundary values. Even though additional technical complications arise in discrete problems compared to continuum ones, we show that a similar approach can also solve initial-boundary-value problems for linear and integrable nonlinear differential-difference equations. We demonstrate the method by solving initial-boundary-value problems for the discrete analogue of both the linear and the nonlinear Schrödinger equations, comparing the solution to those of the corresponding continuum problems. In the linear case we also explicitly discuss Robin-type boundary conditions not solvable by Fourier series. In the nonlinear case, we also identify the linearizable boundary conditions, we discuss the elimination of the unknown boundary datum, we obtain explicitly the linear and continuum limit of the solution, and we write the soliton solutions.