Efficient Finite Difference Research Articles

An Efficient Nonstandard Finite Difference Scheme for Chaotic Fractional-Order Chen System

In this paper, an efficient nonstandard finite difference scheme for the numerical solution of chaotic fractional-order Chen system is developed. In the new method, an appropriate nonlocal framework in conjunction with the Grünwald-Letnikov approximation are applied for the discretization of fractional differential system. By constructing the discretization with the nonstandard finite difference scheme, high resolution of the system can be obtained, and the numerical instabilities of the nonlinear fractional-order Chen chaotic system can be also addressed to some extent. In addition, a new fractional derivative of the Caputo type is employed in the context of fractional-order Chen system to further decrease the computational complexity in the long-term treatment of fractional model. Numerical simulations demonstrate the applicability, accuracy and efficiency of the developed method.

Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium

<p style='text-indent:20px;'>In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.</p>

Modeling the Third-Order Electrodynamic Response of Graphene via an Efficient Finite-Difference Time-Domain Scheme

The third-order electrodynamic response of graphene and the subsequent non-linear effects are comprehensively investigated and numerically modeled in this article. Initially, the third-order conductivity of the medium is analyzed in order to transform it into a convenient expression for its efficient incorporation to explicit computational algorithms. Moreover, the simplified expression of the surface conductivity is imported into a suitably modified 3-D finite-difference time-domain scheme, where graphene is embodied as an equivalent surface current. Numerical examples, considering both an incident plane-wave toward graphene and surface-wave propagation, indicate the successful frequency upconversion, thus validating the proper third-order response modeling.

Numerical Solutions of Conformable Fractional Differential Equations by Taylor and Finite Difference Methods

We drive efficient and reliable finite difference methods for fractional differential equations (FDEs) based on recently defined conformable fractional derivative. We first derive fractional Euler and fractional Taylor methods based on the fractional Taylor expansion. This fractional Taylor series are the generalized fractional Taylor series that are independent of initial point. We show that the proposed methods are more efficient and faster by applying these methods on first order FDEs and second order oscillatory FDEs. Our second approach is based on inverting FDEs to a weakly singular integral equation that is approximated by product integration rule. This new definition has no special functions and thus the proposed numerical methods will be more accurate and easier to implement than existing methods for FDEs. We prove the stability and convergence of the proposed methods. Numerical examples are given to support the theoretical results.

An efficient and high accuracy finite-difference scheme for the acoustic wave equation in 3D heterogeneous media

In this paper we developed a new explicit compact high-order finite difference scheme to solve the 3D acoustic wave equations with spatially variable acoustic velocity. The boundary conditions for the second derivatives of spatial variables have been derived by using the wave equation and the boundary conditions themselves. Theoretical analysis shows that the new scheme has an accuracy order of O(τ2) + O(h4), where τ is the time step and h is the grid size. Combined with Richardson extrapolation or Runge–Kutta method, the new method can be improved to 4th-order accuracy in time, which has been implemented and verified in this paper. Four numerical experiments are conducted to validate the efficiency and accuracy of the new scheme. The stability of the new scheme has been proved by an energy method, which shows that the new scheme is conditionally stable with a Currant–Friedrichs–Lewy (CFL) number which is slightly lower than that of the Padé approximation based method. However, the new scheme is much simpler to implement.

A phase-field moving contact line model with soluble surfactants

A phase-field moving contact line model is presented for a two-phase system with soluble surfactants. With the introduction of some scalar auxiliary variables, the original free energy functional is transformed into an equivalent form, and then a new governing system is obtained. The resulting model consists of two Cahn–Hilliard-type equations and incompressible Navier–Stokes equation with variable densities, together with the generalized Navier boundary condition for the moving contact line. We prove that the proposed model satisfies the total energy dissipation with time. To numerically solve such a complex system, we develop a nonlinearly coupled scheme with unconditional energy stability. A splitting method based on pressure stabilization is used to solve the Navier–Stokes equation. Some subtle implicit–explicit treatments are adopted to discretize convection and stress terms. A stabilization term is artificially added to balance the explicit nonlinear term associated with the surface energy at the fluid–solid interface. We rigorously prove that the proposed scheme can preserve the discrete energy dissipation. An efficient finite difference method on staggered grids is used for the spatial discretization. Numerical results in both two and three dimensions demonstrate the accuracy and energy stability of the proposed scheme. Using our model and numerical scheme, we investigate the wetting behavior of droplets on a solid wall. Numerical results indicate that surfactants can affect the wetting properties of droplet by altering the value of contact angles.

Non-Newtonian effect on natural convection flow over cylinder of elliptic cross section

The non-Newtonian effect in the boundary layer flow over a horizontal elliptical cylinder is investigated numerically. A modified power-law viscosity model is used to correlate the non-Newtonian characteristics of the fluid flow. For natural convection flows, the surface of the cylinder is maintained by the uniform surface temperature (UST) or the uniform heat flux (UHF) condition. The governing equations corresponding to the flow are first transformed into a dimensionless non-similar form using suitable transformations. The resulting equations are solved numerically by an efficient finite difference scheme. The numerical results are presented for the skin friction coefficient and the local Nusselt number with the eccentric angle for different values of the power-law index n. The local skin friction coefficient and the local Nusselt number are found to be higher and lower, respectively, for the shear thickening fluids (n > 1) than the other fluids (n ⩽ 1). The effects of different elliptical configurations on the average Nusselt number are also presented and discussed for both conditions of the surface temperature.

Numerical Computation and Stability Analysis for the Fractional Subdiffusions with Spatial Variable Coefficients

In this paper, we propose an efficient compact finite difference method for a class of time-fractional subdiffusion equations with spatially variable coefficients. Based on the L2-1σ approximation formula of the time-fractional derivative and a fourth-order compact finite difference approximation to the spatial derivative, an efficient compact finite difference method is developed. The local truncation error and the solvability of the developed method are discussed in detail. The unconditional stability of the resulting scheme and also its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Numerical examples are provided to demonstrate the accuracy and the theoretical results.

Efficient and stable generalized auxiliary differential equation FDTD implementation of graphene dispersion

PurposeThe purpose of this paper is to present efficient and stable generalized auxiliary differential equation finite difference time domain (G-ADE-FDTD) implementation of graphene dispersion.Design/methodology/approachA generalized dispersive model is used for describing the graphene’s intraband and interband conductivities in the terahertz and infrared frequencies. In addition, the von Neumann method combined with the Routh-Hurwitz criterion are used for studying the stability of the given implementation.FindingsThe presented G-ADE-FDTD implementation allows modeling graphene’s dispersion using the minimal number of additional auxiliary variables, which will reduce both the CPU time and memory storage requirements. In addition, the stability of the implementation retains the standard non-dispersive Courant–Friedrichs–Lewy (CFL) constraint.Practical implicationsThe given implementation is conveniently applicable for most commonly used dispersive models, such as Debye, Lorentz, complex-conjugate pole residue, etc.Originality/valueThe presented G-ADE-FDTD implementation not only unifies the implementation of both graphene’s intraband and interband conductivities, with the minimal computational requirements but also retains the standard non-dispersive CFL time step stability constraint.

Efficient discontinuous finite difference meshes for 3-D Laplace–Fourier domain seismic wavefield modelling in acoustic media with embedded boundaries

SUMMARYSimulation of acoustic wave propagation in the Laplace–Fourier (LF) domain, with a spatially uniform mesh, can be computationally demanding especially in areas with large velocity contrasts. To improve efficiency and convergence, we use 3-D second- and fourth-order velocity-pressure finite difference (FD) discontinuous meshes (DM). Our DM algorithm can use any spatial discretization ratio between meshes. We evaluate direct and iterative parallel solvers for computational speed, memory requirements and convergence. Benchmarks in realistic 3-D models and topographies show more efficient and stable results for DM with direct solvers than uniform mesh results with iterative solvers.

Micropolar fluid flow between a non-porous disk and a porous disk with slip: Keller-box solution

Abstract In the present analysis, a steady, laminar, incompressible and two-dimensional micropolar fluid flow between a porous disk and a nonporous disk is considered. Similarity transformations are suitably applied to reduce the complex governing equations into a set of nonlinear boundary value problems with velocity slip conditions. An efficient finite difference technique called Keller-box method is employed to obtain the numerical solutions. The aim of this investigation is to study the influence of non-zero tangential slip velocity on velocity, microrotation profiles, and shear stress for different pertinent parameters. The results are then presented in the form of tables and graphs to analyze velocity and microrotation profiles for different physical micropolar parameters, Reynolds number, and slip coefficient. The results are found to be in excellent agreement with earlier findings in the absence of slip.

An efficient high-order compact finite difference scheme based on proper orthogonal decomposition for the multi-dimensional parabolic equation

In this paper, the combination of efficient sixth-order compact finite difference scheme (E-CFDS6) based proper orthogonal decomposition and Strang splitting method (E-CFDS6-SSM) is constructed for the numerical solution of the multi-dimensional parabolic equation (MDPE). For this purpose, we first develop the CFDS6 to attain a high accuracy for the one-dimensional parabolic equation (ODPE). Then, by the Strang splitting method, we have converted the MDPE into a series of one-dimensional ODPEs successfully, which is easier to implement and program with CFDS6 than an alternating direction implicit method. Finally, we employ proper orthogonal decomposition techniques to improve the computational efficiency of CFDS6 and build the E-CFDS6-SSM with fewer unknowns and sufficiently high accuracy. Six numerical examples are presented to demonstrate that the E-CFDS6-SSM not only can largely alleviate the computational load but also hold a high accuracy and simplify the process of the program for the numerical solution of MDPE.

Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications

In this paper, we propose accurate and efficient finite difference methods to discretize the two- and three-dimensional fractional Laplacian (−Δ)α2 (0<α<2) in hypersingular integral form. The proposed methods provide a fractional analogue of the central difference schemes to the fractional Laplacian. As α→2−, they collapse to the central difference schemes of the classical Laplace operator −Δ. We prove that our methods are consistent if u∈C⌊α⌋,α−⌊α⌋+ε(Rd), and the local truncation error is O(hε), with ε>0 a small constant and ⌊⋅⌋ denoting the floor function. If u∈C2+⌊α⌋,α−⌊α⌋+ε(Rd), they can achieve the second order of accuracy for any α∈(0,2). These results hold for any dimension d≥1 and thus improve the existing error estimates of the one-dimensional cases in the literature. Extensive numerical experiments are provided and confirm our analytical results. We then apply our method to solve the fractional Poisson problems and the fractional Allen–Cahn equations. Numerical simulations suggest that to achieve the second order of accuracy, the solution of the fractional Poisson problem should at most satisfy u∈C1,1(Rd). One merit of our methods is that they yield a multilevel Toeplitz stiffness matrix, an appealing property for the development of fast algorithms via the fast Fourier transform (FFT). Our studies of the two- and three-dimensional fractional Allen–Cahn equations demonstrate the efficiency of our methods in solving the high-dimensional fractional problems.

Inertial gravity current produced by the drainage of a cylindrical reservoir from an outer or inner edge

We consider the time-dependent flow of a fluid of density $\unicode[STIX]{x1D70C}_{1}$ in a vertical cylindrical container embedded in a fluid of density $\unicode[STIX]{x1D70C}_{2}~({&lt;}\unicode[STIX]{x1D70C}_{1})$ whose side boundary is suddenly removed and the fluid drains freely from the edge. We show that in the inertial–buoyancy regime (large initial Reynolds number) the flow is modelled by the shallow-water equations and bears similarities to a gravity current released from a lock (the dam-break problem) driven by the reduced gravity $g^{\prime }=(1-\unicode[STIX]{x1D70C}_{2}/\unicode[STIX]{x1D70C}_{1})g$. This formulation is amenable to an efficient finite-difference solution. Moreover, we demonstrate that similarity solutions exist, and show that the flow created by the dam break approaches the predicted self-similar behaviour when the volume ratio ${\mathcal{V}}(t)/{\mathcal{V}}(0)\approx 1/2$ where $t$ is time elapsed from the dam break. We considered two cases of drainage: (i) outward from the outer boundary in a full-radius reservoir; and (ii) inward from the inner radius in an annular-shaped reservoir. For the first case the similarity solution is expressed analytically, while the second case is more complicated and requires a numerical solution. In both cases ${\mathcal{V}}(t)/{\mathcal{V}}(0)$ decays like $t^{-2}$, but the details are different. The similarity solutions admit an adjustable virtual-origin constant, which we determine by matching with the finite-difference solution. The analysis is valid for both Boussinesq and non-Boussinesq systems, and a wide range of geometric parameters (inner and outer radii, and height). The importance of the neglected viscous terms increases with time, and eventually the inertial–buoyancy model becomes invalid. An estimate for this occurrence is also provided. The predictions of the model are compared to results of direct numerical simulations; there is good agreement for the position of the interface and for the averaged radial velocity, and excellent agreement for ${\mathcal{V}}(t)/{\mathcal{V}}(0)$. A box model is used for estimating the effect of a partial (over a sector) dam break. This study is an extension of the work for a rectangular reservoir of Momen et al. (J. Fluid Mech., vol. 827, 2017, pp. 640–663). We demonstrate that there are some similarities, but also significant differences, between the rectangular and the cylindrical reservoirs concerning the velocity, shape of the interface and rate of drainage, which are of interest in applications. The overall conclusion is that this simple model captures very well the flow field under consideration.

A high-order compact difference method for time fractional Fokker–Planck equations with variable coefficients

A high-order compact finite difference method is proposed for time fractional Fokker–Planck equations with variable convection coefficients. This method leads to a very simple and yet efficient compact finite difference scheme with high-order accuracy. It is also very convenient for us to give the corresponding analysis of stability and convergence using a discrete energy method. The proposed method is unconditionally stable and convergent with the convergence order $$\mathcal{O}(\tau ^{2}+h^{4})$$, where $$\tau $$ and h are the step sizes in time and space, respectively. Thus, it improves the convergence order of some recently developed methods. Numerical results confirm the theoretical analysis and demonstrate the high efficiency of this novel method.

A conservative numerical method for the fractional nonlinear Schrödinger equation in two dimensions

This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schrodinger (NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grunwald-Letnikov difference (WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schrodinger (FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations (ODEs) in matrices formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson (CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor (cIIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O(τ2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.

A compact high order Alternating Direction Implicit method for three-dimensional acoustic wave equation with variable coefficient

Efficient and accurate numerical simulation of seismic wave propagation is important in various Geophysical applications such as seismic full waveform inversion (FWI) problem. However, due to the large size of the physical domain and requirement on low numerical dispersion, many existing numerical methods are inefficient for numerical modelling of seismic wave propagation in an heterogeneous media. Despite the great efforts that have been devoted during the past decades, it still remains a challenging task in the development of efficient and accurate finite difference method for multi-dimensional acoustic wave equation with variable velocity. In this paper we proposed a Padé approximation based finite difference scheme for solving the acoustic wave equation in three-dimensional heterogeneous media. The new method is obtained by combining the Padé approximation and a novel algebraic manipulation. The efficiency of the new algorithm is further improved through the Alternative Directional Implicit (ADI) method. The stability of the new algorithm has been theoretically proved by energy method. The new method is conditionally stable with a better Courant–Friedrichs–Lewy condition (CFL) condition, which has been verified numerically. Extensive numerical examples have been solved, which demonstrated that the new method is accurate, efficient and stable.

Iterative realization of finite difference schemes in the fictitious domain method for elliptic problems with mixed derivatives

Development of efficient finite difference schemes and iterative methods for solving anisotropic diffusion problems in an arbitrary geometry domain is considered. To simplify the formulation of the Neumann boundary conditions, the method of fictitious domains is used. On the example of a two-dimensional model problem of potential distribution in an isolated anisotropic ring conductor a comparative efficiency analysis of some promising finite-difference schemes and iterative methods in terms of their compatibility with the fictitious domain method is carried out. On the basis of numerical experiments empirical estimates of the asymptotic dependence of the convergence rate of the biconjugate gradient method with Fourier – Jacobi and incomplete LU factorization preconditioners on the step size and the value of the small parameter determining the continuation of the conductivity coefficient in the fictitious domain method are obtained. It is shown, that for one of the considered schemes the Fourier – Jacobi preconditioner is spectrally optimal and allows to eliminate the asymptotical dependence of the iterations number to achieve a given accuracy both on the value of the step size and the value of the small parameter in the fictitious domain method.

A hybrid finite-difference/low-rank solution to anisotropy acoustic wave equations

P-wave extrapolation in anisotropic media suffers from SV-wave artifacts and computational dependency on the complexity of anisotropy. The anisotropic pseudodifferential wave equation cannot be solved using an efficient time-domain finite-difference (FD) scheme directly. The wavenumber domain allows us to handle pseudodifferential operators accurately; however, it requires either smoothly varying media or more computational resources. In the limit of elliptical anisotropy, the pseudodifferential operator reduces to a conventional operator. Therefore, we have developed a hybrid-domain solution that includes a space-domain FD solver for the elliptical anisotropic part of the anisotropic operator and a wavenumber-domain low-rank scheme to solve the pseudodifferential part. Thus, we split the original pseudodifferential operator into a second-order differentiable background and a pseudodifferential correction term. The background equation is solved using the efficient FD scheme, and the correction term is approximated by the low-rank approximation. As a result, the correction wavefield is independent of the velocity model, and, thus, it has a reduced rank compared with the full operator. The total computation cost of our method includes the cost of solving a spatial FD time-step update plus several fast Fourier transforms related to the rank. The accuracy of our method is of the order of the FD scheme. Applications to a simple homogeneous tilted transverse isotropic (TTI) medium and modified BP TTI models demonstrate the effectiveness of the approach.

Numerical Solution of Nonlinear Reaction–Advection–Diffusion Equation

In the present article, the advection–diffusion equation (ADE) having a nonlinear type source/sink term with initial and boundary conditions is solved using finite difference method (FDM). The solution of solute concentration is calculated numerically and also presented graphically for conservative and nonconservative cases. The emphasis is given for the stability analysis, which is an important aspect of the proposed mathematical model. The accuracy and efficiency of the proposed method are validated by comparing the results obtained with existing analytical solutions for a conservative system. The novelty of the article is to show the damping nature of the solution profile due to the presence of the nonlinear reaction term for different particular cases in less computational time by using the reliable and efficient finite difference method.