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.

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.

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.

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.

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.

Abstract: In this paper, by combining the dimension splitting method and the improved complex variable element‐free Galerkin method, the dimension splitting and improved complex variable element‐free Galerkin (DS‐ICVEFG) method is presented for 3‐dimensional (3D) transient heat conduction problems. Using the dimension splitting method, a 3D transient heat conduction problem is translated into a series of 2‐dimensional ones, which can be solved with the improved complex variable element‐free Galerkin (ICVEFG) method. In the ICVEFG method for each 2‐dimensional problem, the improved complex variable moving least‐square 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 1‐dimensional direction, and the Galerkin weak form of 3D transient heat conduction problem is used to obtain the final discretized equations. Then, the DS‐ICVEFG method for 3D transient heat conduction problems is presented. Four numerical examples are given to show that the new method has higher computational precision and efficiency.

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...

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

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

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.

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

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.

Abstract: This paper is devoted to present an accurate numerical procedure to solve fractional (Caputo sense) Korteweg-de Vries, Korteweg-de Vries-Burgers and Burgers equations by using the spectral Chebyshev collocation method and finite difference method (FDM). The proposed problem is reduced to a system of ODEs with the help of the properties of Chebyshev polynomials of the third kind. This system is solved by using the FDM. Some theorems about the convergence analysis are stated and proved. A numerical simulation and a comparison with the previous work are presented.

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.

Abstract: In the present theoretical investigation, we have numerically simulated the problem of blood flow through an overlapping stenosed arterial blood vessel under the action of externally applied body acceleration and the periodic pressure gradient. The rheology of blood is characterized by the Sutterby fluid model. The blood is considered as an electrically conducting fluid. A steady uniform magnetic field is applied in the radial direction of the blood vessel. The governing nonlinear partial differential equations of the present flow together with prescribed boundary conditions are solved by employing explicit finite difference scheme. Results concerning the temporal distribution of velocity, flow rate, shear stress and resistance to the flow are displayed through graphs. The effects of various emerging parameters on the flow variables are analyzed and discussed in detail. The analysis reveals that the applied magnetic field and periodic body acceleration have considerable effects on the flow field.

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

Abstract: The flow and heat transfer characteristics of both single-wall and multi-wall carbon nanotubes (CNTs) with water and kerosene as base fluid on a moving plate with slip effect are studied numerically. By employing similarity transformation, governing equations are transformed into a set of nonlinear ordinary equations. These equations are solved numerically using the bvp4c solver in Matlab which is a very efficient finite difference method. The influence of numerous parameters such as nanoparticle volume fraction, velocity ratio parameter and first order slip parameter on velocity, temperature, skin friction and heat transfer rate are further explored and discussed in the form of graphical and tabular forms. The results reveal that dual solutions exist when the plate and free stream move in the opposite direction and slip parameter was found to widen the range of the possible solutions. However, skin friction coefficients decrease, whereas the heat transfer increases in the presence of slip parameter. Single-wall carbon nanotubes (SWCNTs) give higher skin friction and heat transfer compared to multi-wall carbon nanotubes (MWCNTs) due to the fact that they have higher density and thermal conductivity. A stability analysis is carried out to determine the stability of the solutions obtained.

Abstract: In this paper, we develop a novel, linearly implicit and local energy-preserving scheme for the sine-Gordon equation. The basic idea is from the invariant energy quadratization approach to construct energy stable schemes for gradient systems, which are energy dispassion. We here take the sine-Gordon equation as an example to show that the invariant energy quadratization approach is also an efficient way to construct linearly implicit and local energy-conserving schemes for energy-conserving systems. Utilizing the invariant energy quadratization approach, the sine-Gordon equation is first reformulated into an equivalent system, which inherits a modified local energy conservation law. The new system are then discretized by the conventional finite difference method and a semi-discretized system is obtained, which can conserve the semi-discretized local energy conservation law. Subsequently, the linearly implicit structure-preserving method is applied for the resulting semi-discrete system to arrive at a fully discretized scheme. We prove that the resulting scheme can exactly preserve the discrete local energy conservation law. Moveover, with the aid of the classical energy method, an unconditional and optimal error estimate for the scheme is established in discrete $H_h^1$-norm. Finally, various numerical examples are addressed to confirm our theoretical analysis and demonstrate the advantage of the new scheme over some existing local structure-preserving schemes.

Abstract: We consider finite difference methods for solving nonlinear fractional differential equations in the Caputo fractional derivative sense with non-uniform meshes. Under the assumption that th...

Abstract: This paper describes a new formulation of the dual reciprocity boundary element method (DRBEM) for two-dimensional transient convection–diffusion–reaction problems with variable velocity. The formulation decomposes the velocity field into an average and a perturbation part, with the latter being treated using a dual reciprocity approximation to convert the domain integrals arising in the boundary element formulation into equivalent boundary integrals. The integral representation formula for the convection–diffusion–reaction problem with variable velocity is obtained from the Green’s second identity, using the fundamental solution of the corresponding steady-state equation with constant coefficients. A finite difference method (FDM) is used to simulate the time evolution procedure for solving the resulting system of equations. Numerical applications are included for three different benchmark examples for which analytical solutions are available, to establish the validity of the proposed approach and to demonstrate its efficiency. Finally, results obtained show that the DRBEM results are in excellent agreement with the analytical solutions and do not present oscillations or damping of the wave front, as it appears in other numerical techniques.