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.

New Finite Difference Mapped WENO Schemes with Increasingly High Order of Accuracy

In this paper,a new type of finite difference mapped weighted essentially non-oscillatory(MWENO)schemes with unequal-sized stencils,such as the seventh-order and ninthorder versions,is constructed for solving hyperbolic conservation laws.For the purpose of designing increasingly high-order finite difference WENO schemes,the equal-sized stencils are becoming more and more wider.The more we use wider candidate stencils,the bigger the probability of discontinuities lies in all stencils.Therefore,one innovation of these new WENO schemes is to introduce a new splitting stencil methodology to divide some fourpoint or five-point stencils into several smaller three-point stencils.By the usage of this new methodology in high-order spatial reconstruction procedure,we get different degree polynomials defined on these unequal-sized stencils,and calculate the linear weights,smoothness indicators,and nonlinear weights as specified in Jiang and Shu(J.Comput.Phys.126:202228,1996).Since the difference between the nonlinear weights and the linear weights is too big to keep the optimal order of accuracy in smooth regions,another crucial innovation is to present the new mapping functions which are used to obtain the mapped nonlinear weights and decrease the difference quantity between the mapped nonlinear weights and the linear weights,so as to keep the optimal order of accuracy in smooth regions.These new MWENO schemes can also be applied to compute some extreme examples,such as the double rarefaction wave problem,the Sedov blast wave problem,and the Leblanc problem with a normal CFL number.Extensive numerical results are provided to illustrate the good performance of the new finite difference MWENO schemes.

Equivalent low-order angular flux nonlinear finite difference equation of MOC transport calculation

The key issue in accelerating method of characteristics(MOC)transport calculations is in obtaining a completely equivalent low-order neutron transport or diffusion equation.Herein,an equivalent low-order angular flux nonlinear finite difference equation is proposed for MOC transport calculations.This method comprises three essential features:(1)the even parity discrete ordinates method is used to build a low-order angular flux nonlinear finite difference equation,and different boundary condition treatments are proposed;(2)two new defined factors,i.e.,the even parity discontinuity factor and odd parity discontinuity factor,are strictly defined to achieve equivalence between the low-order angular flux nonlinear finite difference method and MOC transport calculation;(3)the energy group and angle are decoupled to construct a symmetric linear system that is much easier to solve.The equivalence of this low-order angular flux nonlinear finite difference equation is analyzed for two-dimensional(2D)pin,2D assembly,and 2D C5G7 benchmark problems.Numerical results demonstrate that a low-order angular flux nonlinear finite difference equation that is completely equivalent to the pin-resolved transport equation is established.

A new finite difference scheme for a dissipative cubic nonlinear Schrdinger equation

This paper considers the one-dimensional dissipative cubic nonlinear SchrSdinger equation with zero Dirichlet boundary conditions on a bounded domain. The equation is discretized in time by a linear implicit three-level central difference scheme, which has analogous discrete conservation laws of charge and energy. The convergence with two orders and the stability of the scheme are analysed using a priori estimates. Numerical tests show that the three-level scheme is more efficient.

A TVD-WAF-based hybrid finite volume and finite difference scheme for nonlinearly dispersive wave equations

A total variation diminishing-weighted average flux （TVD-WAF）-based hybrid numerical scheme for the enhanced version of nonlinearly dispersive Boussinesq-type equations was developed. The one-dimensional governing equations were rewritten in the conservative form and then discretized on a uniform grid. The finite volume method was used to discretize the flux term while the remaining terms were approximated with the finite difference method. The second-order TVD-WAF method was employed in conjunction with the Harten-Lax-van Leer （HLL） Riemann solver to calculate the numerical flux, and the variables at the cell interface for the local Riemann problem were reconstructed via the fourth- order monotone upstream-centered scheme for conservation laws （MUSCL）. The time marching scheme based on the third-order TVD Runge- Kutta method was used to obtain numerical solutions. The model was validated through a series of numerical tests, in which wave breaking and a moving shoreline were treated. The good agreement between the computed results, documented analytical solutions, and experimental data demonstrates the correct discretization of the governing equations and high accuracy of the proposed scheme, and also conforms the advantages of the proposed shock-capturing scheme for the enhanced version of the Boussinesq model, including the convenience in the treatment of wave breaking and moving shorelines and without the need for a numerical filter.

Modified characteristic finite difference fractional step method for moving boundary value problem of nonlinear percolation system

A fractional step scheme with modified characteristic finite differences run- ning in a parallel arithmetic is presented to simulate a nonlinear percolation system of multilayer dynamics of fluids in a porous medium with moving boundary values. With the help of theoretical techniques including the change of regions, piecewise threefold quadratic interpolation, calculus of variations, multiplicative commutation rule of differ- ence operators, multiplicative commutation rule of difference operators, decomposition of high order difference operators, induction hypothesis, and prior estimates, an optimal order in 12 norm is displayed to complete the convergence analysis of the numerical algo- rithm. Some numerical results arising in the actual simulation of migration-accumulation of oil resources by this method are listed in the last section.

Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation

In this paper,two fourth-order compact finite difference schemes are derived to solve the nonlinear fourth-order wave equation which can be viewed as a generalized model from the nonlinear beam equation.Differing from the existing compact finite difference schemes which preserve the total energy in a recursive sense,the new schemes are proved to per-fectly preserve the total energy in the discrete sense.By using the standard energy method and the cut-off function technique,the optimal error estimates of the numerical solutions are established,and the convergence rates are of O(h^(4)+τ^(2))with mesh-size h and time-step τ.In order to improve the computational efficiency,an iterative algorithm is proposed as the outer solver and the double sweep method for pentadiagonal linear algebraic equations is introduced as the inner solver to solve the nonlinear difference schemes at each time step.The convergence of the iterative algorithm is also rigorously analyzed.Several numerical results are carried out to test the error estimates and conservative properties.

A Finite Difference Scheme for Blow-Up Solutions of Nonlinear Wave Equations

We consider a finite difference scheme for a nonlinear wave equation, whose solutions may lose their smoothness in finite time, i.e., blow up in finite time. In order to numerically reproduce blow-up solutions, we propose a rule for a time-stepping,which is a variant of what was successfully used in the case of nonlinear parabolic equations. A numerical blow-up time is defined and is proved to converge, under a certain hypothesis, to the real blow-up time as the grid size tends to zero.

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.

A Finite-Difference Approach to the Time-Dependent Mild-Slope Equation

A finite-difference approach is used to develop a time-dependent mild-slope equation incorporating the effects of bottom dissipation and nonlinearity. The Enler predietor-corrector method and the three-point finite-difference method with varying spatial steps are adopted to discretize the time derivatives and the two-dimensional horizontal ones, respectively, thus leading both the time and spatial derivatives to the second-order accuracy. The boundary conditions for the present model are treated on the basis of the general conditions for open and fixed boundaries with an arbitrary reflection coefficient and phase shift. Both the linear and nonlinear versions of the numerical model are applied to the wave propagation and transformation over an elliptic shoal on a sloping beach, respectively, and the linear version is applied to the simulation of wave propagation in a fully open rectangular harbor. From comparison of numerical results with theoretical or experimental ones, it is found that they are in reasonable agreement.

Finite Difference Method of First Boundary Problem of a Generalized Diffusion Model in Population

The solution of nonlinear parabolic equation arising from population dynamics with boundary and initial value are established by the finite difference method,as well as it denotes the unique generalized global solution.

ECONOMICAL DIFFERENCE SCHEME FOR ONE MULTI-DIMENSIONAL NONLINEAR SYSTEM

The multi-dimensional system of nonlinear partial differential equations is considered. In two-dimensional case, this system describes process of vein formation in higher plants. Variable directions finite difference scheme is constructed. The stability and convergence of that scheme are studied. Numerical experiments are carried out. The appropriate graphical illustrations and tables are given.

A Compact Difference Scheme for Time-Space Fractional Nonlinear Diffusion-Wave Equations with Initial Singularity

In this paper,we present a linearized compact difference scheme for onedimensional time-space fractional nonlinear diffusion-wave equations with initial boundary value conditions.The initial singularity of the solution is considered,which often generates a singular source and increases the difficulty of numerically solving the equation.The Crank-Nicolson technique,combined with the midpoint formula and the second-order convolution quadrature formula,is used for the time discretization.To increase the spatial accuracy,a fourth-order compact difference approximation,which is constructed by two compact difference operators,is adopted for spatial discretization.Then,the unconditional stability and convergence of the proposed scheme are strictly established with superlinear convergence accuracy in time and fourth-order accuracy in space.Finally,numerical experiments are given to support our theoretical results.

Time-Domain Nonlinear Wave-Current Interaction with A Steep Wave Riser Considering Internal Flow Effect

The nonlinear dynamic response induced by the wave-current interaction on a deepwater steep wave riser(SWR)is numerically investigated based on a three-dimensional(3 D)time-domain finite element method(FEM).The governing equation considering internal flow is established in the global coordinate system.The whole SWR consists of three segments:the decline segment,buoyancy segment and hang-off segment,in which the buoyancy segment is wrapped by several buoyancy modules in the middle section,leading to the arch bend and sag bend.A Newmark-β iterative scheme is adopted for the accurate analysis to solve the governing equation and update the dynamic response at each time step.The proposed method is verified through the published results for the dynamic response of steel catenary riser(SCR)and static configuration of steel lazy wave riser(SLWR).Simulations are executed to study the influence of wave height,current velocity/direction,internal flow density/velocity and top-end pressure on the tension,configuration and bending moment of the SWR.The results indicate that the influence of the current on the configuration and mechanical behavior of the SWR is greater than that of the wave,especially in the middle section.With increasing current velocity,the suspending height of the middle section drops,meanwhile,its bending moment decreases accordingly,but the tension increases significantly.For a fixed external load,the increasing internal flow density induces the amplification of the tension at the hang-off segment and the mitigation at the decline segment,while the opposite trend occurs at the bending moment.

Error Estimate of a New Conservative Finite Difference Scheme for the Klein-Gordon-Dirac System

In this paper,we derive and analyze a conservative Crank-Nicolson-type finite difference scheme for the Klein-Gordon-Dirac(KGD)system.Differing from the derivation of the existing numerical methods given in literature where the numerical schemes are proposed by directly discretizing the KGD system,we translate the KGD equations into an equivalent system by introducing an auxiliary function,then derive a nonlinear Crank-Nicolson-type finite difference scheme for solving the equivalent system.The scheme perfectly inherits the mass and energy conservative properties possessed by the KGD,while the energy preserved by the existing conservative numerical schemes expressed by two-level’s solution at each time step.By using energy method together with the‘cut-off’function technique,we establish the optimal error estimate of the numerical solution,and the convergence rate is O(τ^(2)+h^(2))in l∞-norm with time stepτand mesh size h.Numerical experiments are carried out to support our theoretical conclusions.

Application of the Hybrid Differential Transform Method to the Nonlinear Equations

In this paper, a hybrid method is introduced briefly to predict the behavior of the non-linear partial differential equations. The method is hybrid in the sense that different numerical methods, differential transform and finite differences, are used in different subdomains. Our aim of this approach is to combine the flexibility of differential transform and the efficiency of finite differences. An explicit hybrid method for the transient response of inhomogeneous nonlinear partial differential equations is presented;applying finite difference scheme on the fixed grid size is used to approximate the space discretisation, whereas the differential transform method is used for time operator. Comparison of the efficiency of the different approaches is a very important aspect of this study. In our test cases, the hybrid approach is faster than the corresponding highly optimized finite difference method in two dimensional computations. We compared our hybrid approach’s results with the exact and/or numerical solutions of PDE which obtained from Adomian Decomposition Method. Results show that the hybrid approach may be an important tool to reduce the execution time and memory requirements for large scale computations and get remarkable results in predicting the solutions of nonlinear initial value problems.

The finite difference method for the three-dimensional nonlinear coupled system of dynamics of fluids in porous media

For the three-dimensional coupled system of multilayer dynamics of fluids in porous media, the second-order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward. Some techniques, such as calculus of variations, energy method,multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates are adopted. Optimal order estimates in l2 norm are derived to determine the error in the second-order approximate solution. These methods have already been applied to the numerical simulation of migration-accumulation of oil resources.

LONG-TIME BEHAVIOR OF FINITE DIFFERENCE SOLUTIONS OF A NONLINEAR SCHRODINGER EQUATION WITH WEAKLY DAMPED

A weakly demped Schrodinger equation possessing a global attractor are considered. The dynamical properties of a class of finite difference scheme axe analysed. The exsitence of global attractor is proved for the discrete system. The stability of the difference scheme and the error estimate of the difference solution are obtained in the autonomous system case. Finally, long-time stability and convergence of the class of finite difference scheme also are analysed in the nonautonomous system case.

全光通信网络非线性突变频率干扰检测算法

【目的】针对全光通信网络非线性突变频率干扰问题进行分析时,主要依托于估计信道协方差直接展开干扰检测,并未展开网络通信信号变换处理,使得检测结果的F1⁃score值较低,为此提出了基于时域有限差分的全光通信网络非线性突变频率干扰检测算法。【方法】结合数据包抓包和镜像两种方式采集全光通信网络流量数据,并进行清理、转换和规约处理。依托于时域有限差分工作原理,在时域和频域空间内描述信号的时宽和带宽,应用导数与傅里叶变换算法对实时采集信号进行处理,利用变换后的信号分析非线性突变频率干扰情况,再针对变换后的信号进行检测分析。在时间⁃频率联合特征分析方法辅助下,提取干扰信号的时域、频域特征,依托于反向传播算法和最小化损失函数,简化非线性突变频率干扰检测过程,采用特征距离函数替换网络损失函数,并将其输入基于孪生网络的干扰识别模型中,得出非线性突变频率干扰检测结果。【结果】在不同噪声条件下,所提算法非线性突变频率干扰检测结果的F1⁃score值保持在0.95以上,检测时间低于40 ms。【结论】应用时域有限差分方法的新型检测方法,能够更准确地反映当前通信网络的干扰情况,保证通信网络正常运行,更好地满足了全光通信网络干扰检测要求。

Nonlinear Schrodinger equation with a Dirac delta potential:finite difference method

The nonlinear Schr?dinger equation with a Dirac delta potential is considered in this paper.It is noted that the equation can be transformed into an equation with a drift-admitting jump.Then following the procedure proposed in Chen and Deng(2018 Phys.Rev.E 98033302),a new second-order finite difference scheme is developed,which is justified by numerical examples.