In the present work, we develop in detail the process leading to reduction of models in chemical kinetics when using the Method of Invariant Grids (MIG). To this end, reduced models (invariant grids) are obtained by refining initial approximations of slow invariant manifolds, and used for integrating smaller and less stiff systems of equations capable to recover the detailed description with high accuracy. Moreover, we clarify the role played by thermodynamics in model reduction, and carry out a comparison between detailed and reduced solutions for a model hydrogen oxidation reaction.

In this paper, we present an immersed boundary method for simulating moving contact lines with surfactant. The governing equations are the incompressible Navier-Stokes equations with the usual mixture of Eulerian fluid variables and Lagrangian interfacial markers. The immersed boundary force has two components: one from the nonhomogeneous surface tension determined by the distribution of surfactant along the fluid interface, and the other from unbalanced Young's force at the moving contact lines. An artificial tangential velocity has been added to the Lagrangian markers to ensure that the markers are uniformly distributed at all times. The corresponding modified surfactant equation is solved in a way such that the total surfactant mass is conserved. Numerical experiments including convergence analysis are carefully conducted. The effect of the surfactant on the motion of hydrophilic and hydrophobic drops are investigated in detail.

We develop an efficient numerical method to compute single slit or double slit diffraction patterns from high frequency wave in inhomogeneous media. We approximate the high frequency asymptotic solution to the Helmholtz equation using the Eulerian Gaussian beam summation proposed in [20, 21]. The emitted rays from a slit are embedded in the phase space using an open segment. The evolution of this open curve is accurately computed using the recently developed Grid Based Particle Method [24] which results in a very efficient computational algorithm. Following the grid based particle method we proposed in [23, 24], we represent the open curve or the open surface by meshless Lagrangian particles sampled according to an underlying fixed Eulerian mesh. The end-points of the open curve are tracked explicitly and consistently with interior particles. To construct the overall wavefield, each of these sampling particles also carry necessary quantities that are obtained by solving advection-reaction equations. Numerical experiments show that the resulting method can model diffraction patterns in inhomogeneous media accurately, even in the occurrence of caustics.

We study high-resolution central schemes in Lagrangian coordinates for the one-dimensional system of conservation laws describing the evolution of two gases in slab geometry separated by an interface. By using Lagrangian coordinates, the interface is transformed to a fixed coordinate in the computational domain and, as a consequence, the movement of the interface is obtained as a byproduct of the numerical solution. The main contribution is the derivation of a special equation of state to be imposed at the interface in order to avoid non-physical oscillations. Suitable boundary conditions at the piston that guarantee second order convergence are described. We compare the solution of the piston problem to other results available in the literature and to a reference solution obtained within the adiabatic approximation. A shock-interface interaction problem is also treated. The results on these tests are in good agreement with those obtained by other methods.

Accurate simulations of metamaterial devices are very important in the analysis of their electromagnetic properties. However, it is very difficult to make full-wave simulations of three-dimensional (3D) metamaterial devices due to the huge memory requirements and long computing time. In this paper, we present an efficient finite-element method (FEM) to analyze 3D axisymmetric electromagnetic devices designed by the transformation-optics approach, such as invisibility cloaks and concentrators. In the proposed method, we use the edge-based vector basis functions to expand the transverse field components, and the node-based scalar basis functions to expand the angular component. The FEM mesh is truncated with a cylindrical perfectly matched layer. We have applied the method to investigate the scattering from spherical and ellipsoidal invisibility cloaks and circularly cylindrical concentrators, in which the permittivity and permeability are both inhomogeneous and anisotropic. Numerical results are presented to show the validity and efficiency of the method.

In this paper, we propose a simple model of opinion dynamics to construct social networks, based on the algorithm of link rewiring of local attachment (RLA) and global attachment (RGA). Generality, the system does reach a steady state where all individuals' opinion and the complex network structure are fixed. The RGA enhances the ability of consensus of opinion formation. Furthermore, by tuning a model parameter p, which governs the proportion of RLA and RGA, we find the formation of hierarchical structure in the social networks for p > p_c. Here, p_c is related to the complex network size N and the minimal coordination number 2K. The model also reproduces many features of large social networks, including the "weak links" property.

This article is concerned with the numerical detection of bifurcation points of nonlinear partial differential equations as some parameter of interest is varied. In particular, we study in detail the numerical approximation of the Bratu problem, based on exploiting the symmetric version of the interior penalty discontinuous Galerkin finite element method. A framework for a posteriori control of the discretization error in the computed critical parameter value is developed based upon the application of the dual weighted residual (DWR) approach. Numerical experiments are presented to highlight the practical performance of the proposed a posteriori error estimator.

To obtain fine distributing calculation about the time, space and energy of neutron flux for a kind of non-stationary particle transport problem, the scheme of global Monte Carlo variance reduction is developed. In order to provide the foundation for this scheme, it is necessary to analyze its effectiveness before putting it to use. This paper fulfills this through analyzing the effectiveness of its core which is Monte Carlo transport importance biased calculation. By decomposing the arithmetic and calculating the representative objectives, the effectiveness of the Monte Carlo transport importance biased calculation is demonstrated.

This work deals with the simulation of two-dimensional Lagrangian hydrodynamics problems. Our objective is the development of an artificial viscosity that is to be used in conjunction with a staggered placement of variables: thermodynamics variables are centered within cells and position and fluid velocity at vertices. In [J. Comput. Phys., 228 (2009), 2391-2425], Maire develops a high-order cell-centered scheme for solving the gas dynamics equations. The numerical results show the accuracy and the robustness of the method, and the fact that very few Hourglass-type deformations are present. Our objective is to establish the link between the scheme of Maire and the introduction of artificial viscosity in a Lagrangian code based on a staggered grid. Our idea is to add an extra degree of freedom to the numerical scheme, which is an approximation of the fluid velocity within cells. Doing that, we can locally come down to a cell-centered approximation and define the Riemann problem associated to discrete variable discontinuities in a very natural way. This results in a node-centered artificial viscosity formulation. Numerical experiments show the robustness and the accuracy of the method, which is very easy to implement.

In this paper, we present an algorithm to simulate a Brownian motion by coupling two numerical schemes: the Euler scheme with the random walk on the hyper-rectangles. This coupling algorithm has the advantage to be able to compute the exit time and the exit position of a Brownian motion from an irregular bounded domain (with corners at the boundary), and being of order one with respect to the time step of the Euler scheme. The efficiency of the algorithm is studied through some numerical examples by comparing the analytical solution with the Monte Carlo solution of some Poisson problems. The Monte Carlo solution of these PDEs requires simulating Brownian motions of different types (natural, reflected or drifted) over an irregular domain.

To solve the electromagnetic scattering problem for homogeneous dielectric bodies of revolution (BOR), a fast inhomogeneous plane wave algorithm is developed. By using the Weyl identity and designing a proper integration path, the aggregation and disaggregation factors can be derived analytically. Compared with the traditional method of moments (MoM), both the memory and CPU time requirements are reduced for large-scale homogeneous dielectric BOR problems. Numerical results are given to demonstrate the validity and the efficiency of the proposed method.

We present a new numerical method for solving two-dimensional Stokes flow with deformable interfaces such as dynamics of suspended drops or bubbles. The method is based on a boundary integral formulation for the interfacial velocity and is spectrally accurate in space. We analyze the singular behavior of the integrals (single-layer and double-layer integrals) appearing in the equations. The interfaces are formulated in the tangent angle and arc-length coordinates and, to reduce the stiffness of the evolution equation, the marker points are evenly distributed in arc-length by choosing a proper tangential velocity along the interfaces. Examples of Stokes flow with bubbles are provided to demonstrate the accuracy and effectiveness of the numerical method.