Analysis of Extended Fisher-Kolmogorov Equation in 2D Utilizing the Generalized Finite Difference Method with Supplementary Nodes

In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)problem.The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolson scheme.Following temporal discretization,the generalized finite difference method(GFDM)with supplementary nodes is utilized to address the nonlinear boundary value problems at each time node.These supplementary nodes are distributed along the boundary to match the number of boundary nodes.By incorporating supplementary nodes,the resulting nonlinear algebraic equations can effectively satisfy the governing equation and boundary conditions of the EFK equation.To demonstrate the efficacy of our approach,we present three numerical examples showcasing its performance in solving this nonlinear problem.

Finite Difference-Peridynamic Differential Operator for Solving Transient Heat Conduction Problems

Transient heat conduction problems widely exist in engineering.In previous work on the peridynamic differential operator(PDDO)method for solving such problems,both time and spatial derivatives were discretized using the PDDO method,resulting in increased complexity and programming difficulty.In this work,the forward difference formula,the backward difference formula,and the centered difference formula are used to discretize the time derivative,while the PDDO method is used to discretize the spatial derivative.Three new schemes for solving transient heat conduction equations have been developed,namely,the forward-in-time and PDDO in space(FT-PDDO)scheme,the backward-in-time and PDDO in space(BT-PDDO)scheme,and the central-in-time and PDDO in space(CT-PDDO)scheme.The stability and convergence of these schemes are analyzed using the Fourier method and Taylor’s theorem.Results show that the FT-PDDO scheme is conditionally stable,whereas the BT-PDDO and CT-PDDO schemes are unconditionally stable.The stability conditions for the FT-PDDO scheme are less stringent than those of the explicit finite element method and explicit finite difference method.The convergence rate in space for these three methods is two.These constructed schemes are applied to solve one-dimensional and two-dimensional transient heat conduction problems.The accuracy and validity of the schemes are verified by comparison with analytical solutions.

High-Order Bound-Preserving Finite Difference Methods for Multispecies and Multireaction Detonations

In this paper,we apply high-order finite difference(FD)schemes for multispecies and multireaction detonations(MMD).In MMD,the density and pressure are positive and the mass fraction of the ith species in the chemical reaction,say zi,is between 0 and 1,withΣz_(i)=1.Due to the lack of maximum-principle,most of the previous bound-preserving technique cannot be applied directly.To preserve those bounds,we will use the positivity-preserving technique to all the zi'is and enforceΣz_(i)=1 by constructing conservative schemes,thanks to conservative time integrations and consistent numerical fluxes in the system.Moreover,detonation is an extreme singular mode of flame propagation in premixed gas,and the model contains a significant stiff source.It is well known that for hyperbolic equations with stiff source,the transition points in the numerical approximations near the shocks may trigger spurious shock speed,leading to wrong shock position.Intuitively,the high-order weighted essentially non-oscillatory(WENO)scheme,which can suppress oscillations near the discontinuities,would be a good choice for spatial discretization.However,with the nonlinear weights,the numerical fluxes are no longer“consistent”,leading to nonconservative numerical schemes and the bound-preserving technique does not work.Numerical experiments demonstrate that,without further numerical techniques such as subcell resolutions,the conservative FD method with linear weights can yield better numerical approximations than the nonconservative WENO scheme.

Efficient Finite Difference/Spectral Method for the Time Fractional Ito Equation Using Fast Fourier Transform Technic

A finite difference/spectral scheme is proposed for the time fractional Ito equation.The mass conservation and stability of the numerical solution are deduced by the energy method in the L^(2)norm form.To reduce the computation costs,the fast Fourier transform technic is applied to a pair of equivalent coupled differential equations.The effectiveness of the proposed algorithm is verified by the first numerical example.The mass conservation property and stability statement are confirmed by two other numerical examples.

Numerical Study by Imposing the Finite Difference Method for Unsteady Casson Fluid Flow with Heat Flux

This article presents an investigation into the flow and heat transfer characteristics of an impermeable stretching sheet subjected to Magnetohydrodynamic Casson fluid. The study considers the influence of slip velocity, thermal radiation conditions, and heat flux. The investigation is conducted employing a robust numerical method that accounts for the impact of thermal radiation. This category of fluid is apt for characterizing the movement of blood within an industrial artery, where the flow can be regulated by a material designed to manage it. The resolution of the ensuing system of ordinary differential equations (ODEs), representing the described problem, is accomplished through the application of the finite difference method. The examination of flow and heat transfer characteristics, including aspects such as unsteadiness, radiation parameter, slip velocity, Casson parameter, and Prandtl number, is explored and visually presented through tables and graphs to illustrate their impact. On the stretching sheet, calculations, and descriptions of the local skin-friction coefficient and the local Nusselt number are conducted. In conclusion, the findings indicate that the proposed method serves as a straightforward and efficient tool for exploring the solutions of fluid models of this kind.

Investigation of three-pulse photon echo in thick crystal using finite-difference time-domain method

This paper investigates the phenomenon of three-pulse photon echo in thick rare-earth ions doped crystal whose thickness is far larger than 0.002 cm which is adopted in previous works.The influence of thickness on the three-pulse photon echo＇s amplitude and efficiency is analyzed with the Maxwell-Bloch equations solved by finite-difference timedomain method.We demonstrate that the amplitude of three-pulse echo will increase with the increasing of thickness and the optimum thickness to generate three-pulse photon echo is 0.3 cm for Tm^（3＋）：YAG when the attenuation of the input pulse is taken into account.Meanwhile,we find the expression 0.09 exp（α＇L）,which is previously employed to describe the relationship between echo＇s efficiency and thickness,should be modified as 1.3 · 0.09 exp（2.4 ·α＇L） with the propagation of echo considered.

Optical simulation of in-plane-switching blue phase liquid crystal display using the finite-difference time-domain method

The finite-difference time-domain method is used to simulate the optical characteristics of an in-plane switching blue phase liquid crystal display.Compared with the matrix optic methods and the refractive method,the finite-difference timedomain method,which is used to directly solve Maxwell＇s equations,can consider the lateral variation of the refractive index and obtain an accurate convergence effect.The simulation results show that e-rays and o-rays bend in different directions when the in-plane switching blue phase liquid crystal display is driven by the operating voltage.The finitedifference time-domain method should be used when the distribution of the liquid crystal in the liquid crystal display has a large lateral change.

USE OF FINITE-DIFFERENCE TIME-DOMAIN METHOD FOR CALCULATING EM ABSORPTION IN LOSSY DIELECTRIC SCATTERER

The problem for calculating EM energy absorption by lossy dielectric scatterer ir-radiated by plane wave are discussed.The factors affecting the accuracy of computation arediscussed.The calculated results of EM energy absorption and its distribution in homogeneousand layered homogenous lossy dielectric spheres are presented,and a comparison of these resultswith analytical solution is given.The calculation is carried out for dielectric cylinder on conduct-ing ground as well,and the results are compared with the image theory.All the computationsshew that the finite-difference time-domain method can give satisfactory results.

An efficient locally one-dimensional finite-difference time-domain method based on the conformal scheme

An efficient conformal locally one-dimensional finite-difference time-domain（LOD-CFDTD） method is presented for solving two-dimensional（2D） electromagnetic（EM） scattering problems. The formulation for the 2D transverse-electric（TE） case is presented and its stability property and numerical dispersion relationship are theoretically investigated. It is shown that the introduction of irregular grids will not damage the numerical stability. Instead of the staircasing approximation, the conformal scheme is only employed to model the curve boundaries, whereas the standard Yee grids are used for the remaining regions. As the irregular grids account for a very small percentage of the total space grids, the conformal scheme has little effect on the numerical dispersion. Moreover, the proposed method, which requires fewer arithmetic operations than the alternating-direction-implicit（ADI） CFDTD method, leads to a further reduction of the CPU time. With the total-field/scattered-field（TF/SF） boundary and the perfectly matched layer（PML）, the radar cross section（RCS） of two2 D structures is calculated. The numerical examples verify the accuracy and efficiency of the proposed method.

Improved finite difference method for pressure distribution of aerostatic bearing

An improved finite difference method （FDM）is described to solve existing problems such as low efficiency and poor convergence performance in the traditional method adopted to derive the pressure distribution of aerostatic bearings. A detailed theoretical analysis of the pressure distribution of the orifice-compensated aerostatic journal bearing is presented. The nonlinear dimensionless Reynolds equation of the aerostatic journal bearing is solved by the finite difference method. Based on the principle of flow equilibrium, a new iterative algorithm named the variable step size successive approximation method is presented to adjust the pressure at the orifice in the iterative process and enhance the efficiency and convergence performance of the algorithm. A general program is developed to analyze the pressure distribution of the aerostatic journal bearing by Matlab tool. The results show that the improved finite difference method is highly effective, reliable, stable, and convergent. Even when very thin gas film thicknesses （less than 2 Win）are considered, the improved calculation method still yields a result and converges fast.

An explicit finite element－finite difference method for analyzing the effect of visco－elastic local topography on the earthquake motion

An explicit finite element-finite difference method for analyzing the effects of two-dimensional visco-elastic localtopography on earthquake ground motion is prOPosed in this paper. In the method, at first, the finite elementdiscrete model is formed by using the artificial boundary and finite element method, and the dynamic equationsof local nodes in the discrete model are obtained according to the theory of the special finite element method similar to the finite difference method, and then the explicit step-by-step integration formulas are presented by usingthe explicit difference method for solving the visco-elastic dynamic equation and Generalized Multi-transmittingBoundary. The method has the advantages of saving computing time and computer memory space, and it is suitable for any case of topography and has high computing accuracy and good computing stability.

COMPACT FINITE DIFFERENCE-FOURIER SPECTRAL METHOD FOR THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported.

A method of solving the stiffness problem in Biot＇s poroelastic equations using a staggered high-order finite-difference

In modelling elastic wave propagation in a porous medium, when the ratio between the fluid viscosity and the medium permeability is comparatively large, the stiffness problem of Blot＇s poroelastic equations will be encountered. In the paper, a partition method is developed to solve the stiffness problem with a staggered high-order finite-difference. The method splits the Biot equations into two systems. One is stiff, and solved analytically, the other is nonstiff, and solved numerically by using a high-order staggered-grid finite-difference scheme. The time step is determined by the staggered finite-difference algorithm in solving the nonstiff equations, thus a coarse time step may be employed. Therefore, the computation efficiency and computational stability are improved greatly. Also a perfect by matched layer technology is used in the split method as absorbing boundary conditions. The numerical results are compared with the analytical results and those obtained from the conventional staggered-grid finite-difference method in a homogeneous model, respectively. They are in good agreement with each other. Finally, a slightly more complex model is investigated and compared with related equivalent model to illustrate the good performance of the staggered-grid finite-difference scheme in the partition method.

Implicit finite difference method for fractional percolation equation with Dirichlet and fractional boundary conditions

An implicit finite difference method is developed for a one-dimensional frac- tional percolation equation （FPE） with the Dirichlet and fractional boundary conditions. The stability and convergence are discussed for two special cases, i.e., a continued seep- age flow with a monotone percolation coefficient and a seepage flow with the fractional Neumann boundary condition. The accuracy and efficiency of the method are checked with two numerical examples.

Numerical storm surge model with higher order finite difference method of lines for the coast of Bangladesh

In this study, the method of lines (MOLs) with higher order central difference approximation method coupled with the classical fourth order Runge-Kutta (RK(4,4)) method is used in solving shallow water equations (SWEs) in Cartesian coordinates to foresee water levels associated with a storm accurately along the coast of Bangladesh. In doing so, the partial derivatives of the SWEs with respect to the space variables were discretized with 5-point central difference, as a test case, to obtain a system of ordinary differential equations with time as an independent variable for every spatial grid point, which with initial conditions were solved by the RK(4,4) method. The complex land-sea interface and bottom topographic details were incorporated closely using nested schemes. The coastal and island boundaries were rectangularized through proper stair step representation, and the storing positions of the scalar and momentum variables were specified according to the rules of structured C-grid. A stable tidal regime was made over the model domain considering the effect of the major tidal constituent, M2 along the southern open boundary of the outermost parent scheme. The Meghna River fresh water discharge was taken into account for the inner most child scheme. To take into account the dynamic interaction of tide and surge, the generated tidal regime was introduced as the initial state of the sea, and the surge was then made to come over it through computer simulation. Numerical experiments were performed with the cyclone April 1991 to simulate water levels due to tide, surge, and their interaction at different stations along the coast of Bangladesh. Our computed results were found to compare reasonable well with the limited observed data obtained from Bangladesh Inland Water Transport Authority (BIWTA) and were found to be better in comparison with the results obtained through the regular finite difference method and the 3-point central difference MOLs coupled with the RK(4,4) method with regard to the root mean square error values.

Finite difference method for dynamic response analysis of anchorage system

Based on some assumptions, the dynamic analysis model of anchorage system is established. The dynamic governing equation is expressed as finite difference format and programmed by using MATLAB language. Compared with theoretical method, the finite difference method has been verified to be feasible by a case study. It is found that under seismic loading, the dynamic response of anchorage system is synchronously fluctuated with the seismic vibration. The change of displacement amplitude of material points is slight, and comparatively speaking, the displacement amplitude of the outside point is a little larger than that of the inside point, which shows amplification effect of surface. While the axial force amplitude transforms considerably from the inside to the outside. It increases first and reaches the peak value in the intersection between the anchoring section and free section, then decreases slowly in the free section. When considering damping effect of anchorage system, the finite difference method can reflect the time attenuation characteristic better, and the calculating result would be safer and more reasonable than the dynamic steady-state theoretical method. What is more, the finite difference method can be applied to the dynamic response analysis of harmonic and seismic random vibration for all kinds of anchor, and hence has a broad application prospect.

Using finite difference method to simulate casting thermal stress

Thermal stress simulation can provide a scientific reference to eliminate defects such as crack,residual stress centralization and deformation etc.,caused by thermal stress during casting solidification.To study the thermal stress distribution during casting process,a unilateral thermal-stress coupling model was employed to simulate 3D casting stress using Finite Difference Method(FDM),namely all the traditional thermal-elastic-plastic equations are numerically and differentially discrete.A FDM/FDM numerical simulation system was developed to analyze temperature and stress fields during casting solidification process.Two practical verifications were carried out,and the results from simulation basically coincided with practical cases.The results indicated that the FDM/FDM stress simulation system can be used to simulate the formation of residual stress,and to predict the occurrence of hot tearing.Because heat transfer and stress analysis are all based on FDM,they can use the same FD model,which can avoid the matching process between different models,and hence reduce temperature-load transferring errors.This approach makes the simulation of fluid flow,heat transfer and stress analysis unify into one single model.

Numerical simulation of standing wave with 3D predictor-corrector finite difference method for potential flow equations

A three-dimensional （3D） predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically.

Applying Finite Difference Method to Simulate the Performance of a Perforated Breakwater Under Regular Waves

Using a discretized finite difference method, a numerical model was developed to study the interaction of regular waves with a perforated breakwater. Considering a non-viscous, non-rotational fluid, the governing equations of Laplacian velocity potential were developed, and specific conditions for every single boundary were defined. The final developed model was evaluated based on an existing experimental result. The evaluated model was used to simulate the condition for various wave periods from 0.6 to 2 s. The reflection coefficient and transmission coefficient of waves were examined with different breakwater porosities, wave steepnesses, and angular frequencies. The results show that the developed model can suitably present the effect of the structural and hydraulic parameters on the reflection and transmission coefficients. It was also found that with the increase in wave steepness, the reflection coefficient increased logarithmically, while the transmission coefficient decreased logarithmically.

Finite difference time domain method forward simulation of complex geoelectricity ground penetrating radar model

The ground penetrating radar(GPR) forward simulation all aims at the singular and regular models, such as sandwich model, round cavity, square cavity, and so on, which are comparably simple. But as to the forward of curl interface underground or “v” figure complex model, it is difficult to realize. So it is important to forward the complex geoelectricity model. This paper takes two Maxwell’s vorticity equations as departure point, makes use of the principles of Yee’s space grid model theory and the basic principle finite difference time domain method, and deduces a GPR forward system of equation of two dimensional spaces. The Mur super absorbed boundary condition is adopted to solve the super strong reflection on the interceptive boundary when there is the forward simulation. And a self-made program is used to process forward simulation to two typical geoelectricity model.