Implicit difference methods for Hamilton-Jacobi functional differential equations

Classical solutions of initial boundary value problems are approximated by solutions of associated implicit difference functional equations. A stability result is proved by using a comparison technique with nonlinear estimates of the Perron type for given functions. The Newton method is used to numerically solve nonlinear equations generated by implicit difference schemes. It is shown that there are implicit difference schemes which are convergent whereas the corresponding explicit difference methods are not. The results obtained can be applied to differential integral problems and differential equations with deviated variables.     

Implicit Solution of a Two-Dimensional Parabolic Inverse Problem with Temperature Overspecification

Three different implicit finite difference schemes for solving the two-dimensional parabolic inverse problem with temperature overspecification are considered. These schemes are developed for indentifying the control parameter which produces, at any given time, a desired temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the second-order (5,1) Backward Time Centered Space (BTCS) implicit formula, and the second-order (5,5) Crank-Nicolson implicit finite difference formula and the fourth-order (9,9) implicit scheme. These finite difference schemes are unconditionally stable. The (9,9) implicit formula takes a huge amount of CPU time, but its fourth-order accuracy is significant. The results of a numerical experiment are presented, and the accuracy and central processor (CPU) times needed for each of the methods are discussed and compared. The implicit finite difference schemes use more central processor times than the explicit finite difference techniques, but they are stable for every diffusion number.     

对流扩散方程区域分裂特征差分方法

1引言对流扩散方程是许多物理问题的数学模型,研究其稳定的数值解法具有重要的应用价值．而标准的差分法和有限元法通常会失效,出现数值振荡．80年代,Douglas和Russel提出了特征线方法,在一定程度上克服了数值振荡,保证了数值的稳定,尤其对“对流占优”问题,更能突出特征法的优越性,并有了大量的理论成果[1,2,3]．区域分裂是一种解决大规模的科学与工程计算问题的有效方法,Dawson,Du和Dupont对热传导方程给出了非重叠区域分裂格式及分析,由于内边界的显格式,需要一定的稳定性条件Δt≤CH2;而Du等在[5]给出了抛物方程的几种区域分裂格式,对区域分裂法的     

Uniformly maximumnorm stable difference schemes

Sufficient conditions for maximumnorm stability, uniform in the mesh-widths, of implicit two-level difference schemes with constant coefficients are given. Uniform stability is a necessary and sufficient condition for consistent difference schemes to be convergent, when the mesh-widths are unrelated.     

Implicit Difference Schemes for the Generalized Non-Linear Schrödinger System

In this paper we prove under certain weak conditions that two classes of implicit difference schemes for the generalized non-linear schrödinger system are convergent and that an iteration method for the corresponding non-linear difference equation is convergent. Therefore, quite a complete theoretical foundation of implicit schemes for the generalized non-linear Schrödinger system is established in this paper.     

非线性波动方程的弱隐式与显式差分方法

广泛出现于物理、化学、机械动力学、生物、几何学等领域的非线性波动方程已经有很多的研究工作,Sine-Gordon方程和非线性受迫振动方程就是典型的例子.周毓麟教授在[1]中研究了非线性波动方程组     

An inverse problem of finding a source parameter in a semilinear parabolic equation

An inverse problem concerning diffusion equation with source control parameter is considered. Several finite-difference schemes are presented for identifying the control parameter. These schemes are based on the classical forward time centred space (FTCS) explicit formula, and the 5-point FTCS explicit method and the classical backward time centred space (BTCS) implicit scheme, and the Crank–Nicolson implicit method. The classical FTCS explicit formula and the 5-point FTCS explicit technique are economical to use, are second-order accurate, but have bounded range of stability. The classical BTCS implicit scheme and the Crank–Nicolson implicit method are unconditionally stable, but these schemes use more central processor (CPU) times than the explicit finite difference mehods. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. The results of a numerical experiment are presented, and the accuracy and CPU time needed for this inverse problem are discussed.     

二维非线性Schr\"{o}dinger方程显式格式的最大模误差分析

本文讨论了数值求解二维非线性Schr\"{o}dinger方程周期边值问题的Du Fort-Frankel格式和蛙跳格式. 以解函数的一个广义时间导数作为独立变量, 将非线性方程初边值问题改写成一个混合方程组形式, 应用我们最新提出的离散能量技巧讨论这两个三层显式格式的收敛性. 分析表明, 在必要的网格条件下, 差分解在最大模意义下二阶收敛. 数值算例验证了理论分析结果.     

Second-order characteristic schemes in time and age for a nonlinear age-structured population model

In this paper, we analyze two new second-order characteristic schemes in time and age for an age-structured population model with nonlinear diffusion and reaction. By using the characteristic difference to approximate the transport term and the average along the characteristics to treat the nonlinear spatial diffusion and reaction terms, an implicit second-order characteristic scheme is proposed. To compute the nonlinear approximation system, an explicit second-order characteristic scheme in time and age is further proposed by using the extrapolation technique. The global existence and uniqueness of the solution of the nonlinear approximation scheme are established by using the theory of variation methods, Schauder’s fixed point theorem, and the technique of prior estimates. The optimal error estimates of second order in time and age are strictly proved for both the implicit and the explicit characteristic schemes. Numerical examples are given to illustrate the performance of the methods.     

Stable non-standard implicit finite difference schemes for non-linear heat transfer in a thin finite rod

In this study, first, three non-standard implicit finite difference schemes are proposed for solving the initial-boundary value problem involving a quartic non-linearity that arises in heat transfer involving conduction with thermal radiation. A thin finite rod exposed to radiating heat across its lateral surface into a medium of constant temperature and convection is ignored. Stability and consistency of the third scheme is proved. Numerical results are compared with non-standard explicit finite difference schemes that show fully stability of our third proposed scheme. Then, three non-standard implicit and three non-standard explicit finite difference schemes are proposed for solving the heat transfer problem with additional convection term. It is shown that in the second case when the model involves conduction, radiation and convection terms, the rod reaches steady state sooner. Numerical results for implicit and explicit schemes are compared and the effect of the convection term is discussed.     

Alternating triangular schemes for convection–diffusion problems

Explicit–implicit approximations are used to approximate nonstationary convection–diffusion equations in time. In unconditionally stable two-level schemes, diffusion is taken from the upper time level, while convection, from the lower layer. In the case of three time levels, the resulting explicit–implicit schemes are second-order accurate in time. Explicit alternating triangular (asymmetric) schemes are used for parabolic problems with a self-adjoint elliptic operator. These schemes are unconditionally stable, but conditionally convergent. Three-level modifications of alternating triangular schemes with better approximating properties were proposed earlier. In this work, two- and three-level alternating triangular schemes for solving boundary value problems for nonstationary convection–diffusion equations are constructed. Numerical results are presented for a two-dimensional test problem on triangular meshes, such as Delaunay triangulations and Voronoi diagrams.     

Implicit difference methods for evolution functional differential equations

A general theory of implicit difference schemes for nonlinear functional differential equations with initial boundary conditions is presented. A theorem on error estimates of approximate solutions for implicit functional difference equations of the Volterra type with an unknown function of several variables is given. This general result is employed to investigate the stability of implicit difference schemes generated by first-order partial differential functional equations and by parabolic problems. A comparison technique with nonlinear estimates of the Perron type for given functions with respect to the functional variable is used.     

钝体绕流非定常解

本文结合目前流场显示的研究课题,对方块物体和山形物体的钝体绕流在起动阶段的运动情况,进行相应的数值模拟.并用有限差分方法求解二维不可压缩流体运动的N-S方程的非定常解.对差分格式中的显式,隐式和交替方向隐式几种格式进行了讨论.最后用显式和交替方向隐式方法计算了山形物体和方块物体在起动阶段的运动情况.     

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.     

Upwind finite difference schemes for linear conservation law with memory

In this article we consider upwind finite difference schemes for a class of linear conservation laws with memory. Assuming the positivity of the kernel, it is proved by using the energy estimates that the upwind finite difference scheme, explicit or implicit, is stable and convergent to the real solution. The numerical results of some examples, including Burger's equation with memory, are reported; the effect of memory is also discussed based on the numerical results. © 1994 John Wiley & Sons, Inc.     

Parameterized families of finite difference schemes for the wave equation

This article is devoted to an analysis of simple families of finite difference schemes for the wave equation. These families are dependent on several free parameters, and methods for obtaining stability bounds as a function of these parameters are discussed in detail. Access to explicit stability bounds such as those derived here may, it is hoped, lead to optimization techniques for so‐called spectral‐like methods, which are difference schemes dependent on many free parameters (and for which maximizing the order of accuracy may not be the defining criterion). Though the focus is on schemes for the wave equation in one dimension, the analysis techniques are extended to two dimensions; implicit schemes such as ADI methods are examined in detail. Numerical results are presented. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 463–480, 2004.     

三维两相渗流驱动问题迎风区域分裂显隐差分法

对三维两相渗流驱动问题提出了两种迎风区域分裂显隐差分格式．压力方程采用了七点差分格式,为了能达到实际并行计算的要求,对饱和度方程采用了迎风区域分裂差分法,内边界处和各子区域分别对应显隐格式．得到了离散l2模收敛性分析,最后给出数值试验,支撑了理论分析结果．     

带吸收边界条件的声波方程显式差分格式的稳定性分析

１．引言 在进行无界或半无界区域上各种波动方程的数值求解时,需引进入工边界以限制计算范围,在这些边界上应加相应的人工边界条件．这种边界条件应保证所求得的有界区域上的解很好地逼近原来无界区域上的解．对波动方程来说,就是在边界上人工反射应尽可能地小,使之对区域内部解的影响在允许的误差范围以内.因而它们被称为无反射边界条件或吸收边界条件．这种边界条件还应保证所形成的有界区域上的微分方程定解问题是适定的．这也是各种数值方法稳定的必要条件。 近二十多年来,已发展了声波方程的各种类型的吸收边界条件,其中以Ｃｌ…     

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.     

Estimate of solutions for differential and difference functional equations with applications to difference methods

The theorems on the estimate of solutions for nonlinear second-order partial differential functional equations mainly of parabolic type with Dirichlet’s condition and for the suitable explicit finite difference functional schemes are proved. The proofs are based on the comparison technique. The convergent difference method given is considered without an assumption of the global generalized Perron condition on the functional variable but with local one in some sense only. It is a consequence of our estimate theorems. The functional dependence is of the Volterra type.