一类非线性Schr dinger方程的高精度守恒数值格式

1引言本文讨论下面非线性Schr(?)dinger方程(NLS)方程的初边值问题:i(?)u/(?)t (?)~2u/(?)x~2 2|u~2|u=0,(1) u(x_l,t)=u(x_r,t)=0,t>0,(2) u(x,0)=u_0(x),x_l≤x≤x_r,(3)其中u(x,t)是复值函数,u_0(x)为已知的复值函数,i~2=-1.该问题有着如下的电荷与能量守恒关系:     

双曲型和抛物型微分不等式解的三曲线定理及估计

本文对于系数满足条件(A)(见§3)的 Laplace 双曲型微分不等式(?)~2u/(?)_x(?)_y+a(x,y,)(?)_u/(?)_x+b(x,y)(?)~u/(?)~y+c(x,y)u≥0(c≤0)的解以及抛物型微分不等式(?)~2v/(?)x~2-(?)v/(?)t+c(x,t)v≥0(c≤0)的解,分别在 c≡0和 c(?)0的情形建立了各自的一般形式的三曲线定理;在 c(?)0,且所考虑的函数预先给定的初值-边值取非正值的情形,给出了一种建立更有效的估计的方法.此外,本文还改进了 Agmon-Nirenberg-Protter 关于 Laplace 双曲型微分不等式的一个最大值原理.     

双曲型和抛物型微分不等式解的三曲线定理及估计

本文对于系数满足条件(A)(见§3)的 Laplace 双曲型微分不等式(?)~2u/(?)_x(?)_y a(x,y,)(?)_u/(?)_x b(x,y)(?)~u/(?)~y c(x,y)u≥0(c≤0)的解以及抛物型微分不等式(?)~2v/(?)x~2-(?)v/(?)t c(x,t)v≥0(c≤0)的解,分别在 c≡0和 c(?)0的情形建立了各自的一般形式的三曲线定理;在 c(?)0,且所考虑的函数预先给定的初值-边值取非正值的情形,给出了一种建立更有效的估计的方法.此外,本文还改进了 Agmon-Nirenberg-Protter 关于 Laplace 双曲型微分不等式的一个最大值原理.     

一类退化抛物方程解的几何性质

本文讨论非线性退化抛物方程u_t=△φ(u)的Cauchy问题弱解u(x,t)的正则性与几何性质.本文证明:若正数β足够大,则曲面ψ=ψ(x,t)=[φ(u)]~β是随时间t的连续变化而漂浮于空间R~(n+1)中的n维完备黎曼流形,它与实欧氏空R~n相切于低维流形(?)H_n(t),而H_u(t)={x∈R~n:u(x,t)0);函数ψ(x,t)在经典的意义下满足另一退化抛物方程.     

带非均匀项的Sine—Gordon方程

文中得出了x-SG方程(1)的B(?)cklund变换和反散射形式。通过方程(1)的反散射解研究,我们得到了当特征问题(2.4)的位势u(x,t)(q(x,t)=-1/2 u_x(x,t))满足方程(1)时的散射量随时间的演化规律,并分别利用B(?)cklund变换和反散射方法,我们求出了方程(1)的孤子解,且它们是一致的。     

方程~2u/x~1x~1 ~2u/x~2x~2=f(u)的Bcklund变换

本文利用Wahlquist-Estabrook过程(WEP)研究了方程(?)~2u/(?)x~1(?)x~1 (?)~2u/(?)x~2(?)x~2=f(u)(这里f是任意函数)的B(?)cklund变换.我们发现该方程存在B(?)cklund变换的充分条件是d~2f/du~2=λf.我们所得到的结果的一个特殊情况就是Leibbrandt的结论.     

不可压Navier-Stokes方程迎风格式及后验误差估计

1引言不可压Navier-Stokes方程作为流体力学的基本方程,其数值计算一直是科学与工程计算关心的问题．本文考虑定常问题: -ε△u (u·▽)u ▽p = f x∈Ω,▽·u=0 x∈Ω, (1) u =0 x∈(?)Ω．这里ε=1／Re是Reynolds数的倒数,u=(u1,u2,…,ud)为待求流速场,p是待求压力场,f=(f1,f2,…,fd)是给定的体力．Ωv(?) Rd(d=2,3)是有界区域,且具有分片Lipschitz连续边界(?)Ω．     

时间周期折现Hamilton-Jacobi方程里1-周期解的存在性

主要运用PDE方法,在时间1-周期的哈密尔顿函数H(x,t,p)关于(x,t,p)连续、关于p强制且关于t,x周期、关于t线性的条件下,证明了比较定理,从而得到了时间周期折现Hamilton-Jacobi方程λu(x,t)+u_t(x,t)+H(x,t,D_xu(x,t))=0里唯一1-周期解的存在性.     

Bakry-Emery型Ricci曲率下两类抛物方程正解的梯度估计

本文考虑完备黎曼流形上,在Bakry-Emery型Ricci曲率有下界的条件下两类抛物方程?u/?t=△Vu+au log u 和(△v-?/?t)u(x,t)+p(x,t)uβ(x,t)+q(x,t)u(x,t)=0正解的梯度估计,这里α,β ∈(R),△V(·):=△+(V,▽(·)).由于引入了 △V,相应地,在...     

半线性蜕缩椭圆型方程的Dirichlet问题

设Ω(?)R~n(n≥2)是光滑有界区域.讨论如下的半线性蜕缩椭圆型方程的Dirichlet问题Lu ≡-sum from i,j=1 to n((?)/(?)x_i)(aij(x)((?)u/(?)xj)=g(x,u) (x,u),在Ω中,u=0,在(?)Ω上。(1)这里,且sum from i,j=1 to n(aijξiξj≥k sum from i=1 to n(ρ~a_i(x)ξ_i~2),(?)x∈(?),(?)ξ∈R~n,(2)     

Compact difference scheme for two-dimensional fourth-order nonlinear hyperbolic equation

High-order compact finite difference method for solving the two-dimensional fourth-order nonlinear hyperbolic equation is considered in this article. In order to design an implicit compact finite difference scheme, the fourth-order equation is written as a system of two second-order equations by introducing the second-order spatial derivative as a new variable. The second-order spatial derivatives are approximated by the compact finite difference operators to obtain a fourth-order convergence. As well as, the second-order time derivative is approximated by the central difference method. Then, existence and uniqueness of numerical solution is given. The stability and convergence of the compact finite difference scheme are proved by the energy method. Numerical results are provided to verify the accuracy and efficiency of this scheme.     

美式看跌期权定价的两种有限差分格式

基于Black-Scholes模型,采用指数拟合有限差分法与外推的指数拟合有限差分法对美式看跌期权价值进行了数值计算,对这两种数值方法及其与已往的显式、隐式、C-N等有限差分的优缺点进行了比较,并给出数值算例,通过对此算例做的一系列数值试验,验证了算法的有效性,并得到了一些在期权交易的实际操作中有用的结果.     

A 5-instant finite difference formula to find discrete time-varying generalized matrix inverses,matrix inverses,and scalar reciprocals

Finite difference schemes have been widely studied because of their fundamental role in numerical analysis. However, most finite difference formulas in the literature are not suitable for discrete time-varying problems because of intrinsic limitations and their relatively low precision. In this paper, a high-precision 1-step-ahead finite difference formula is developed. This 5-instant finite difference (5-IFD) formula is used to approximate and discretize first-order derivatives, and it helps us to compute discrete time-varying generalized matrix inverses. Furthermore, as special cases of generalized matrix inverses, time-varying matrix inversion, and scalar reciprocals are generally deemed as independent problems and studied separately, which are solved unitedly in this paper. The precision of the 5-IFD formula and the convergence behavior of the corresponding discrete-time models are derived theoretically and shown in numerical experiments. Conventional useful formulas, such as the Euler forward finite difference (EFFD) formula and the 4-instant finite difference (4-IFD) formula are also used for comparisons and to show the superiority of the 5-IFD formula.

     

Two-level finite difference scheme of improved accuracy order for time-dependent problems of mathematical physics

In the theory of finite difference schemes, the most complete results concerning the accuracy of approximate solutions are obtained for two- and three-level finite difference schemes that converge with the first and second order with respect to time. When the Cauchy problem is numerically solved for a system of ordinary differential equations, higher order methods are often used. Using a model problem for a parabolic equation as an example, general requirements for the selection of the finite difference approximation with respect to time are discussed. In addition to the unconditional stability requirements, extra performance criteria for finite difference schemes are presented and the concept of SM stability is introduced. Issues concerning the computational implementation of schemes having higher approximation orders are discussed. From the general point of view, various classes of finite difference schemes for time-dependent problems of mathematical physics are analyzed.     

对称正则长波方程的拟紧致守恒差分逼近

该文对称正则长波方程的初边值问题进行了数值研究, 提出了一个两层隐式拟紧差分格式,格式很好地模拟了初值问题的守恒性质. 得到了差分解的存在唯一性, 并在先验估计基础上运用能量方法分析了格式的稳定性及二阶收敛性.     

Analysis of new conservative difference scheme for two-dimensional Rosenau-RLW equation

In this article, numerical solution for the Rosenau-RLW equation in 2D is considered and a conservative Crank–Nicolson finite difference scheme is proposed. Existence of the numerical solutions for the difference scheme has been shown by Browder fixed point theorem. A priori bound and uniqueness as well as conservation of discrete mass and discrete energy for the finite difference solutions are discussed. Unconditional stability and a second-order accuracy on both space and time of the difference scheme are proved. Numerical experiments are given to support our theoretical results.     

The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions

Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space. For the approximation to this equation with the initial and boundary value conditions using the finite difference method, the difficulty is how to construct matched finite difference schemes at all the inner grid points. In this paper, two finite difference schemes are constructed for the problem. The accuracy is second-order in time and first-order in space. The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme. The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme. The conservation, boundedness, stability, convergence of these schemes are discussed and analyzed by the energy method together with other techniques. The two-level nonlinear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent. Some numerical examples illustrate the efficiency of the proposed finite difference schemes.     

Numerical methods for fourth‐order nonlinear elliptic boundary value problems

The aim of this article is to present several computational algorithms for numerical solutions of a nonlinear finite difference system that represents a finite difference approximation of a class of fourth‐order elliptic boundary value problems. The numerical algorithms are based on the method of upper and lower solutions and its associated monotone iterations. Three linear monotone iterative schemes are given, and each iterative scheme yields two sequences, which converge monotonically from above and below, respectively, to a maximal solution and a minimal solution of the finite difference system. This monotone convergence property leads to upper and lower bounds of the solution in each iteration as well as an existence‐comparison theorem for the finite difference system. Sufficient conditions for the uniqueness of the solution and some techniques for the construction of upper and lower solutions are obtained, and numerical results for a two‐point boundary‐value problem with known analytical solution are given. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:347–368, 2001     

Mesh-centered finite differences from nodal finite elements for elliptic problems

After it is shown that the classical five-point mesh-centered finite difference scheme can be derived from a low-order nodal finite element scheme by using nonstandard quadrature formulae, higher-order block mesh-centered finite difference schemes for second-order elliptic problems are derived from higher-order nodal finite elements with nonstandard quadrature formulae as before, combined to a procedure known as “transverse integration.” Numerical experiments with uniform and nonuniform meshes and different types of boundary conditions confirm the theoretical predictions, in discrete as well as continuous norms. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 439–465, 1998     

Finite difference calculations of buoyant convection in an enclosure: verification of the nonlinear algorithm

Solutions are presented to nonlinear finite difference equations used to represent fire-driven buoyant convection in enclosures. The solutions depend upon the fact that these difference equations permit the decomposition of the discretized velocity field into solenoidal and irrotational components. The irrotational field is shown to satisfy a finite difference analog of Bernoulli's equation when the density is constant. This leads to a three-dimensional time-dependent solution to the difference equations. The solenoidal field is shown to possess steady-state two-dimensional solutions corresponding to a constant non-zero value of the discretized vorticity. The two solutions, together with results presented elsewhere describing finite difference approximations to linear internal waves in enclosures, have been used in the development and testing of the computer-based algorithms used to solve these equations. They have proved particularly useful in assessing the accuracy of finite difference approximations to the equations of inviscid fluid mechanics, as well as in debugging the computer codes implementing these algorithms.