In the present manuscript we consider the practical problem of wave interaction with a vertical wall. However, the novelty here consists in the fact that the wall can move horizontally due to a system of springs. The water wave evolution is described with the free surface potential flow model. Then, a semi-analytical numerical method is presented. It is based on a mapping technique and a finite difference scheme in the transformed domain. The idea is to pose the equations on a fixed domain. This method is thoroughly tested and validated in our study. By choosing specific values of spring parameters, this system can be used to damp (or in other words to extract the energy of) incident water waves.

A type of hybrid WCNS and CPR method for solving conservation laws on hybrid structured and unstructured meshes is proposed. WCNS on structured grid is conjoined with CPR on unstructured grid through CPR on structured grid. The main hybrid technique becomes coupling WCNS and CPR on curvilinear structured grid. Calculation of grid metrics, interpolation methods of physical coordinates and state variables, and computation of Riemann flux near coupling interface are designed to maintain the expected high order accuracy and to satisfy discrete geometric conservation laws in the whole computational domain. Third-order schemes and fifth-order schemes are considered. Numerical simulations show that the proposed hybrid WCNS-CPR schemes can obtain designed accuracy, satisfy geometric conservation law and have good balance of computational efficiency and grid flexibility.

Based on the Boltzmann model equation, the Gas-Kinetic Unified Algorithm (GKUA) will be developed to simulate the two-dimensional micro-scale gas flows with irregular configuration. The numerical scheme for the direct evaluation of the unified velocity distribution function in the computable model of the Boltzmann equation and the multi-block grid docking technology are constructed, and the numerical procedures of characteristic-based boundary conditions are presented to model the gas-surface interaction and the inlet/outlet boundaries for the two-dimensional micro-channel flows. The two-dimensional Couette flow, the pressure-driven micro-channel flows, and the irregular micro-orifice flows in different scales are numerically solved from high rarefied free-molecule to near-continuum flow with the Knudsen numbers of $Kn$ = 100−0.01. The computed results are compared and validated with the DSMC data in the transitional flow regime and the slip N-S solutions in the near-continuum flow regime, in which the GKUA is verified accurately and smoothly to simulate the two-dimensional micro-channel flows with strong adaptability and good precision. The micro-channel flow features with the wide range of $Kn$ numbers in the near-continuum slip and transitional flow regimes are revealed, and it is probable to provide a way in developing a new numerical algorithm for micro-scale flows.

We propose a direct imaging method based on reverse time migration (RTM) algorithm for imaging extended targets using electromagnetic waves at a fixed frequency in the rectangular waveguide. The imaging functional is defined as the imaginary part of the cross-correlation of the Green function for Helmholtz equation and the back-propagated electromagnetic field. The resolution of our RTM method for penetrable extended targets is studied by virtue of Helmholtz-Kirchhoff identity in the rectangular domain, which implies that the imaging functional always peaks in the target. Numerical examples are provided to demonstrate the powerful imaging quality and confirm our theoretical results.

Throughout this paper, we study the spectrum of a periodic Schrödinger operator on a zigzag super carbon nanotube, which is a generalization of the zigzag carbon nanotube. We prove that its absolutely continuous spectrum has the band structure. Moreover, we show that its eigenvalues with infinite multiplicities consisting of the Dirichlet eigenvalues and points embedded in the spectral band for some corresponding Hill operator. We also give the asymptotics for the spectral band edges.

Coarse-grained molecular dynamics simulations of DPPC lipid bilayers were performed with different system sizes at T = 323 K for a period of 1 µs. The structural properties of the systems were demonstrated by examining the area and volume per lipid, electron density profile, order parameter, and the lipid bilayer thickness. It was shown that the finite system size has a negligible effect on the ensemble averages of the area and volume per lipid, the order parameter, and the bilayer thickness. However, the electron density profiles become smoother and wider at a larger system size due to the increasing surface/interface fluctuation from undulations. On the other hand, the lipid dynamics was quantified by computing the lateral diffusion coefficients of DPPC molecules. It was revealed that, the effective lateral diffusion coefficient of DPPC increases initially by 19% as the bilayer increases from 64 to 256 lipids per leaflet, and then it changes slightly and fluctuates around a steady value as the system further expands.

In this paper, we propose a new gradient recovery method for elliptic interface problem using body-fitted meshes. Due to the lack of regularity of the solution at the interface, standard gradient recovery methods fail to give superconvergent results and thus will lead to overrefinement when served as a posteriori error estimators. This drawback is overcome by designing a new gradient recovery operator. We prove the superconvergence of the new method on both mildly unstructured meshes and adaptive meshes. Several numerical examples are presented to verify the superconvergence and its robustness as a posteriori error estimator.

In this paper we develop conservative finite-difference schemes (FDS) for the process of femtosecond pulse interaction with semiconductor. This process is described by the set of 2D dimensionless differential equations concerning concentrations of both free electrons and ionized donors, and potential of electric field, induced by laser pulse and laser beam intensity changing. The electron mobility, electron diffusion, nonlinear dependence of absorption coefficient on semiconductor characteristics are taken into account also.
For the problem under consideration we have constructed and compared two conservative FDS. One of them is based on the well known split-step method, the second one is based on the original two-stage iteration process. We paid the special attention to the 2D Poisson equation solution. This equation is solved by using an additional iteration process. Thus, to solve the problem under consideration it is necessary to achieve a convergence of two iteration processes.
As follows from computer simulation provided by us, the criterion choice for the iteration process convergence can significantly affect on the equations solution accuracy. We used the criterion based on assessment of an absolute and relative error of the solution obtained on iterations. This criterion is also used for Poisson equation solving. However, the iteration convergence criterion, based on discrepancy estimating, is more effective for using in this case.
Computer simulation results showed that the developed conservative FDS on the base of two-stage iteration process is an effective tool for investigation of complicated modes of semiconductor characteristics changing.

We consider a system of liquid crystal modeled by hard spherocylinders. In certain range of the pressure, the system exhibits two metastable phases: the isotropic phase and the nematic phase. In the isotropic phase, the spherocylinders are randomly packed. In contrast, the spherocylinders are well-ordered in the nematic phase. The isotropic-nematic phase transition is a rare event because it involves the crossing of energy barrier(s). This makes direct simulations, e.g. using molecular dynamics, of the transition event infeasible. In this paper, we study the phase transition in a coarse-grained space formed by two collective variables: the order parameter of the spherocylinders and the volume of the system. We compute the free energy in the collective variable space, the minimum free energy path (MFEP) between the isotropic phase and the nematic phase, and the transition state. Our results reveal the multilayer structure of the critical nucleus. The nucleus will grow further and evolve to the nematic phase after it crosses the energy barrier.

The steady-state Poisson-Nernst-Planck (ssPNP) equations are an effective model for the description of ionic transport in ion channels. It is observed that an ion channel exhibits voltage-dependent switching between open and closed states. Different conductance states of a channel imply that the ssPNP equations probably have multiple solutions with different level of currents. We propose numerical approaches to study multiple solutions to the ssPNP equations with multiple ionic species. To find complete current-voltage (I-V) and current-concentration (I-C) curves, we reformulate the ssPNP equations into four different boundary value problems (BVPs). Numerical continuation approaches are developed to provide good initial guesses for iteratively solving algebraic equations resulting from discretization. Numerical continuations on V, I, and boundary concentrations result in S-shaped and double S-shaped (I-V and I-C) curves for the ssPNP equations with multiple species of ions. There are five solutions to the ssPNP equations with five ionic species, when an applied voltage is given in certain intervals. Remarkably, the current through ion channels responds hysteretically to varying applied voltages and boundary concentrations, showing a memory effect. In addition, we propose a useful computational approach to locate turning points of an I-V curve. With obtained locations, we are able to determine critical threshold values for hysteresis to occur and the interval for V in which the ssPNP equations have multiple solutions. Our numerical results indicate that the developed numerical approaches have a promising potential in studying hysteretic conductance states of ion channels.

A backward doubly stochastic differential equation (BDSDE) based nonlinear filtering method is considered. The solution of the BDSDE is the unnormalized density function of the conditional expectation of the state variable with respect to the observation filtration, which solves the nonlinear filtering problem through the Kallianpur formula. A first order finite difference algorithm is constructed to solve the BSDES, which results in an accurate numerical method for nonlinear filtering problems. Numerical experiments demonstrate that the BDSDE filter has the potential to significantly outperform some of the well known nonlinear filtering methods such as particle filter and Zakai filter in both numerical accuracy and computational complexity.

In this paper, we present a detailed numerical implementation of the multicomponent potential theory of adsorption which is among the most accurate gas mixtures adsorption models. The implementation uses the NIST Refprop database to describe fluid properties and applies to pure gases and mixtures in both subcritical and supercritical regimes. The limitations of the model and the issues encountered with its implementation are discussed. The adsorption isotherms of CH₄/CO₂ mixture are modeled and parameterized as implementation examples.