From Sequential to Parallel Growth of Cities： Theory and Evidence from Canada

This paper examines city growth patterns and the corresponding city size distribution evolution over long periods of time using a simple New Economic Geography(NEG) model and urban population data from Canada. The main findings are twofold. First, there is a transition from sequential to parallel growth of cities over long periods of time: city growth shows a sequential mode in the stage of rapid urbanization, i.e., the cities with the best development conditions will take the lead in growth, after which the cities with higher ranks will become the fastest-growing cities; in the late stage of urbanization, city growth converges according to Gibrat′s law, and exhibits a parallel growth pattern. Second, city size distribution is found to have persistent structural characteristics: the city system is self-organized into multiple discrete size groups; city growth shows club convergence characteristics, and the cities with similar development conditions eventually converge to a similar size. The results will not only enhance our understanding of urbanization process, but will also provide a timely and clear policy reference for promoting the healthy urbanization of developing countries.

On the Computations of Gas-Solid Mixture Two-Phase Flow

This paper presents high-resolution computations of a two-phase gas-solid mixture using a well-defined mathematical model.The HLL Riemann solver is applied to solve the Riemann problem for the model equations.This solution is then employed in the construction of upwind Godunov methods to solve the general initial-boundary value problem for the two-phase gas-solid mixture.Several representative test cases have been carried out and numerical solutions are provided in comparison with existing numerical results.To demonstrate the robustness,effectiveness and capability of these methods,the model results are compared with reference solutions.In addition to that,these results are compared with the results of other simulations carried out for the same set of test cases using other numerical methods available in the literature.The diverse comparisons demonstrate that both the model equations and the numerical methods are clear in mathematical and physical concepts for two-phase fluid flow problems.

Recovering Navier–Stokes Equations from Asymptotic Limits of the Boltzmann Gas Mixture Equation

This paper is devoted to the derivation of macroscopic fluid dynamics from the Boltzmann mesoscopic dynamics of a binary mixture of hard-sphere gas particles.Specifically the hydrodynamics limit is performed by employing different time and space scalings.The paper shows that,depending on the magnitude of the parameters which define the scaling,the macroscopic quantities(number density,mean velocity and local temperature)are solutions of the acoustic equation,the linear incompressible Euler equation and the incompressible Navier–Stokes equation.The derivation is formally tackled by the recent moment method proposed by[C.Bardos,et al.,J.Stat.Phys.63(1991)323]and the results generalize the analysis performed in[C.Bianca,et al.,Commun.Nonlinear Sci.Numer.Simulat.29(2015)240].