Showing papers on "Finite difference method published in 2016"
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30 Jun 2016
TL;DR: In this paper, a front tracking reservoir simulator was used to simulate the water coning problem in the modeling of hydrocarbon recovery. But the simulation was limited to a single reservoir and finite element and finite difference methods for continuous flows in porous media.
Abstract: Problems arising in the modeling of processes for hydrocarbon recovery Finite element and finite difference methods for continuous flows in porous media A front tracking reservoir simulator Five-spot validation studies and the water coning problem Statistical fluid dynamics: The influence of geometry on surface instabilities Some numerical methods for discontinuous flows in porous media.
348 citations
TL;DR: In this paper, a nonconforming Virtual Element Method (VEM) was proposed for the approximation of second order elliptic problems. But the method is not suitable for finite element methods.
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.
262 citations
TL;DR: This article proposes to discretize the problem of linear elastic homogenization by finite differences on a staggered grid and introduces fast and robust solvers and reduces the memory consumption of the Moulinec–Suquet algorithms by 50%.
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.
158 citations
TL;DR: This is the first proof for the stability of the ($3-\alpha$)-order scheme for the time-fractional diffusion equation and the theoretical result is validated by a number of numerical tests.
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.
137 citations
TL;DR: In this article, finite difference methods with non-uniform meshes for solving nonlinear fractional differential equations are presented, where the non-equidistant stepsize is non-decreasing and the rectangle formula and trapezoid formula are proposed based on theNon- uniform meshes.
130 citations
TL;DR: This work reformulates CPR methods using summation-by-parts (SBP) operators with simultaneous approximation terms (SATs), a framework popular for finite difference methods, and proves entropy stability for Burgers' equation is proved for general SBP CPR methods not including boundary nodes.
123 citations
TL;DR: In this paper, Wang et al. derived the fractional boundary layer governing equations for nonlinear coupled equations with mixed time-space derivatives in the convection terms, which are solved by a newly developed finite difference method combined with an L1-algorithm.
100 citations
TL;DR: The Mimetic Finite Difference method and Finite Volume Method are used to improve the numerical solution of the embedded discrete fracture model and can be used to simulate fractured reservoir with complex geometrical shape, which fails to be solved by the primal method based on the finite difference method.
99 citations
TL;DR: This scheme is a phase-field model for two-phase incompressible flows, consisting a Cahn-Hilliard-type diffusion equation and a Navier-Stokes equation, which involves adaptive mesh, adaptive time and a nonlinear multigrid finite difference method.
93 citations
TL;DR: In this paper, a linear implicit finite difference scheme for solving the generalized time fractional burgers equation is proposed, which is shown to be globally stable and convergent, and the finite difference method is proved to be unconditional globally stable.
92 citations
TL;DR: In this paper, a method for analyzing metasurfaces, modeled as zero-thickness electromagnetic sheets, in finite difference (FD) techniques is presented, which consists in describing the spatial discontinuity induced by the metasuran surface as a virtual structure, located between nodal rows of the Yee grid, using an FD version of generalized sheet transition conditions.
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.
TL;DR: In this article, the authors validate the experimental work carried out in the winter of 2015 on the concentrator in the city of Algerian city "Blida" by a numerical simulation, where the tap water used as a heat carrier fluid.
TL;DR: This paper reviews many recent developments and further applications of nonstandard finite difference (NSFD) methods encountered in the past decade and gives a detailed account on various definitions/notions of NSFD methods appeared in the literature in past two decades.
Abstract: In this paper, we review many recent developments and further applications of nonstandard finite difference (NSFD) methods encountered in the past decade. In particular, it is a follow up article of the one published in 2005 [K.C. Patidar, On the use of non-standard finite difference methods, J. Differ. Equ. Appl. 11 (2005), pp. 735–758]. It also includes those research contributions in this field that are very significant and published prior to the above article but were not included in the above paper simply because we did not have access to them when we wrote the above article. We also give a detailed account on various definitions/notions of NSFD methods appeared in the literature in past two decades. All contributions are listed chronologically except that in some instances we have grouped certain works to show connectivity in those fields. While categorizing these research contributions, we considered a number of different application areas. Moreover, due to space limitations, firstly, we have not i...
TL;DR: A scheme for high order finite difference methods for two-dimensional fractional differential equations with temporal Caputo and spatial Riemann-Liouville derivatives is proposed and it is shown that it converges with second order in time and fourth order in space.
Abstract: We consider high order finite difference methods for two-dimensional fractional differential equations with temporal Caputo and spatial Riemann-Liouville derivatives in this paper We propose a scheme and show that it converges with second order in time and fourth order in space The accuracy of our proposed method can be improved by Richardson extrapolation Approximate solution is obtained by the generalized minimal residual (GMRES) method A preconditioner is proposed to improve the efficiency for the implementation of the GMRES method
TL;DR: In this paper Galerkin finite element approximation of optimal control problems governed by time fractional diffusion equations is investigated and fully discrete first order optimality condition is developed based on 'first discretize, then optimize' approach.
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.
TL;DR: In this article, a coupled discrete element-finite difference model of the SLS process is proposed, where the powder particles are modeled as discrete, thermally and mechanically interacting spheres and the solid, underneath substrate is modeled via the finite difference method.
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.
TL;DR: The magnitude of the artificially diffusive term is numerically investigated and optimized using Fourier analysis and shows that the optimal coefficient increases as the coarse cell optical thickness increases, indicating that the pCMFD method overcorrects the diffusion coefficient which increases the spectral radius.
TL;DR: In this article, the authors have numerically solved a benchmark heat transfer nanofluid problem using three different widely used numerical approaches: Finite Element Method (FEM), Lattice Boltzmann Method (LBM) and Finite Difference Method(FDM).
TL;DR: In this paper, a nonlinear fractional boundary layer governing equations of nanofluid are formulated with time dependent fractional derivatives in the convection terms, which are solved by finite difference method combined with an L1-algorithm.
TL;DR: In this article, a numerical scheme to solve space fractional order diffusion equation is proposed using shifted Chebyshev polynomials of the third kind, where fractional differential derivatives are expressed in terms of the Caputo sense.
TL;DR: This work presents a high order accurate finite difference method for enforcing nonlinear friction laws, in a consistent and provably stable manner, suitable for efficient explicit time integration and shows numerical simulations on band limited self-similar fractal faults revealing the complexity of rupture dynamics on rough faults.
TL;DR: Numerical simulations for this novel multi-strain TB model of variable-order fractional derivatives, which incorporates three strains: drug-sensitive, emerging multi-drug resistant (MDR) and extensively drug-resistant (XDR), conclude that, NSFDM preserves the positivity of the solutions and numerically stable in large regions than SFDM.
TL;DR: Energy stability of the coupled numerical method is proven for the case of curved, nonconforming block-to-block interfaces and a provably energy stable coupling between curvilinear finite difference methods and a curved-triangle discontinuous Galerkin method is demonstrated.
Abstract: A methodology for handling block-to-block coupling of nonconforming, multiblock summation-by-parts finite difference methods is proposed. The coupling is based on the construction of projection operators that move a finite difference grid solution along an interface to a space of piecewise defined functions; we specifically consider discontinuous, piecewise polynomial functions. The constructed projection operators are compatible with the underlying summation-by-parts energy norm. Using the linear wave equation in two dimensions as a model problem, energy stability of the coupled numerical method is proven for the case of curved, nonconforming block-to-block interfaces. To further demonstrate the power of the coupling procedure, we show how it allows for the development of a provably energy stable coupling between curvilinear finite difference methods and a curved-triangle discontinuous Galerkin method. The theoretical results are verified through numerical simulations on curved meshes as well as eigenval...
TL;DR: In this article, the authors investigated numerical solutions of the regularized long wave (RLW) equation by using Haar wavelet (HW), combined with finite difference method, and the results of computations are compared with exact solutions and those already published.
Abstract: In this paper, we are going to investigate numerical solutions of the regularized long wave (RLW) equation by using Haar wavelet (HW), combined with finite difference method. The motion of a single solitary wave, interaction of two solitary waves, Maxwellian initial condition and wave undulation are our test problems for measuring performance of the proposed method. The results of computations are compared with exact solutions and those already published. \({L_{2}}\) and \({L_{\infty}}\) error norms and the numerical conservation laws are computed for discussing the accuracy and efficiency of the proposed method.
TL;DR: In this paper, the eccentricity effects and depth of investigation (DOI) of two typical types of azimuthal resistivity Logging-While-Drilling (ARLWD) tools (Type I: an axial transmitter with a transverse receiver; Type II: a tilted receiver) are investigated.
TL;DR: The numerical approximation of the distributed order time fractional reaction-diffusion equation on a semi-infinite spatial domain by a fully discrete scheme based on finite difference method in time and spectral approximation using Laguerre functions in space is proposed.
TL;DR: In this article, the authors investigated 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; their goal is to get the steady temperature distribution in the fluid.
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.
01 Jan 2016
TL;DR: In this article, a general procedure for deriving difference approximations to spatial derivatives using Taylor series expansions is presented, and the error incurred in the approximation, the truncation or discretization error is thoroughly analyzed.
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).
TL;DR: In this paper, a hybrid implicit-explicit-finite-difference time domain (HIE-FDTD) method is used to simulate a graphene-based polarizer.
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.
TL;DR: In this article, a numerical treatment of nonconforming grid interfaces and non-conforming mesh blocks is presented, where the authors use high order finite difference methods to solve the wave equation in the second order form.
Abstract: We use high order finite difference methods to solve the wave equation in the second order form. The spatial discretization is performed by finite difference operators satisfying a summation-by-parts property. The focus of this work is on numerical treatments of non-conforming grid interfaces and non-conforming mesh blocks. Interface conditions are imposed weakly by the simultaneous approximation term technique in combination with interface operators, which move discrete solutions between grids at an interface. In particular, we consider an interpolation approach and a projection approach with corresponding operators. A norm-compatible condition of the interface operators leads to energy stability for first order hyperbolic systems. By imposing an additional constraint on the interface operators, we derive an energy estimate of the numerical scheme for the second order wave equation. We carry out eigenvalue analyses to investigate the additional constraint and its relation to stability. In addition, a truncation error analysis is performed, and discussed in relation to convergence properties of the numerical schemes. In the numerical experiments, stability and accuracy properties of the numerical scheme are further explored, and the practical usefulness of non-conforming grid interfaces and mesh blocks is discussed in two practical examples.