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An interesting fact which was used in the construction of targeted essentially non-oscillatory (TENO) schemes is that a $(n+1)$th order accurate scheme can be written as a linear combination of two $n$th order accurate schemes. Taking advantage of this fact, we propose a formula to determine the ideal weights used in developing finite difference WENO/WENO-like schemes. Such quantitative results are helpful for further discussion on finite difference WENO/WENO-like schemes.

Following recent theoretical results, it is suggested that positronium (Ps) might undergo spontaneous oscillations between two 4D space–time sheets whenever subjected to constant irrotational magnetic vector potentials. We show that these oscillations that would come together with o-Ps/p-Ps oscillations should have important consequences on Ps decay rates. Experimental setup and conditions are also suggested for demonstrating in nonaccelerator experiments this new invisible decay mode.

In recent papers, a model of a two-sheeted space–time M₄ ×Z₂ was introduced and the quantum dynamics of massive fermions was studied in this framework. In the present study, we show that the physical predictions of the model are perfectly consistent with observations and most importantly, it can solve the puzzling problem of the four-dimensional localization of the fermion species in multidimensional space–times. It is demonstrated that fermion localization on the sheets arises from the combination of the discrete bulk structure and environmental interactions. The mechanism described in this paper can be seen as an alternative to the domain wall localization arising in continuous five-dimensional space–times. Although tightly constrained, motions between the sheets are, however, not completely prohibited. As an illustration, a resonant mechanism through which fermion oscillations between the sheets might occur is described.

In this paper, a well-known finite difference technique is combined with thermal lattice Boltzmann method to solve 2-dimensional incompressible thermal fluid flow problems. A small number of microvelocity components are applied for the calculation of temperature field. The combination of finite difference with lattice Boltzmann method is found to be an efficient and stable approach for the simulation at high Rayleigh number of natural convection in a square cavity.

In the present work, the optical properties of GaAs/AlGaAs semiparabolic quantum wells (QWs) are studied under the effect of applied electric field and magnetic field by using the compact-density-matrix method. The energy eigenvalues and their corresponding eigenfunctions of the system are calculated by using the differential method. Simultaneously, the nonlinear optical rectification (OR) and optical absorption coefficients (OACs) are investigated, which are modulated by the applied electric field and magnetic field. It is found that the position and the magnitude of the resonant peaks of the nonlinear OR and OACs can depend strongly on the applied electric field, magnetic field and confined potential frequencies. This gives a new way to control the device applications based on the intersubband transitions of electrons in this system.

In this paper, a solution-adaptive finite difference method is presented for the simulation of flow past a pair of side-by-side circular cylinders. The method consists of an adaptive 5-point stencil algorithm, a central-difference spatial discretization and an immersed boundary representation of complex geometry. Numerical results indicate that the method is capable of solving the flow problems with complex geometry in a multilevel manner.

Based on the elasto-/hydro-dynamic model the dynamic properties of the five-fold symmetry quasicrystals with point groups 5, are investigated, by using the finite difference method. The problems including dynamic initiation of crack growth and fast crack propagation of this material are studied. The results show that the phonon–phason coupling effect plays an important role to the dynamic properties of the quasicrystals.

A family of inexpensive discretization schemes for diffusion problems on unstructured polygonal and polyhedral meshes is introduced. The material properties are described by a full tensor. The theoretical results are confirmed with numerical experiments.

The main purpose of this work is to explore a general cell–fluid model which is based on a mixture theory formulation that accounts for the interplay between oxytactically (chemotaxis toward gradient in oxygen) moving bacteria cells in water and the buoyance forces caused by the difference in density between cells and fluid. The model involves two mass balance and two general momentum balance equations, respectively, for the cell and fluid phase, combined with a convection–diffusion–reaction equation for oxygen. In particular, the momentum balance equations include interaction terms which describe the cell–fluid drag force effect. Hence, the model is an extension of the classical Navier–Stokes equation in two different ways: (i) inclusion of two phases (cell and fluid) instead of one; (ii) inclusion of a chemotactic transport mechanism. The model can be understood as a natural generalization of the much studied chemotaxis-Stokes model explored by [I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA102 (2005) 2277–2282]. First, we explore the model for parameters in a range which ensures that it lies close to the previously studied chemotaxis-Stokes model (essentially very low cell volume fraction). Main observations are (i) formation of sinking finger-shaped plumes and (ii) convergence to plumes that possibly can be stationary (i.e. persist over time). The general cell–fluid model provides new insight into the role played by the cell–fluid interaction term which controls the competition between gravity segregation and chemotaxis effect on the formation of cell plumes. Second, we explore cases with large cell volume fraction (far beyond the regime captured by the chemotaxis-Stokes model), which gives rise to rich pattern-formation behavior. The general cell–fluid model opens for exploring a hierarchy of different “submodels”. Hence, it seems to be an interesting model for further investigations of various, general cell–fluid spatio-temporal evolution dynamics, both from an experimental and mathematical point of view.

A finite-difference algorithm is developed in the 3D cylindrical coordinate system. Synthetic waveforms excited by monopole and dipole acoustic sources are calculated for acoustic logging in a tilted layered formation composed of two poroelastic media. It is revealed that the existence of a tilted layer drives more acoustic energy out and therefore reduces the reflection back to the wellbore. Numerical results show that the amplitudes and the arrival times of the shear waves and flexural modes are influenced by the borehole inclination. The Stoneley modes, however, show little sensitivity to the orientation of the layer interface.

We solve, by using a monotone and stable approximation, the fully nonlinear degenerate parabolic equation derived by Cheridito, Soner and Touzi [8] from the stochastic control problem of super-replicating a contingent claim under gamma constraints. We present some numerical results.

A semi-analytical solution method, called the Finite Difference–Distributed Transfer Function Method, is developed for static and dynamic problems of two-dimensional elastic bodies composed of multiple rectangular subregions. In the development, the original two-dimensional elasticity problem is first reduced into a one-dimensional boundary-value problem by finite difference; the exact solution of the reduced problem is then obtained by using the distributed transfer functions of the elastic continuum. The proposed technique, which combines the simplicity of finite difference and the closed form of analytical solutions, is capable of handling arbitrary boundary conditions, delivers highly accurate solutions for static and dynamic problems, and is computationally efficient. The proposed method is illustrated on a square region and an L-shaped region.

This paper presents a novel dimension-adaptive numerical integration method for dynamic analysis of structures with stochastic parameters subjected to deterministic excitations. First, an efficient dimension-adaptive algorithm is proposed to detect the importance of each random parameter involved in the structural model, based on which the quadrature nodes used for numerical integration can be collocated more reasonably. Then, the Gaussian quadrature formulas are used to evaluate the structural response statistics. To further improve the robustness and efficiency of the proposed method, the dimension-adaptive integration is only used to calculate the structural displacement response statistics. The velocity and acceleration response statistics are further evaluated using the finite difference formulas based on the concept of stochastic difference. Such a strategy is especially attractive when evaluating the response statistics of the derivative processes requires more quadrature nodes than that of the original process. Finally, two numerical examples encountered in civil engineering, including a shear frame with stochastic parameters subjected to a seismic ground motion and an Euler beam with unidimensional stochastic field of material properties (discretized via the Karhunen–Loève expansion) subjected to a moving load are studied to illustrate the performance of the proposed method. Via the numerical results, the accuracy and efficiency of the proposed method are verified.

In this paper, a cascadic Newton’s method is designed to solve the Monge–Ampère equation. In the process of implementing the cascadic multigrid, we use the Full-Local type interpolation as prolongation operator and Newton iteration as smoother. In order to obtain Full-Local type interpolation, we provide several finite difference stencils. Especially, the skewed finite difference methods are first applied by us for the elliptic Monge–Ampère equation. Based on Full-Local interpolation techniques and cascade principle, the new algorithm can save a large amount of computation time. Some numerical experiments are provided to confirm the efficiency of our proposed method.

This paper proposes a new numerical method to solve the 1D time-dependent Schrödinger equations based on the finite difference scheme by means of multiquadrics (MQ) and inverse multiquadrics (IMQ) radial basis functions. The numerical examples are given to confirm the good accuracy of the proposed methods.

An algorithm has been developed to identify the position of voids in structures by coupling level set method and finite element method (FEM). The identification problem is transformed into a minimum problem whose objective function is defined as a least square form of displacement error. A perimeter constraint term is also added in the objective function to make the solution well posed. The level set is applied in the present algorithm to represent the position and geometry of the voids. The velocity field of level set function is obtained by analyzing the shape derivative of objective function. FEM based on Euler description is employed for solving the forward problem. The same fixed meshes adopted by the solution of forward problem are used for finite difference computation of the level set function. The procedure of this algorithm has been applied to the voids identification of two-dimensional (2D) and three-dimensional (3D) structures, the examples of single void and multiple voids are considered. The results indicate that the voids in structure can be identified effectively by the present algorithm and the algorithm is also stable to noise.

A new approach to extend the applicability of different Boussinesq models [Nwogu, 1993; Schäffer and Madsen, 1995; Gobbi and Kirby, 1996] in deep water is introduced. This approach is based on optimizing the new defined performance index for linear wave dispersion property. The index measures the capability of a water wave model to simulate transformations of irregular wave over general topography. By optimizing this index at different relative water depths, the validity range for these models is extended. Optimizations process is based on calculating some selected free parameters in every model such that performance index is minimized. All optimized parameters are given in plotted figures verses the dimensionless peak frequency of the wave train. Comparisons among the optimized models are given. The approach applicability is investigated compared with Nwogu's model. Finite difference scheme over staggered grid is used to implement a high order numerical algorithm for the model.

A new numerical multiphase model is presented to study wave propagation over arbitrary shaped submerged obstacles. The high order space accurate weighted essentially non-oscillatory (WENO) method is adopted along with relatively coarse Cartesian uniform grids. Viscosity effects are included and the free surface is tracked using the level set method. The model is validated via application to solitary and progressive wave motion over rectangular, trapezoidal and semicircular obstacles. The complex flow field features induced near a large sized obstacle, including separation vortices and large free surface deformations, are accurately reproduced. Compared to other relatively complicated models, the present model is efficient and produces enhanced results.

Two higher-order compact finite difference approaches, the Hermitian and the Lax-Wendroff are examined by applying them to the viscous Burgers' equation. The difference equations obtained by the two methods were integrated in time through the third-order Strong-Stability-Preserving Runge-Kutta scheme. Absolute errors are computed by using an exact solution. The results are also compared with a second order central difference scheme. The Hermitian approach is far easier to implement. On uniform grids the Lax-Wendroff scheme produces smaller errors during the initial stages, but both methods are equally good for larger durations of integration. The convergence rate of the Hermitian scheme is slightly higher than the Lax-Wendroff scheme although both are of fourth order. Both schemes are unstable beyond a certain step size in time and space. When numerical boundary conditions are imposed, second-order conditions produce the best results whereas linear extrapolation proved to be the worst. It was also observed that large domains were required to implement the numerical boundary conditions properly. There was no detrimental effect on the accuracy of the results obtained through either of the two schemes when the size of the domain was greatly increased. Both schemes showed remarkable improvement in accuracy when clustered grids were employed. However much smaller time steps are required for stable solutions.

In this paper we consider Prandtl's boundary layer problem for incompressible laminar flow past a plate with transfer of fluid through the surface of the plate. When the Reynolds number is large the solution of this problem has a parabolic boundary layer. In a neighbourhood of the plate the solution of the problem has an additional singularity which is caused by the absence of the compartability conditions. To solve this problem outside nearest neighbourhood of the leading edge, we construct a direct numerical method for computing approximations to the solution of the problem using a piecewise uniform mesh appropriately fitted to the parabolic boundary layer. To validate this numerical method, the model Prandtl problem with self-similar solution was examined, for which a reference solution can be computed using the Blasius problem for a nonlinear ordinary differential equation. For the model problem, suction/blowing of the flow rate density is v₀(x)=-v_i2^-1/2Re^1/2x^-1/2, where the Reynolds number Re can be arbitrarily large and v_i is the intensity of the mass transfer with arbitrary values in the segment [-.3,.3]. We considered the Prandtl problem in a finite rectangle excluding the leading edge of the plate for various values of Re which can be arbitrary large and for some values of v_i, when meshes with different number of mesh points were used. To find reference solutions for the the velocity components and their derivatives with required accuracy, we solved the Blasius problem using a semi–analytical numerical method. By extensive numerical experiments we showed that the direct numerical method constructed in this paper allows us to approximate both the solution and its derivatives Re–uniformly for different values of v_i.