Showing papers on "Finite difference method published in 2019"

Difference Equations and Inequalities: Theory, Methods, and Applications

Abstract: Preliminaries linear initial value problems miscellaneous difference equations difference inequalities qualitative properties of solutions of difference systems qualitative properties of solutions of higher order difference equations qualitative properties of solutions of neutral difference equations boundary value problems for linear systems boundary value problems for nonlinear systems miscellaneous properties of solutions of higher order linear difference equations boundary value problems for higher order difference equations Sturm-Liouville problems and related inequalities difference inequalities in several independent variables.

Numerical Solutions with Linearization Techniques of the Fractional Harry Dym Equation

Abstract: In this study, numerical solutions of the fractional Harry Dym equation are investigated. Linearization techniques are utilized for non-linear terms existing in the fractional Harry Dym equation. The error norms L2 and L∞ are computed. Stability of the finite difference method is studied with the aid of Von Neumann stabity analysis

Numerical solutions of the fractional Fisher's type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods.

Abstract: The main objective of this paper is to investigate an accurate numerical method for solving a biological fractional model via Atangana-Baleanu fractional derivative. We focused our attention on linear and nonlinear Fisher’s equations. We use the spectral collocation method based on the Chebyshev approximations. This method reduced the nonlinear equations to a system of ordinary differential equations by using the properties of Chebyshev polynomials and then solved them by using the finite difference method. This is the first time that this method is used to solve nonlinear equations in Atangana-Baleanu sense. We present the effectiveness and accuracy of the proposed method by computing the absolute error and the residual error functions. The results show that the given procedure is an easy and efficient tool to investigate the solution of nonlinear equations with local and non-local singular kernels.

Energy stability and convergence of SAV block-centered finite difference method for gradient flows

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.

On the Variable Two-Step IMEX BDF Method for Parabolic Integro-differential Equations with Nonsmooth Initial Data Arising in Finance

Abstract: In this paper the implicit-explicit (IMEX) two-step backward differentiation formula (BDF2) method with variable step-size, due to the nonsmoothness of the initial data, is developed for solving pa...

Numerical investigation of fractional-order unsteady natural convective radiating flow of nanofluid in a vertical channel

Abstract: In the current article, we analyzed the unsteady natural convection with the help of fractional approach. Firstly, the unsteady natural convection radiating flow in an open ended vertical channel beside the magnetic effects. We assumed the channel is stationary with non-uniform temperature. Secondly, we utilized a fractional calculus approach for the constitutive relationship of a fluid model. The modeled problem is transformed into nondimensional form via viable non-dimensional variables. In order to investigate the numerical solutions of non-dimensional system of partial differential equations finite difference approach coupled with Crank Nicolson method is developed and successfully applied. The beauty of Crank Nicolson finite difference scheme is, this scheme is unconditionally stable. A very careful survey of literate witnesses that this scheme has never been reported in the literary for fluid problems. The physical changes are discussed with the help of graphics. The expression for both velocity field and temperature distribution has been made via said scheme. A comprehensive discussion about the influence of various related dimensionless parameters upon the flow properties disclosed our work. It is observed that velocity field decreases as enhancing the magnetic field effects. Heat transfer enhanced as enhancing the nanoparticle volume fraction parameter. Velocity field and heat transfer shows the dominant behavior for the case of Cu-based nanofluid as compare to Al2O3 based nanofluid. Comparative study also included to show the accuracy of the proposed finite difference scheme. It is to be highlighted that the proposed scheme is very efficient and well-matched to investigate the solutions of modeled problem and can be extended to diversify problems of physical nature.

A meshfree generalized finite difference method for surface PDEs

Abstract: In this paper, we propose a novel meshfree Generalized Finite Difference Method (GFDM) approach to discretize PDEs defined on manifolds. Derivative approximations for the same are done directly on the tangent space, in a manner that mimics the procedure followed in volume-based meshfree GFDMs. As a result, the proposed method not only does not require a mesh, it also does not require an explicit reconstruction of the manifold. In contrast to some existing methods, it avoids the complexities of dealing with a manifold metric, while also avoiding the need to solve a PDE in the embedding space. A major advantage of this method is that all developments in usual volume-based numerical methods can be directly ported over to surfaces using this framework. We propose discretizations of the surface gradient operator, the surface Laplacian and surface Diffusion operators. Possibilities to deal with anisotropic and discontinuous surface properties with large jumps are also introduced, and a few practical applications are presented.

Numerical solutions of waves-current interactions by generalized finite difference method

Abstract: In this paper, a meshless numerical wave flume, based on the generalized finite difference method (GFDM), is adopted to accurately and efficiently simulate the interactions of water waves and current. The GFDM, a newly-developed meshless method, is truly free from mesh generation and numerical quadrature. The proposed meshless numerical wave flume is the combination of the GFDM, the second-order Runge–Kutta method, the semi-Lagrangian approach, the sponge layer and the ramping function. The problems of wave-current interactions in flumes with horizontal and inclined bottoms are accurately and stably investigated by the proposed meshless scheme, respectively. The changes of waveform can be obviously found, while the cases of coplanar, opposing and no currents are stably simulated. Besides, the distribution of steady current in the flume with inclined bottom, which is governed by an inverse Cauchy problem, is acquired by the GFDM in a stable manner. Numerical results of wave-current interactions are compared with other solutions to verify the accuracy of the proposed meshless scheme. Additionally, different parameters of the proposed meshless numerical scheme are examined to validate the consistency and stability of the proposed numerical wave flume for solutions of wave-current interactions.

Error estimates of a regularized finite difference method for the logarithmic Schrödinger equation

Abstract: We present a regularized finite difference method for the logarithmic Schrodinger equation (LogSE) and establish its error bound. Due to the blowup of the logarithmic nonlinearity, i.e., $\ln \rho\...

Optimal Energy Flow for Integrated Energy Systems Considering Gas Transients

Abstract: This letter presents an optimal energy flow (OEF) for integrated energy systems considering the transient process of natural gas flows, whose main dynamic features are efficiently approximated in our setup. To achieve this, new discretization criteria are proposed to divide the gas flow equations into reasonable temporal and spatial segments to accurately model the gas flow dynamics in a set of difference equations. Simulation results validate the effectiveness of the OEF based on the proposed discretization criteria and its advantages over existing methods.

Multiple Slip Effects on MHD Unsteady Flow Heat and Mass Transfer Impinging on Permeable Stretching Sheet with Radiation

Abstract: This paper reports multiple slip effects on MHD unsteady flow heat and mass transfer over a stretching sheet with Soret effect; suction/injection and thermal radiation are numerically analyzed. We consider a time-dependent applied magnetic field and stretching sheet which moves with nonuniform velocity. Suitable similarity variables are used to transform governing partial differential equations into a system of coupled nonlinear ordinary differential equations. The transformed equations are then solved numerically by applying an implicit finite difference method with quasi-linearization technique. The influences of the various parameters on the velocity temperature and concentration profiles as well as on the skin friction coefficient and Sherwood and Nusselt numbers are discussed by the aid of graphs and tables.

The dimension splitting element-free Galerkin method for 3D transient heat conduction problems

Abstract: By transforming a 3D problem into some related 2D problems, the dimension splitting element-free Galerkin (DSEFG) method is proposed to solve 3D transient heat conduction problems. The improved element-free Galerkin (IEFG) method is used for 2D transient heat conduction problems, and the finite difference method is applied in the splitting direction. The discretized system equation is obtained based on the Galerkin weak form of 2D problem; the essential boundary conditions are imposed with the penalty method; and the finite difference method is employed in the time domain. Four exemplary problems are chosen to verify the efficiency of the DSEFG method. The numerical solutions show that the efficiency and precision of the DSEFG method are greater than ones of the IEFG method for 3D problems.

Biological interactions between Carreau fluid and microswimmers in a complex wavy canal with MHD effects

Abstract: The efficient magnetic swimming of actual or mechanically designed microswimmers within bounded regions is reliant on several factors: the actuation of these swimmers via magnetic field, rheology of surrounding liquid (with dominant viscous forces), nature of medium (either porous or non-porous), position (either straight, inclined or declined) and state (either active or passive) of the narrow passage. To witness these interactions, we utilize Carreau fluid with Taylor swimming sheet model under magnetic and porous effects. Moreover, the cervical canal is approximated as a two-dimensional complex wavy channel inclined at certain angle with the horizontal. The momentum equations are reduced by means of lubrication assumption, which finally leads to a fourth-order differential equation. MATLAB’s built-in bvp4c function is employed to solve the resulting boundary value problem. The solution obtained via bvp4c is further verified by finite difference method. In both these methods, the refined values of flow rate and cell speed are computed by utilizing modified Newton–Raphson method. These realistic pairs are further utilized to calculate the energy delivered by the microswimmer. The numerical results are plotted and discussed at the end of the article. Our study explains that the optimum speed of the microorganism can be achieved by means of exploiting the fluid rheology and with the suitable application of the magnetic field. The peristaltic nature of the channel walls and porous medium may also serve as alternative factors to control the speed of the propeller.

Stratified electromagnetohydrodynamic flow of nanofluid supporting convective role

Abstract: This study numerically examined unsteady double stratified EMHD mixed convection flow of nanofluid via permeable stretching sheet. It also looked at the convective heat and mass boundary conditions as well as the Navier velocity slip. In the thermal field, the effects of radiative heat transfer, heat generation/absorption, viscous dissipation, together with Ohmic heating (both magnetic and electric fields) were considered. The concentration field accounts for the chemical reaction. These show the physical behavior of electromagnetohydrodynamic flow associated with the problem formulation. The characteristics in regard to convective heat and mass, Navier slips conditions, as well as double stratification, were imposed. Such structure arises in energy efficiency and performance, which is achievable without higher pumping power, serves in the extrusion manufacturing process involving the thermal system for efficient devices particularly in polymeric, paper production, and food processing. The governing equations, which are nonlinear partial differential equations, were modelled by ordinary differential equations using suitable transformations. The ODEs were solved numerically, using implicit finite difference method (Keller box method). The physical implications deliberated on the behavior via the velocity, thermal energy, and concentration fields as well as the skin friction coefficient; the Nusselt and Sherwood numbers were scrutinized in relation to several parameters via mathematical model. The analysis shows that thermal and concentration stratifications decrease the distributions adjacent to the sheet surface, indicating decrease in the concentration nanoparticles and reduction in thermal energy. Augmentation occurs with convective heat and mass Biot numbers with the fields. The electric and magnetic parameters exhibit opposite flow behavior to the velocity and temperature. Chemical reaction and viscous dissipation weaken the concentration profile. Numerical results were compared with the published data available in the literature for limiting cases, and good agreement was noticed.