Showing papers on "Finite difference method published in 2018"
TL;DR: In this article, a model has been established for deep borehole heat exchanger (DBHE) with coaxial tubes, considering coupled heat transfer in the tubes and surrounding subsurface, and an algorithm for direct solution of resulted algebraic equation set is used so as to achieve very efficient computation.
115 citations
TL;DR: In this paper, the q-homotopy analysis transform method was used to compute the approximate solutions for the fractional cubic isothermal auto-catalytic chemical system with Caputo-Fabrizio and Atangana-Baleanu fractional time derivatives in Liouville-Caputo sense.
Abstract: In this paper, we obtain analytical solutions for the fractional cubic isothermal auto-catalytic chemical system with Caputo–Fabrizio and Atangana–Baleanu fractional time derivatives in Liouville–Caputo sense. We utilize the q-homotopy analysis transform method to compute the approximate solutions. We find the optimal values of h so we assure the convergence of the approximate solutions. Finally, we compare our results numerically with the finite difference method and excellent agreement is found.
105 citations
TL;DR: In this article, numerical analysis has been performed to study natural convection heat transfer combined with entropy generation of ferrofluid in a square cavity with a non-uniformly heated centered horizontal plate under the influence of inclined uniform magnetic field.
96 citations
TL;DR: In this article, a new numerical technique for solving the fractional order diffusion equation is introduced, which basically depends on the Non-Standard finite difference method (NSFD) and Chebyshev collocation method.
Abstract: In this paper, a new numerical technique for solving the fractional order diffusion equation is introduced. This technique basically depends on the Non-Standard finite difference method (NSFD) and Chebyshev collocation method, where the fractional derivatives are described in terms of the Caputo sense. The Chebyshev collocation method with the (NSFD) method is used to convert the problem into a system of algebraic equations. These equations solved numerically using Newton’s iteration method. The applicability, reliability, and efficiency of the presented technique are demonstrated through some given numerical examples.
90 citations
TL;DR: In this paper, a square open cavity filled with two different porous blocks under the influence of uniform horizontal magnetic field is numerically studied, where porous blocks of different thermal properties, permeability and porosity are located on the bottom wall.
84 citations
TL;DR: In this paper, a generalization of the Gray-Scott model by using variable-order fractional differential equations is proposed and the variable-orders were set as smooth functions bounded in ( 0, 1 ] and, specifically, the Liouville-Caputo and the Atangana-Baleanu -Caputo fractional derivatives were used to express the time differentiation.
Abstract: A reaction–diffusion system can be represented by the Gray–Scott model. The reaction–diffusion dynamic is described by a pair of time and space dependent Partial Differential Equations (PDEs). In this paper, a generalization of the Gray–Scott model by using variable-order fractional differential equations is proposed. The variable-orders were set as smooth functions bounded in ( 0 , 1 ] and, specifically, the Liouville–Caputo and the Atangana–Baleanu–Caputo fractional derivatives were used to express the time differentiation. In order to find a numerical solution of the proposed model, the finite difference method together with the Adams method were applied. The simulations results showed the chaotic behavior of the proposed model when different variable-orders are applied.
77 citations
TL;DR: A novel finite difference method to discretize the fractional Laplacian ( − Δ ) α / 2 in hypersingular integral form by introducing a splitting parameter, which is then approximated by the weighted trapezoidal rule.
75 citations
TL;DR: In this article, a two-dimensional numerical analysis of the performance of a phase change materials (PCM) unit is presented, where the enthalpy method is used to solve the governing equations for the melting process in PCMs.
74 citations
TL;DR: In this article, an efficient numerical method is introduced for solving the fractional (Caputo sense) Fisher equation, which presents the problem of biological invasion and occurs, e.g., in ecology, physiology, and in general phase transition problems and others.
Abstract: In this paper, an efficient numerical method is introduced for solving the fractional (Caputo sense) Fisher equation. This equation presents the problem of biological invasion and occurs, e.g., in ecology, physiology, and in general phase transition problems and others. We use the spectral collocation method which is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce the proposed problem to a system of ODEs, which is solved by using finite difference method (FDM). Some theorems about the convergence analysis are stated. A numerical simulation and a comparison with the previous work are presented. We can apply the proposed method to solve other problems in engineering and physics.
71 citations
TL;DR: In this article, a finite-difference time-domain modeling of finite-size zero thickness space-modulated Huygens' metasurfaces based on generalized sheet transition conditions is proposed and numerically demonstrated.
Abstract: A finite-difference time-domain modeling of finite-size zero thickness space–time-modulated Huygens’ metasurfaces based on generalized sheet transition conditions is proposed and numerically demonstrated. A typical all-dielectric Huygens’ unit cell is taken as an example and its material permittivity is modulated in both space and time, to emulate a traveling-type spatio-temporal perturbation on the metasurface. By mapping the permittivity variation onto the parameters of the equivalent Lorentzian electric and magnetic susceptibility densities, $\chi _{\text {ee}}$ and $\chi _{\text {mm}}$ , the problem is formulated into a set of second-order differential equations in time with nonconstant coefficients. The resulting field solutions are then conveniently solved using an explicit finite-difference technique and integrated with a Yee-cell-based propagation region to visualize the scattered fields taking into account the various diffractive effects from the metasurface of finite size. Several examples are shown for both linear and space–time varying metasurfaces which are excited with normally incident plane and Gaussian beams, showing detailed scattering field solutions. While the time-modulated metasurface leads to the generation of new collinearly propagating temporal harmonics, these harmonics are angularly separated in space, when an additional space modulation is introduced in the metasurface.
71 citations
TL;DR: A family of explicit isogeometric collocation methods for structural dynamics that are obtained from predictor–multicorrector schemes that exhibit similar levels of computational cost, ease of implementation, and ease of parallelization to finite difference methods.
TL;DR: In this article, the results of the two methods were compared and this was shown that: 1) these two methods can draw the surface profiles and flow hydrograph as well; 2) the mesh size in finite difference method can be larger than that one; 4) the difference between two methods are increased by increasing the time and distance.
Abstract: The unsteady flow can be analysed by Saint-Venant equations These equations can be solved by characteristics and finite difference methods The Saint-Venant equations are changed into four complete differential equations in characteristics method and these equations are solved by drawing two characteristics lines The Saint-Venant equations are changed into set nonlinear equations and are solved using Preissman scheme in finite difference method This set of equation is changed into linear equation using Newton-Rafson method and can be solved using Sparce method In this research, the results of the two method were compared and this was shown that: 1) these two methods can draw the surface profiles and flow hydrograph as well; 2) the finite difference method is more accurate than that one; 3) the mesh size in finite difference method can be larger than that one; 4) the difference between two methods are increased by increasing the time and distance
TL;DR: In this paper, the authors investigated the nonlinear time-fractional Korteweg-de Vries (KdV) equation by using the (1/G′)-expansion method and the finite forward difference method.
Abstract: In this study, we investigate the nonlinear time-fractional Korteweg–de Vries (KdV) equation by using the (1/G′)-expansion method and the finite forward difference method. We first obtain the exact...
Posted Content•
TL;DR: Construction and analysis of a block-centered finite difference method for the spatial discretization of the scalar auxiliary variable Crank-Nicolson scheme (SAV/CN-BCFD) for gradient flows are presented, and it is shown rigorously that scheme is second-order in both time and space in various discrete norms.
Abstract: We present in this paper construction and analysis of a block-centered finite difference method for the spatial discretization of the scalar auxiliary variable Crank-Nicolson scheme (SAV/CN-BCFD) for gradient flows, and show rigorously that scheme is second-order in both time and space in various discrete norms. When equipped with an adaptive time strategy, the SAV/CN-BCFD scheme is accurate and extremely efficient. Numerical experiments on typical Allen-Cahn and Cahn-Hilliard equations are presented to verify our theoretical results and to show the robustness and accuracy of the SAV/CN-BCFD scheme.
TL;DR: In this article, the stochastic behavior of hydrodynamic journal bearings is investigated by solving the Reynolds equation with random parameters numerically based on finite difference method, and the steady state and dynamic characteristics are quantified considering random variabilities in eccentricity and surface roughness.
Book•
08 Feb 2018TL;DR: In this paper, the authors apply the difference analogue of the energy method to establish the unconditional stability of two types of difference approximations to parabolic differential equations, the (implicit) alternating direction methods of DOUGLAS, PEACEMAN, and RACHFORD, and a new semi-explicit method.
Abstract: In previous papers [12], [13], [14] the author developed a difference analogue of the energy method for determining the stability of difference approximations to partial differential equations with variable coefficients. The purpose of this paper is to apply this method to establish the unconditional stability of two types of difference approximations to parabolic differential equations, the (implicit) alternating direction methods of DOUGLAS, PEACEMAN, and RACHFORD [3], [5], [15], and a new semi-explicit method. For the model problem, the first boundary value problem for the heat conduction equation in a rectangular domain, the unconditional stability of the alternating direction methods was proved in [3] and [3]. The proof consists in showing, with the aid of Fourier analysis, that the yon Neumann stability condition [4], [11] is always satisfied. It can be shown [1], however, that this method of proof cannot be extended beyond the model problem. With the aid of the energy method we prove that the results in E3] and [6] can be extended beyond the model problem. We first treat, as a typical case, the heat conduction equation in a cylindrical domain with an essentially arbitrary, bounded base. Then we indicate briefly the extension to parabolic equations with variable coefficients. The second type of difference method we term the semi-explicit method, because it is an explicit method only for certain orderings of the net points. The idea for this difference method comes from the observation that there is a formal correspondence between parabolic difference equations and iterative methods for solving elliptic difference equations; the semi-explicit method corresponds to the well known method of successive displacements [7]. The only other known example of an unconditionally stable explicit difference method is due to Do FORT and FRANKEL [6]. Their method, however, requires two lines of initial data to start the solution, while the semi-explicit method is self-starting. On the other hand, both of these methods involve a similar local truncation error, and they must be subjected to a mild mesh ratio condition in order to be consistent [11] with the differential equation being approximated. For other applications of the energy method to the stability problem for partial difference equations, see FRIEDRICHS [8], KREISS [9], LAX [10] and LEES
TL;DR: In this article, a hybrid improved complex variable element-free Galerkin (H-ICVEFG) method is presented for three-dimensional advection-diffusion problems.
Abstract: In this paper, combining the dimension splitting method with the improved complex variable element-free Galerkin method, a hybrid improved complex variable element-free Galerkin (H-ICVEFG) method is presented for three-dimensional advection-diffusion problems. Using the dimension splitting method, a three-dimensional advection-diffusion problem is transformed into a series of two-dimensional ones which can be solved with the improved complex variable element-free Galerkin (ICVEFG) method. In the ICVEFG method, the improved complex variable moving least-squares (ICVMLS) approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions. Finite difference method is used in the one-dimensional direction. And Galerkin weak form of three-dimensional advection-diffusion problems is used to obtain the final discretized equations. Then the H-ICVEFG method for three-dimensional advection-diffusion problems is presented. Numerical examples are provided to discuss the influences of the weight functions, the effects of the scale parameter, the penalty factor, the number of nodes, the step number and the time step on the numerical solutions. And the advantages of the H-ICVEFG method with higher computational accuracy and efficiency are shown.
TL;DR: In this article, the authors investigated the vibration characteristics of pipes conveying fluid in the supercritical range by using Timoshenko beam theory for the first time and derived the nonlinear transverse vibration governing equation.
12 Nov 2018
TL;DR: In this paper, a pre-processor is used to input the problem geometry, generate the grid, and define the flow parameter and the boundary conditions to the code, which is then used to solve the governing equations of the flow subject to the conditions provided.
Abstract: Computational fluid dynamics (CFD) provides numerical approximation to the equations that govern fluid motion. Application of the CFD to analyze a fluid problem requires the following steps. First, the mathematical equations describing the fluid flow are written. These are usually a set of partial differential equations. These equations are then discretized to produce a numerical analogue of the equations. The domain is then divided into small grids or elements. Finally, the initial conditions and the boundary conditions of the specific problem are used to solve these equations. All CFD codes contain three main elements: (1) A pre-processor, which is used to input the problem geometry, generate the grid, and define the flow parameter and the boundary conditions to the code. (2) A flow solver, which is used to solve the governing equations of the flow subject to the conditions provided. There are four different methods used as a flow solver: (i) finite difference method; (ii) finite element method, (iii) finite volume method. (3) A post-processor, which is used to massage the data and show the results in graphical and easy to read format.
TL;DR: Second order, fully discrete, energy stable methods on spatially staggered grids for a hydrodynamic phase field model of binary viscous fluid mixtures in a confined geometry subject to both physical and periodic boundary conditions are presented.
Abstract: We present second order, fully discrete, energy stable methods on spatially staggered grids for a hydrodynamic phase field model of binary viscous fluid mixtures in a confined geometry subject to both physical and periodic boundary conditions. We apply the energy quadratization strategy to develop a linear-implicit scheme. We then extend it to a decoupled, linear scheme by introducing an intermediate velocity term so that the phase variable, velocity field, and pressure can be solved sequentially. The two new, fully discrete linear schemes are then shown to be unconditionally energy stable, and the linear systems resulting from the schemes are proved uniquely solvable. Rates of convergence of the two linear schemes in both space and time are verified numerically. The decoupled scheme tends to introduce excessive dissipation compared to the coupled one. The coupled scheme is then used to simulate fluid drops of one fluid in the matrix of another fluid as well as mixing dynamics of binary polymeric, viscous...
TL;DR: A fast finite difference method for distributed-order space-fractional PDEs on a general convex domain in multiple space dimensions that has an optimal order storage requirement and almost linear computational complexity, without any lossy compression.
Abstract: Fractional partial differential equations (PDEs) provide a powerful and flexible tool for modeling challenging phenomena including anomalous diffusion processes and long-range spatial interactions, which cannot be modeled accurately by classical second-order diffusion equations However, numerical methods for space-fractional PDEs usually generate dense or full stiffness matrices, for which a direct solver requires O ( N 3 ) computations per time step and O ( N 2 ) memory, where N is the number of unknowns The significant computational work and memory requirement of the numerical methods makes a realistic numerical modeling of three-dimensional space-fractional diffusion equations computationally intractable Fast numerical methods were previously developed for space-fractional PDEs on multidimensional rectangular domains, without resorting to lossy compression, but rather, via the exploration of the tensor-product form of the Toeplitz-like decompositions of the stiffness matrices In this paper we develop a fast finite difference method for distributed-order space-fractional PDEs on a general convex domain in multiple space dimensions The fast method has an optimal order storage requirement and almost linear computational complexity, without any lossy compression Numerical experiments show the utility of the method
TL;DR: The application of the GFDM to solving different non-linear problems including applications to heat transfer, acoustics and problems of mass transfer are shown.
TL;DR: The convergence rates of the proposed schemes are proved to be at O(h2 + τ2) with mesh-size h and time step τ in the discrete H2-norm, and the analysis method can be directly extended to other linear finite difference schemes for solving the KGS equations in high dimensions.
Abstract: The focus of this paper is on the optimal error bounds of two finite difference schemes for solving the d-dimensional (d = 2, 3) nonlinear Klein-Gordon-Schrodinger (KGS) equations. The proposed finite difference schemes not only conserve the mass and energy in the discrete level but also are efficient in practical computation because only two linear systems need to be solved at each time step. Besides the standard energy method, an induction argument as well as a ‘lifting’ technique are introduced to establish rigorously the optimal H
2-error estimates without any restrictions on the grid ratios, while the previous works either are not rigorous enough or often require certain restriction on the grid ratios. The convergence rates of the proposed schemes are proved to be at O(h
2 + τ
2) with mesh-size h and time step τ in the discrete H
2-norm. The analysis method can be directly extended to other linear finite difference schemes for solving the KGS equations in high dimensions. Numerical results are reported to confirm the theoretical analysis for the proposed finite difference schemes.
TL;DR: It is proved that the rate of convergence of the maximum nodal error on any subdomain that is bounded away from t=0{t=0} is higher than the rate obtained when the maximum nodsal error is measured over the entire space-time domain.
Abstract: Abstract A standard finite difference method on a uniform mesh is used to solve a time-fractional convection-diffusion initial-boundary value problem. Such problems typically exhibit a mild singularity at the initial time t=0{t=0}. It is proved that the rate of convergence of the maximum nodal error on any subdomain that is bounded away from t=0{t=0} is higher than the rate obtained when the maximum nodal error is measured over the entire space-time domain. Numerical results are provided to illustrate the theoretical error bounds.
TL;DR: It is shown that the pressure and velocity are discontinuous across the fracture-interface and the fracture indeed acts as the fast pathway or geological barrier in the aquifer system.
TL;DR: In this paper, the authors employed the powerful sine-Gordon expansion method in obtaining analytical solutions of the Benjamin-Bona-Mahony equation, which has a wide range of applications in modelling long surface gravity waves of small amplitude.
Abstract: In this article, we employed the powerful sine-Gordon expansion method in obtaining analytical solutions of the Benjamin–Bona–Mahony equation. We obtain some new solutions with the hyperbolic function structures. Benjamin–Bona–Mahony equation has a wide range of applications in modelling long surface gravity waves of small amplitude. We also plot the 2- and 3-dimensional graphics of all analytical solutions obtained in this paper. On the other hand, we analyze the finite difference method and operators, we obtain discretize equation using the finite difference operators. We consider one of the analytical solutions to the Benjamin–Bona–Mahony equation with the new initial condition. We observe that finite difference method is stable when Fourier–Von Neumann technique is used. We also analyze the accuracy of the finite difference method with terms of the errors $$L_{2}$$
and $$L_{\infty }$$
. We use the finite difference method in obtaining the numerical solutions of the Benjamin–Bona–Mahony equation. We compare the numerical results and the exact solution that are obtained in this paper, we support this comparison with the graphic plot. We perform all the computations and graphics plot in this study with the help of Wolfram Mathematica 9.
TL;DR: In this paper, the dimension split element-free Galerkin (DSEFG) method for three-dimensional potential problems is presented. And the convergence study and error analysis of the DSEFG method are presented.
Abstract: This paper presents the dimension split element-free Galerkin (DSEFG) method for three-dimensional potential problems, and the corresponding formulae are obtained The main idea of the DSEFG method is that a three-dimensional potential problem can be transformed into a series of two-dimensional problems For these two-dimensional problems, the improved moving least-squares (IMLS) approximation is applied to construct the shape function, which uses an orthogonal function system with a weight function as the basis functions The Galerkin weak form is applied to obtain a discretized system equation, and the penalty method is employed to impose the essential boundary condition The finite difference method is selected in the splitting direction For the purposes of demonstration, some selected numerical examples are solved using the DSEFG method The convergence study and error analysis of the DSEFG method are presented The numerical examples show that the DSEFG method has greater computational precision and computational efficiency than the IEFG method
TL;DR: A finite-difference spatial discretized scheme that preserves the port-Hamiltonian structure of infinite dimensional systems governed by the wave equation is proposed, based on the use of staggered grids for the discretization of different variables of the system.