Simulations mostly by physicists of the competition between adult languages since 2003 are reviewed. The Viviane and Schulze models give good and reasonable agreement, respectively, with the empirical histogram of language sizes. Also the numbers of different languages within one language family is modeled reasonably in an intermediate range. Bilingualism is now incorporated into the Schulze model. Also the rate at which the majority shifts from one language to another is found to be nearly independent of the population size, or to depend strongly on it, according to details of the Schulze model. Other simulations, like Nettle-Culicover-Nowak, are reviewed more briefly.

In this paper we study the behavior of a family of implicit numerical methods applied to stochastic differential equations with multiple time scales. We show by a combination of analytical arguments and numerical examples that implicit methods in general fail to capture the effective dynamics at the slow time scale. This is due to the fact that such implicit methods cannot correctly capture non-Dirac invariant distributions when the time step size is much larger than the relaxation time of the system. 

The aim of this paper is to develop an efficient numerical method to compute the eigenvalues of the stability analysis of a problem describing the motion of a fluid within a cylindrical container heated non-homogeneously from below. An axisymmetric stationary motion settles in, at certain values of the external parameters appearing in the set of partial differential equations modeling the problem. This basic solution is computed by discretizing the equations with a Chebyshev collocation method. Its linear stability is formulated with a generalized eigenvalue problem. The numerical approach (generalized Arnoldi method) uses the idea of preconditioning the eigenvalue problem with a modified Cayley transformation before applying the Arnoldi method. Previous works have dealt with transformations requiring regularity to one of the submatrices. In this article we extend those results to the case in which that submatrix is singular. This method allows a fast computation of the critical eigenvalues which determine whether the steady flow is stable or unstable. The algorithm based on this method is compared to the QZ method and is found to be computationally more efficient. The reliability of the computed eigenvalues in terms of stability is confirmed via pseudospectra calculations.

The conventional vector addition theorem is written in a compact notation. Then a new and succinct derivation of the vector addition theorem is presented that is as close to the derivation of the scalar addition theorem. Newly derived expressions in this new derivation are used to diagonalize the vector addition theorem. The diagonal form of the vector addition theorem is important in the design of fast algorithms for computational wave physics such as computational electromagnetics and computational acoustics.

This paper presents a new finite-volume discretization of a generalised Lattice Boltzmann equation (LBE) on unstructured grids. This equation is the continuum LBE, with the addition of a second order time derivative term (memory), and is derived from a second-order differential form of the semi-discrete Boltzmann equation in its implicit form. The new scheme, named unstructured lattice Boltzmann equation with memory (ULBEM), can be advanced in time with a larger time-step than the previous unstructured LB formulations, and a theoretical demonstration of the improved stability is provided. Taylor vortex simulations show that the viscosity is the same as with standard ULBE and demonstrates that the new scheme improves both stability and accuracy. Model validation is also demonstrated by simulating backward-facing step flow at low and moderate Reynolds numbers, as well as by comparing the reattachment length of the recirculating eddy behind the step against experimental and numerical data available in literature. 

There have been several recent papers on developing moving mesh methods for solving phase-field equations. However, it is observed that some of these moving mesh solutions are essentially different from the solutions on very fine fixed meshes. One of the purposes of this paper is to understand the reason for the differences. We carried out numerical sensitivity studies systematically in this paper and it can be concluded that for the phase-field equations, the numerical solutions are very sensitive to the starting mesh and the monitor function. As a separate issue, an efficient alternating Crank-Nicolson time discretization scheme is developed for solving the nonlinear system resulting from a finite element approximation to the phase-field equations.

This paper aims to explore RLC equivalent circuit synthesis method for reduced-order models of interconnect circuits obtained by Krylov subspace based model order reduction (MOR) methods. To guarantee pure RLC equivalent circuits can be synthesized, both the structures of input and output incidence matrices and the block structure of the circuit matrices should be preserved in the reduced-order models. Block structure preserving MOR methods have been well established. In this paper, we propose an embeddable Input-Output structure Preserving Order Reduction (IOPOR) technique to further preserve the structures of input and output incidence matrices. By combining block structure preserving MOR methods and IOPOR technique, we develop an RLC equivalent circuit synthesis method RLCSYN (RLC SYNthesis). Inline diagonalization and regularization techniques are specifically proposed to enhance the robustness of inductance synthesis. The pure RLC model, high modeling accuracy, passivity guaranteed property and SPICE simulation robustness make RLCSYN more applicable in interconnect analysis, either for digital IC design or mixed signal IC simulation.

We studied the dynamical phase transition in kinetic Ising ferromagnets driven by oscillating magnetic field in meanfield approximation. The meanfield differential equation was solved by sixth order Runge-Kutta-Felberg method. We calculated the transition temperature as a function of amplitude and frequency of oscillating field. This was plotted against field amplitude taking frequency as a parameter. As frequency increases the phase boundary is observed to become inflated. The phase boundary shows an inflection point which separates the nature of the transition. On the dynamic phase boundary a tricritical point (TCP) was found, which separates the nature (continuous/discontinuous) of the dynamic transition across the phase boundary. The inflection point is identified as the TCP and hence a simpler method of determining the position of TCP was found. TCP was observed to shift towards high field for higher frequency. As frequency decreases the dynamic phase boundary is observe to shrink. In the zero frequency limit this boundary shows a tendency to merge to the temperature variation of the coercive field.

This work is concerned with the numerical simulations on the Gierer-Meinhardt activator-inhibitor models. We consider the case when the inhibitor time constant τ is non-zero. In this case, oscillations and pulse splitting are observed numerically. Numerical experiments are carried out to investigate the dynamical behaviors and instabilities of the spike patterns. The numerical schemes used are based upon an efficient moving mesh finite element method which distributes more grid points near the localized spike regions.

We develop a two-relaxation-time (TRT) Lattice Boltzmann model for hydrodynamic equations with variable source terms based on equivalent equilibrium functions. A special parametrization of the free relaxation parameter is derived. It controls, in addition to the non-dimensional hydrodynamic numbers, any TRT macroscopic steady solution and governs the spatial discretization of transient flows. In this framework, the multi-reflection approach [16, 18] is generalized and extended for Dirichlet velocity, pressure and mixed (pressure/tangential velocity) boundary conditions. We propose second- and third-order accurate boundary schemes and adapt them for corners. The boundary schemes are analyzed for exactness of the parametrization, uniqueness of their steady solutions, support of staggered invariants and for the effective accuracy in case of time dependent boundary conditions and transient flow. When the boundary scheme obeys the parametrization properly, the derived permeability values become independent of the selected viscosity for any porous structure and can be computed efficiently. The linear interpolations [5, 46] are improved with respect to this property.

The original ghost fluid method (GFM) developed in [13] and the modified GFM (MGFM) in [26] have provided a simple and yet flexible way to treat two-medium flow problems. The original GFM and MGFM make the material interface ”invisible” during computations and the calculations are carried out as for a single medium such that its extension to multi-dimensions becomes fairly straightforward. The Runge-Kutta discontinuous Galerkin (RKDG) method for solving hyperbolic conservation laws is a high order accurate finite element method employing the useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers, TVD Runge-Kutta time discretizations, and limiters. In this paper, we investigate using RKDG finite element methods for two-medium flow simulations in one and two dimensions in which the moving material interfaces are treated via nonconservative methods based on the original GFM and MGFM. Numerical results for both gas-gas and gas-water flows are provided to show the characteristic behaviors of these combinations.

Polynomial chaos methods (and generalized polynomial chaos methods) have been extensively applied to analyze PDEs that contain uncertainties. However, this approach is rarely applied to hyperbolic systems. In this paper we analyze the properties of the resulting deterministic system of equations obtained by stochastic Galerkin projection. We consider a simple model of a scalar wave equation with random wave speed. We show that when uncertainty causes the change of characteristic directions, the resulting deterministic system of equations is a symmetric hyperbolic system with both positive and negative eigenvalues. A consistent method of imposing the boundary conditions is proposed and its convergence is established. Numerical examples are presented to support the analysis.