Discrete analogue of the Burgers equation

We propose the set of coupled ordinary differential equations dnj/dt = nj−1 − nj as a discrete analogue of the classic Burgers equation. We focus on traveling waves and triangular waves, and find that these special solutions of the discrete system capture major features of their continuous counterpart. In particular, the propagation velocity of a traveling wave and the shape of a triangular wave match the continuous behavior. However, there are some subtle differences. For traveling waves, the propagating front can be extremely sharp as it exhibits double exponential decay. For triangular waves, there is an unexpected logarithmic shift in the location of the front. We establish these results using asymptotic analysis, heuristic arguments, and direct numerical integration. PACS numbers: 05.45.−a, 02.60.Lj, 02.50.−r (Some figures may appear in colour only in the online journal) The classic Burgers equation nt + (n)x = νnxx (1) is the simplest partial differential equation which incorporates both nonlinear advection and diffusive spreading [1–6]. This ubiquitous equation emerges naturally in the presence of dissipation, and it is broadly used to model traffic flows [2], transport processes [7, 8], surface growth [9, 10], and large scale formation of matter in the Universe [11, 12]. The Burgers equation has two important properties. The first is continuity: equation (1) can be written in the form nt + Jx = 0, hence assuring mass conservation in the absence of sources or sinks. If we view the quantity n as a density, then the total mass is a conserved quantity, ∫∞ −∞ dx n(x, t) = const., as long as the density vanishes, n(x) → 0 in the limits x → ±∞. The second property is asymmetry: due to the nonlinear advection term, equation (1) is not invariant with respect to the inversion transformation x → −x. Our goal is to construct a discrete (in space) counterpart of the Burgers equation that maintains these two properties [13–16]. Thus we discretize the spatial coordinate but 1751-8113/12/455003+09$33.00 © 2012 IOP Publishing Ltd Printed in the UK & the USA 1 J. Phys. A: Math. Theor. 45 (2012) 455003 E Ben-Naim and P L Krapivsky keep the time variable continuous, n(x, t) → nj(t), where j is integer. The differential equation dn j dt = f (n j−1)− f (n j) (2) represents a continuity equation on a one-dimensional lattice. Indeed, a finite total mass M = ∑∞j=−∞ n j remains constant, M = const., if two conditions are met: (i) a vanishing density n j → 0 as j → ±∞, and (ii) a vanishing function f at the origin, f (0) = 0. To reproduce the nonlinear advection term in (1), we take a purely quadratic and positive function f (n) = n2. With this choice, we arrive at the set of nonlinear difference-differential equations dn j dt = nj−1 − nj . (3) This system of equations meets the two criteria of mass conservation and asymmetry. Immediately, we can point out an important difference between the discrete equation (3) and the continuous equation (1). Let us treat the spatial variable in (3) as continuous, j → x, and replace the difference with a second order Taylor expansion. The result of these two steps is the continuous equation nt + (n)x = (nnx)x. (4) By construction, the nonlinear advection term is the same as in (1). However, the viscosity equals the density, ν = n, whereas in the original Burgers equation, the viscosity is constant. We restrict our attention to positive densities, n > 0, (5) so that the solutions of (3) are stable (avoiding a negative diffusion instability). We note that transport coefficients often depend on density or temperature; in fluid dynamics [4, 5], for example, transport coefficients vary as √ T for hard-sphere gases. Equation (3) describes the evolution of the probability density in a two-body analogue of the standard Poisson process3. For example, we mention a homophilic network growth process [17]. In the canonical random network model, a pair of nodes are chosen at random and subsequently, the two are connected by a link. This elementary step is repeated indefinitely, and in finite time, a percolating network emerges. As a model of homophilic networks where only similar entities interact, we considered the situation where only nodes with exactly the same degree can be connected [18]. The degree distribution n j(t), that is, the fraction of nodes of degree j at time t, obeys the rate equation (3). The initial condition n j(0) = δ j,0 represents a disconnected set of nodes. In this network context, the quantity nj(t) is a probability density, and mass conservation guarantees proper normalization, ∑∞ j=0 n j = 1. Moreover, the condition (5) reflects that probability distribution functions are by definition positive. In this paper, we discuss the solutions of the discrete equation (3) in view of the well known solutions of the continuous equation (1). Using a combination of theoretical and numerical methods, we analyzed the solutions of the discrete equation for the following standard initial conditions [2]: (i) a step function resulting in a traveling wave, (ii) a localized delta function leading to a triangular wave, and (iii) a complementary step function with an ensuing rarefaction wave. For all of these cases, we find that the discrete analogue faithfully captures the primary features of the continuous Burgers equation. However, we also find subtle and interesting departures from the classical solutions in the first two cases. Therefore, in the rest of this paper, we focus on traveling waves (also known as shock waves) and triangular waves. 3 For the standard Poisson process, j → j + 1, the probability distribution function p j obeys the rate equation dp j/dt = p j−1 − p j . Similarly, the rate equations (3) describe the evolution of the probability density for the ‘two-body’ generalization ( j, j) → ( j + 1, j + 1).