Abstract: We introduce the nonconforming Virtual Element Method (VEM) for the approximation of second order elliptic problems. We present the construction of the new element in two and three dimensions, highlighting the main differences with the conforming VEM and the classical nonconforming finite element methods. We provide the error analysis and establish the equivalence with a family of mimetic finite difference methods. Numerical experiments verify the theory and validate the performance of the proposed method.

Abstract: Summary In this article, we propose to discretize the problem of linear elastic homogenization by finite differences on a staggered grid and introduce fast and robust solvers. Our method shares some properties with the FFT-based homogenization technique of Moulinec and Suquet, which has received widespread attention recently because of its robustness and computational speed. These similarities include the use of FFT and the resulting performing solvers. The staggered grid discretization, however, offers three crucial improvements. Firstly, solutions obtained by our method are completely devoid of the spurious oscillations characterizing solutions obtained by Moulinec–Suquet's discretization. Secondly, the iteration numbers of our solvers are bounded independently of the grid size and the contrast. In particular, our solvers converge for three-dimensional porous structures, which cannot be handled by Moulinec–Suquet's method. Thirdly, the finite difference discretization allows for algorithmic variants with lower memory consumption. More precisely, it is possible to reduce the memory consumption of the Moulinec–Suquet algorithms by 50%. We underline the effectiveness and the applicability of our methods by several numerical experiments of industrial scale. Copyright © 2015 John Wiley & Sons, Ltd.

Abstract: In this paper, we consider a numerical method for the time-fractional diffusion equation. The method uses a high order finite difference method to approximate the fractional derivative in time, resulting in a time stepping scheme for the underlying equation. Then the resulting equation is discretized in space by using a spectral method based on the Legendre polynomials. The main body of this paper is devoted to carry out a rigorous analysis for the stability and convergence of the time stepping scheme. As a by-product and direct extension of our previous work, an error estimate for the spatial discretization is also provided. The key contribution of the paper is the proof of the ($3-\alpha$)-order convergence of the time scheme, where $\alpha$ is the order of the time-fractional derivative. Then the theoretical result is validated by a number of numerical tests. To the best of our knowledge, this is the first proof for the stability of the ($3-\alpha$)-order scheme for the time-fractional diffusion equation.

Abstract: We introduce a rigorous and simple method for analyzing metasurfaces, modeled as zero-thickness electromagnetic sheets, in finite difference (FD) techniques. The method consists in describing the spatial discontinuity induced by the metasurface as a virtual structure, located between nodal rows of the Yee grid, using an FD version of generalized sheet transition conditions. In contrast to previously reported approaches, the proposed method can handle sheets exhibiting both electric and magnetic discontinuities, and represents therefore a fundamental contribution to computational electromagnetics. It is presented here in the framework of the FD frequency domain method, but also applies to the FD time domain scheme. The theory is supported by five illustrative examples.

Abstract: Selective laser sintering (SLS) is an additive manufacturing technology whereby one can 3D print parts out of a powdered material. However, in order to produce defect free parts of sufficient strength, the process parameters (laser power, scan speed, powder layer thickness, etc.) must be carefully optimized depending on material, part geometry, and desired final part characteristics. Computational methods are very useful in the quick optimization of such parameters without the need to run numerous costly experiments. Most published models of this process involve continuum-based techniques, which require the homogenization of the powder bed and thus do not capture the stochastic nature of this process. Thus, the aim of this research is to produce a reduced order computational model of the SLS process which combines the essential physics with fast computation times. In this work the authors propose a coupled discrete element-finite difference model of this process. The powder particles are modeled as discrete, thermally and mechanically interacting spheres. The solid, underneath substrate is modeled via the finite difference method. The model is validated against experimental results in the literature and three-dimensional simulations are presented.

Abstract: In this paper Galerkin finite element approximation of optimal control problems governed by time fractional diffusion equations is investigated. Piecewise linear polynomials are used to approximate the state and adjoint state, while the control is discretized by variational discretization method. A priori error estimates for the semi-discrete approximations of the state, adjoint state and control are derived. Furthermore, we also discuss the fully discrete scheme for the control problems. A finite difference method developed in Lin and Xu (2007) is used to discretize the time fractional derivative. Fully discrete first order optimality condition is developed based on 'first discretize, then optimize' approach. The stability and truncation error of the fully discrete scheme are analyzed. Numerical example is given to illustrate the theoretical findings.

Abstract: This chapter provides in-depth coverage of the finite difference method (FDM) in the context of elliptic boundary value problems. The general procedure for deriving difference approximations to spatial derivatives using Taylor series expansions is first presented. The error incurred in the approximation – the truncation or discretization error – is thoroughly analyzed, and the procedure to develop higher-order approximations to derivatives is outlined. The implementation of the three canonical types of boundary conditions, i.e., Dirichlet, Neumann, and Robin, is discussed. Presentation of the matrix form of the discrete equations is finally followed by extension of the FDM to multidimensional geometries, including those described by the cylindrical coordinate system, and generalized curvilinear coordinates (body-fitted mesh).

Abstract: A lattice-Boltzmann (LB) scheme, based on the Bhatnagar-Gross-Krook (BGK) collision rules is developed for a phase-field model of alloy solidification in order to simulate the growth of dendrites. The solidification of a binary alloy is considered, taking into account diffusive transport of heat and solute, as well as the anisotropy of the solid-liquid interfacial free energy. The anisotropic terms in the phase-field evolution equation, the phenomenological anti-trapping current (introduced in the solute evolution equation to avoid spurious solute trapping), and the variation of the solute diffusion coefficient between phases, make it necessary to modify the equilibrium distribution functions of the LB scheme with respect to the one used in the standard method for the solution of advection-diffusion equations. The effects of grid anisotropy are removed by using the lattices D3Q15 and D3Q19 instead of D3Q7. The method is validated by direct comparison of the simulation results with a numerical code that uses the finite-difference method. Simulations are also carried out for two different anisotropy functions in order to demonstrate the capability of the method to generate various crystal shapes.

Abstract: In Liu and Qiu (J Sci Comput 63:548---572, 2015), we presented a class of finite difference Hermite weighted essentially non-oscillatory (HWENO) schemes for conservation laws, in which the reconstruction of fluxes is based on the usual practice of reconstructing the flux functions. In this follow-up paper, we present an alternative formulation to reconstruct the numerical fluxes, in which we first use the solution and its derivatives directly to interpolate point values at interfaces of computational cells, then we put the point values at interface of cell in building block to generate numerical fluxes. The building block can be arbitrary monotone fluxes. Comparing with Liu and Qiu (2015), one major advantage is that arbitrary monotone fluxes can be used in this framework, while in Liu and Qiu (2015) the traditional practice of reconstructing flux functions can be applied only to smooth flux splitting. Furthermore, these new schemes still keep the effectively narrower stencil of HWENO schemes in the process of reconstruction. Numerical results for both one and two dimensional equations including Euler equations of compressible gas dynamics are provided to demonstrate the good performance of the methods.

Abstract: A dispersion hybrid implicit–explicit-finite-difference time-domain (HIE-FDTD) method is used to simulate a graphene-based polarizer in this paper. The surface conductivity of graphene is incorporated into the conventional HIE-FDTD method directly through an auxiliary difference equation (ADE). The time step size in the proposed method has no relation with the fine spatial cells in the graphene layer. The simulation results show that the calculated result of the dispersion HIE-FDTD method agrees very well with that of the conventional ADE-FDTD method, but its computational time is considerably reduced. By using the dispersion HIE-FDTD method, it numerically validates that graphene can be used as a tunable linear-to-circular polarizer through controlling its chemical potential.

Abstract: This numerical study is aimed at investigating the convective heat transfer and flow fluid inside a horizontal circular tube in a fully developed laminar flow regime under a constant wall temperature boundary condition, commonly called the Graetz problem; our goal is to get the steady temperature distribution in the fluid. The complexity of the partial differential equation that describes the temperature field with the associated linear or non-linear boundary conditions is simplified by means of numerical methods using current computational tools. The simplified energy equation is solved numerically by the orthogonal collocation method followed by the finite difference method (Crank-Nicholson method). The calculations were effected through a FORTRAN computer program, and the results show that the orthogonal collocation method gives better results than the Crank-Nicholson method. In addition, the numerical results were compared to the experimental values obtained on the same tube diameter. It is important to note that the numerical results are in good agreement with the published experimental data.

Abstract: The behavior of a water-based nanofluid containing motile gyrotactic micro-organisms passing an isothermal nonlinear stretching sheet in the presence of a non-uniform magnetic field is studied numerically. The governing partial differential equations including continuity, momentums, energy, concentration of the nanoparticles, and density of motile micro-organisms are converted into a system of the ordinary differential equations via a set of similarity transformations. New set of equations are discretized using the finite difference method and have been linearized by employing the Newton’s linearization technique. The tri-diagonal system of algebraic equations from discretization is solved using the well-known Thomas algorithm. The numerical results for profiles of velocity, temperature, nanoparticles concentration and density of motile micro-organisms as well as the local skin friction coefficient Cfx, the local Nusselt number Nux, the local Sherwood number Shx and the local density number of the motile microorganism Nnx are expressed graphically and described in detail. This investigation shows the density number of the motile micro-organisms enhances with rise of M, Gr/Re2, Pe and Ω but it decreases with augment of Rb and n. Also, Sherwood number augments with an increase of M and Gr/Re2, while decreases with n, Rb, Nb and Nr. To show the validity of the current results, a comparison between the present results and the existing literature has been carried out.

Abstract: Efficient numerical seismic wavefield modelling is a key component of modern seismic imaging techniques, such as reverse-time migration and full-waveform inversion. Finite difference methods are perhaps the most widely used numerical approach for forward modelling, and here we introduce a novel scheme for implementing finite difference by introducing a time-to-space wavelet mapping. Finite difference coefficients are then computed by minimising the difference between the spatial derivatives of the mapped wavelet and the finite difference operator over all propagation angles. Since the coefficients vary adaptively with different velocities and source wavelet bandwidths, the method is capable to maximise the accuracy of the finite difference operator. Numerical examples demonstrate that this method is superior to standard finite difference methods, while comparable to Zhang’s optimised finite difference scheme.

Abstract: This paper presents a numerical study of thermosolutal mixed convection in rectangular enclosure with sliding top lid. The fluid flow is solved by the multiple relaxation time (MRT) lattice Boltzmann method (LBM), whereas the temperature and concentration fields are computed by finite difference method (FDM). The main objective of this study is to investigate the accuracy and the effectiveness of such model to predict thermodynamics for heat and mass transfer in a driven cavity. This model is validated with different numerical methods in the current literature. A good agreement is obtained between our results and previous works. The different comparisons demonstrate the robustness and the accuracy of the proposed approach.

Abstract: Natural convective heat transfer and fluid flow in an open porous cavity filled with a nanofluid is studied numerically using the Tiwari and Das nanofluid model. The transport equations for mass, momentum and energy formulated in dimensionless stream function and temperature are solved numerically using a second-order accurate finite difference method. Particular efforts are focused on the effects of the governing parameters on the heat and fluid flow. It is found that an increase in undulation number of the wavy vertical wall leads to an attenuation of convective flow and a decrease in the heat transfer rate.