Fourth Order Symmetric Finite Difference Schemes for the Acoustic Wave Equation

We present an explicit, symmetric finite difference scheme for the acoustic wave equation on a rectangle with Neumann and/or Dirichlet boundary conditions. The scheme is fourth order accurate both in time and space. It is obtained by mass lumping of a finite element scheme. The accuracy and the difference approximations at the boundary are analyzed in terms of local and global errors. AMS subject classification (2000) 65M10     

Finite Difference Method for Reaction-Diffusion Equation with Nonlocal Boundary Conditions

In this paper, we present a numerical approach to a class of nonlinear reaction-diffusion equations with nonlocal Robin type boundary conditions by finite difference methods. A second-order accurate difference scheme is derived by the method of reduction of order. Moreover, we prove that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete L2-norm. A simple numerical example is given to illustrate the efficiency of the proposed method.     

Finite Difference Method for Reaction-Diffusion Nonlocal Boundary Conditions Equation with

In this paper, we present a numerical approach to a class of nonlinear reactiondiffusion equations with nonlocal Robin type boundary conditions by finite difference methods. A second-order accurate difference scheme is derived by the method of reduction of order. Moreover, we prove that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete L2-norm. A simple numerical example is given to illustrate the efficiency of the proposed method.     

Compact and Monotone Difference Schemes for the Generalized Fisher Equation

Differential Equations - For the generalized Fisher equation with nonlinear convection, monotone and compact difference schemes of $$4+1$$ and $$4+2 $$ approximation orders are constructed and...     

色散方程的四点显式差分格式

本文对色散方程u_t=au>_xxx构造了一类高稳定性的、在中间层涉及四个网格点的三层显式差分格式,其局部截断误差为O(τ+h),其稳定条件为|R|=|α|τ/h3≤0.25至|R|≤10,它们较大地改善了同类格式的稳定条件|R|≤0.25[1].     

Convergence of Discrete Green's Functions for Finite Difference Schemes

AMS(MOS): 65L10

The convergence of the discrete Green's function g_h is studied for finite difference schemes approximating m-th order linear two-point boundary value problems. Schemes of noncompact form and in part of the paper also nonuniform grids are admitted. Sharp convergence results are obtained for the difference quotients of g_h up to order m-1.     

非线性边界条件下二阶奇异差分方程的正解

本文我们考虑如下二阶奇异差分边值问题\begin{equation*}\begin{cases}-\Delta^{2} u(t-1)=\lambda g(t)f(u) ,\ t\in [1,T]_\mathbb{Z},\\u(0)=0,\\ \Delta u(T)+c(u(T+1))u(T+1)=0,\end{cases}\end{equation*}正解的存在性. 其中, $\lambda>0$, $f:(0,\infty)\rightarrow \mathbb{R}$ 是连续的,且允许在~$0$ 处奇异.通过引入一个新的全连续算子, 我们建立正解的存在性.     

对流扩散方程的高效稳定差分格式

基于二阶修正Dennis格式 ,提出了采用时间相关法求解定常对流扩散方程的一种具有节省内存空间和提高定常解收敛速度的有理式型优化半隐和松驰半隐紧致格式 .本文建立的差分格式具有运算量小、无网格雷诺数限制的优点 ,是无条件稳定和无条件单调的。通过对非线性Burgers方程进行的数值计算结果表明 ,文中构造的有理式型优化半隐和松驰半隐紧致格式适合于非线性问题计算 ,且保持了无条件稳定和无条件单调的特性 ,尤其能使定常解收敛速度加快 ,精度提高 .     

解对流方程的加耗散项的差分格式

解对流方程的大多数常见的显式差分格式 ,其稳定性条件是苛刻的 .这一困难可由在常规的显式差分格式中引入耗散项而得到克服 .基于此 ,我们导出一类新的无条件稳定的两层的半显式差分格式及若干具有高稳定性的显式格式 .它们包含了若干已知的具有高稳定性的显式格式 .     

KdV方程的一类本性并行差分格式

对三阶KdV方程给出了—组非对称的差分公式,并用这些差分公式和对称的Crank-Nicolson型公式构造了一类具有本性并行的交替差分格式．证明了格式的线性绝对稳定性．对—个孤立波解、二个孤立波解和三个孤立波解的情况分别进行了数值试验,并对—个孤立波解的数值解的收敛阶和精确性进行了试验和比较．     

二阶半线性中立型差分方程的振动性准则

考虑二阶半线性中立型差分方程给出了方程(1)的解的振动性的充分条件．所有结果推广和改进了关于中立和时滞差分方程已有结果．     

Finite Difference Methods for the Heat Equation with a Nonlocal Boundary Condition

We consider the numerical solution by finite difference methods of the heat equation in one space dimension, with a nonlocal integral boundary condition, resulting from the truncation to a finite interval of the problem on a semi-infinite interval. We first analyze the forward Euler method, and then the $θ$-method for $0 < θ ≤ 1$, in both cases in maximum-norm, showing $O(h^2 + k)$ error bounds, where $h$ is the mesh-width and $k$ the time step. We then give an alternative analysis for the case $θ = 1/2$, the Crank-Nicolson method, using energy arguments, yielding a $O(h^2$ + $k^{3/2}$) error bound. Special attention is given the approximation of the boundary integral operator. Our results are illustrated by numerical examples.     

一类非线性Schr(o)dinger方程的守恒差分法与Fourier谱方法

考察了一类带导数项的非线性Schrodinger方程的周期边值问题，提出了一种守恒的差分格式，在空间方向上采用Fourier谱方法，证明了格式的稳定性和收敛性．数值试验得到了与理论分析一致的结果．     

Efficient Simulation of Wave Propagation with Implicit Finite Difference Schemes

Finite difference method is an important methodology in the approximation of waves. In this paper, we will study two implicit finite difference schemes for the simulation of waves. They are the weighted alternating direction implicit (ADI) scheme and the locally one-dimensional (LOD) scheme. The approximation errors, stability conditions, and dispersion relations for both schemes are investigated. Our analysis shows that the LOD implicit scheme has less dispersion error than that of the ADI scheme. Moreover, the unconditional stability for both schemes with arbitrary spatial accuracy is established for the first time. In order to improve computational efficiency, numerical algorithms based on message passing interface (MPI) are implemented. Numerical examples of wave propagation in a three-layer model and a standard complex model are presented. Our analysis and comparisons show that both ADI and LOD schemes are able to efficiently and accurately simulate wave propagation in complex media.     

Travelling Wave Solutions for Generalized Fisher Equation

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ［１，２］，ＡｒｏｎｓｏｎａｎｄＷｅｉｎｂｅｒｇｅｒｈａｖｅｓｔｕｄｉｅｄｓｙｓｔｅｍａｔｉｃｌｙｔｈｅｓｃａｌａｒｎｏｎｌｉｎｅａｒｄｉｆｆｕ－ｓｉｏｎｅｑｕａｔｉｏｎｉｎｏｎｅｓｐａｃｅｖａｒｉａｂｌｅｕｔ＝ｕｘｘ＋φ（ｕ），（１．１）ｗｈｅｒｅｕ＝ｕ（ｘ，ｔ）ａｎｄφ（ｕ）ｉｓａｎｏｎｌｉｎｅａｒｆｕｎｃｔｉｏｎ．Ｅｑｕａｔｉｏｎ（１．１）ａｒｉｓｅｓｉｎｓｅｖｅｒａｌａｐｐｌｉｃａ－ｔｉｏｎｓ；Ｓｅｅ［１，２］ａｎｄ［３］ｆｏｒｉｎｆｏｒｍａｔｉｏｎａ…     

The evolution of travelling waves in generalized Fisher equations via matched asymptotic expansions: Exponential corrections

In this paper we address an initial boundary value problem for a generalized Fisher equation. In particular we develop the matched asymptotic theory presented in Leach and Needham (2000) to consider the correction terms to the asymptotic approach to the wave-front of speed v = v*( 2) as t (time) . We establish the precise form of these corrections, and demonstrate that the rate of approach to the wave-front is algebraic in t when v* = 2 (there being two cases), but exponential in t when v* > 2.     

Difference Equation for $N$-Body Type Problem

In this paper, the difference equation for $N$-body type problem is established, which can be used to find the generalized solutions by computing the critical points numerically. And its validity is testified by an example from Newtonian Three-body problem with unequal masses.     

New Conservative Schemes for Regularized Long Wave Equation

In this paper, two finite difference schemes are presented for initial-boundary value problems of Regularized Long-Wave(RLW) equation. They all have the advantages that there are discrete energies which are conserved. Convergence and stability of difference solutions with order O(h2 τ2) are proved in the energy norm. Numerical experiment results demonstrate the effectiveness of the proposed schemes.     

A Second-Order Semi-Implicit Method for the Inertial Landau-Lifshitz-Gilbert Equation

Electron spins in magnetic materials have preferred orientations collectively and generate the macroscopic magnetization. Its dynamics spans over a wide range of timescales from femtosecond to picosecond, and then to nanosecond. The Landau-Lifshitz-Gilbert (LLG) equation has been widely used in micromagnetics simulations over decades. Recent theoretical and experimental advances have shown that the inertia of magnetization emerges at sub-picosecond timescales and contributes significantly to the ultrafast magnetization dynamics, which cannot be captured intrinsically by the LLG equation. Therefore, as a generalization, the inertial LLG (iLLG) equation is proposed to model the ultrafast magnetization dynamics. Mathematically, the LLG equation is a nonlinear system of parabolic type with (possible) degeneracy. However, the iLLG equation is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy, and exhibits more complicated structures. It behaves as a hyperbolic system at sub-picosecond timescales, while behaves as a parabolic system at larger timescales spanning from picosecond to nanosecond. Such hybrid behaviors impose additional difficulties on designing efficient numerical methods for the iLLG equation. In this work, we propose a second-order semi-implicit scheme to solve the iLLG equation. The second-order temporal derivative of magnetization is approximated by the standard centered difference scheme, and the first-order temporal derivative is approximated by the midpoint scheme involving three time steps. The nonlinear terms are treated semi-implicitly using one-sided interpolation with second-order accuracy. At each time step, the unconditionallyunique solvability of the unsymmetric linear system is proved with detailed discussions on the condition number. Numerically, the second-order accuracy of the proposed method in both time and space is verified. At sub-picosecond timescales, the inertial effect of ferromagnetics is observed in micromagnetics simulations, in consistency with the hyperbolic property of the iLLG model; at nanosecond timescales, the results of the iLLG model are in nice agreements with those of the LLG model, in consistency with the parabolic feature of the iLLG model.