In this article, we address both recent advances and open questions in some mathematical and computational issues in geophysical fluid dynamics (GFD) and climate dynamics. The main focus is on 1) the primitive equations (PEs) models and their related mathematical and computational issues, 2) climate variability, predictability and successive bifurcation, and 3) a new dynamical systems theory and its applications to GFD and climate dynamics.

The membrane's dynamics is very important for cells. A membrane in 2-dimensional space can be seen as an incompressible closed curve in a plane or a cylindrical surface in 3-dimensional space. In this paper, we design a second-order accurate numerical algorithm to simulate the shape transformation of the membrane. In the algorithm, we use the tangent angles to present the curve and avoid the difficulties from the constraint of curve's incompressible condition. A lot of interesting phenomena are obtained. Some of them are very like the life processes of cells, such as exocytosis and endocytosis. Furthermore, we can see the relation between two dynamic models clearly. At last, considering the influence of the inner incompressible fluids partially, we add a constraint: the area circled by the membrane maintain invariable. The numerical results show the dynamic motions of a curve remaining its local arc length and inner area constant.

Two-component submonolayer growth on triangular lattice is qualitatively studied by kinetic Monte Carlo techniques. The hopping barrier governing surface diffusion of the atoms is estimated using pair interaction potentials. Several degrees of freedoms enhancing the surface diffusion of atoms are also introduced. The main advantages of the presented technique are the reduced number of free parameters and the clear diffusion activated mechanism for the segregation of different types of atoms. The potential of this method is exemplified by reproducing (i) phase-boundary creation and dynamics related to vacancies and stacking faults; (ii) a special co-deposition and segregation process where the segregated atoms of the second component surround the islands formed by the first type of atoms.

The aerodynamics of freely falling objects is one of the most interesting flow mechanics problems. In a recent study, Andersen, Pesavento, and Wang [J. Fluid Mech., vol. 541, pp. 65-90 (2005)] presented the quantitative comparison between the experimental measurement and numerical computation. The rich dynamical behavior, such as fluttering and tumbling motion, was analyzed. However, obvious discrepancies between the experimental measurement and numerical simulations still exist. In the current study, a similar numerical computation will be conducted using a newly developed unified coordinate gas-kinetic method [J. Comput. Phys, vol. 222, pp. 155-175 (2007)]. In order to clarify some early conclusions, both elliptic and rectangular falling plates will be studied. Under the experimental condition, the numerical solution shows that the averaged translational velocity for both rectangular and elliptical plates are almost identical during the tumbling motion. However, the plate rotation depends strongly on the shape of the plates. In this study, the details of fluid forces and torques on the plates and plates movement trajectories will be presented and compared with the experimental measurements.

The maximum principle is a basic qualitative property of the solution of second-order elliptic boundary value problems. The preservation of the qualitative characteristics, such as the maximum principle, in discrete model is one of the key requirements. It is well known that standard linear finite element solution does not satisfy maximum principle on general triangular meshes in 2D. In this paper we consider how to enforce discrete maximum principle for linear finite element solutions for the linear second-order self-adjoint elliptic equation. First approach is based on repair technique, which is a posteriori correction of the discrete solution. Second method is based on constrained optimization. Numerical tests that include anisotropic cases demonstrate how our method works for problems for which the standard finite element methods produce numerical solutions that violate the discrete maximum principle.

We explore the use of PDE constrained nonlinear optimization techniques to optimize and design electromagnetic crystals which exhibit frozen mode behavior. This is characterized by Van Hove singularities in the dispersion relation, e.g., stationary reflection points and degenerate band edge points. Hence, the optimization process modifies the dispersion relation by adjusting the geometries and material parameters. The resulting algorithm is found to be capable of recovering all known crystal configurations as well as many new configurations, some of which display dramatically improved properties over previously used configuration. We investigate both gyrotropic photonic crystals and degenerate band edge crystals as well as the more complex case of the oblique incidence. In this latter case, we extend the investigation to the three-dimensional case to identify the first three-dimensional crystal exhibiting frozen mode behavior.

To enhance the stability of traffic flow, a new car-following model is proposed by taking into account the support of intelligent transportation system (ITS) information, which includes both the headway and the velocity difference of multiple preceding cars. The new model is based on the Optimal Velocity (OV) model and its extended models. The stability condition of the model is obtained by using the linear stability theory. Through nonlinear analysis, the modified Korteweg-de Vries equation is constructed and solved, and the traffic flow is classified into three types, i.e. stable, metastable, and unstable. The jam phase can thus be described by the kink-antikink soliton solution for the mKdV equation. The numerical simulation results show that compared with previous models considering only one of the ITS information, the proposed model can suppress traffic jams more efficiently when both headway and velocity difference of arbitrary preceding cars are taken into account. The results of numerical simulation coincide with the theoretical ones.

We present discretization and solver methods for a model of the solid-gas phase in a crystal growth apparatus. The model equations are coupled Eulerian and heat-transfer equations with flux boundary conditions. For a more detailed discussion we consider simpler equations and present time- and space-decomposition methods as solver methods to decouple the multi-physics processes. We present the error analysis for the discretization and solver methods. Numerical experiments are performed for the Eulerian and heat-transfer equation using decomposition methods. We present a real-life application of a crystal growth apparatus, based on underlying stationary heat conduction. Finally we discuss further error analysis and application to a more complex model of crystal growth.

Social networks play an important role in determining the dynamics and outcome of language change. Early empirical studies only examine small-scale local social networks, and focus on the relationship between the individual speakers' linguistic behaviors and their characteristics in the network. In contrast, computer models can provide an efficient tool to consider large-scale networks with different structures and discuss the long-term effect of individuals' learning and interaction on language change. This paper presents an agent-based computer model which simulates language change as a process of innovation diffusion, to address the threshold problem of language change. In the model, the population is implemented as a network of agents with age differences and different learning abilities, and the population is changing, with new agents born periodically to replace old ones. Four typical types of networks and their effect on the diffusion dynamics are examined. When the functional bias is sufficiently high, innovations always diffuse to the whole population in a linear manner in regular and small-world networks, but diffuse quickly in a sharp S-curve in random and scale-free networks. The success rate of diffusion is higher in regular and small-world networks than in random and scale-free networks. In addition, the model shows that as long as the population contains a small number of statistical learners who can learn and use both linguistic variants statistically according to the impact of these variants in the input, there is a very high probability for linguistic innovations with only small functional advantage to overcome the threshold of diffusion.

This paper presents a further numerical study of the interaction dynamics for solitary waves in a nonlinear Dirac model with scalar self-interaction, the Soler model, by using a fourth order accurate Runge-Kutta discontinuous Galerkin method. The phase plane method is employed for the first time to analyze the interaction of Dirac solitary waves and reveals that the relative phase of those waves may vary with the interaction. In general, the interaction of Dirac solitary waves depends on the initial phase shift. If two equal solitary waves are in-phase or out-of-phase initially, so are they during the interaction; if the initial phase shift is far away from 0 and π, the relative phase begins to periodically evolve after a finite time. In the interaction of out-of-phase Dirac solitary waves, we can observe: (a) full repulsion in binary and ternary collisions, depending on the distance between initial waves; (b) repulsing first, attracting afterwards, and then collapse in binary and ternary collisions of initially resting two-humped waves; (c) one-overlap interaction and two-overlap interaction in ternary collisions of initially resting waves. 

In this short note we clarify some issues regarding the existence of minimizers for the Thomas-Fermi-von Weiszacker energy functional in orbital-free density functional theory, when the Wang-Teter corrections are included.