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     

Unconditional stability of parallel alternating difference schemes for semilinear parabolic systems

The general alternating schemes with intrinsic parallelism for semilinear parabolic systems are studied. First we prove the a priori estimates in the discrete H¹ space of the difference solution for these schemes. Then the existence of the difference solution for these schemes follows from the fixed point principle. Finally the unconditional stability of the general alternating schemes is proved. The alternating group explicit scheme, the alternating segment explicit–implicit scheme and the alternating segment Crank–Nicolson scheme are the special cases of the general alternating schemes.     

The Entropy Condition for Implicit TVD Schemes

A class of implicit trapezoidal TVD schemes is proven to satisfy a discrete convex entropy inequality and the solution sequence of such implicit trapezoidal schemes converges to the physically relevant solution for genuinely nonlinear scalar conservation laws. The results are extended for a class of generalized implicit one-leg TVD schemes.     

On the perturbation algorithm for the semidiscrete scheme for the evolution equation and estimation of the approximate solution error using semigroups

In a Banach space, for the approximate solution of the Cauchy problem for the evolution equation with an operator generating an analytic semigroup, a purely implicit three-level semidiscrete scheme that can be reduced to two-level schemes is considered. Using these schemes, an approximate solution to the original problem is constructed. Explicit bounds on the approximate solution error are proved using properties of semigroups under minimal assumptions about the smoothness of the data of the problem. An intermediate step in this proof is the derivation of an explicit estimate for the semidiscrete Crank–Nicolson scheme. To demonstrate the generality of the perturbation algorithm as applied to difference schemes, a four-level scheme that is also reduced to two-level schemes is considered.     

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.     

Monotonicity criteria for difference schemes designed for hyperbolic equations

Previously formulated monotonicity criteria for explicit two-level difference schemes designed for hyperbolic equations (S.K. Godunov’s, A. Harten’s (TVD schemes), characteristic criteria) are extended to multileveled, including implicit, stencils. The characteristic monotonicity criterion is used to develop a universal algorithm for constructing high-order accurate nonlinear monotone schemes (for an arbitrary form of the desired solution) based on their analysis in the space of grid functions. Several new fourth-to-third-order accurate monotone difference schemes on a compact three-level stencil and nonexpanding (three-point) stencils are proposed for an extended system, which ensures their monotonicity for both the desired function and its derivatives. The difference schemes are tested using the characteristic monotonicity criterion and are extended to systems of hyperbolic equations.     

Adjoint-coordinated approximations and economical discrete implementations in a dynamic problem of the linear theory of elasticity

For dynamic three-dimensional problems of the elasticity theory, we construct a new class of economical implicit difference schemes with high degree of parallelism. They include difference schemes whose parallelism degree is the same as for ordinary explicit schemes. So far, even the very existence of implicit schemes with the same parallelism degree has been strongly doubted.     

On the role of conservation laws in the problem on the occurrence of unstable solutions for quasilinear parabolic equations and their approximations

For a broad class of initial-boundary value problems for quasilinear parabolic equations with nonlinear source and their approximations, we show that if the initial energy is negative, then the solution always blows up in finite time. This is especially important for finding sufficiently simple and easy-to-verify conditions guaranteeing the presence of physical effects such as heat localization in peaking modes or thermal explosions and for deriving two-sided estimates of the solution lifespan. We construct the corresponding new classes of difference schemes for which grid analogs of integral conservation laws hold. We show that, to obtain efficient two-sided estimates for the blow-up time of the solution of the differential problem, in practice, one should use difference schemes with explicit as well as implicit approximation to the source.     

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 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.     

A numerical method for solving partial differential algebraic equations

Linear systems of partial differential equations with constant coefficient matrices are considered. The matrices multiplying the derivatives of the sought vector function are assumed to be singular. The structure of solutions to such systems is examined. The numerical solution of initialboundary value problems for such equations by applying implicit difference schemes is discussed.     

Some properties of difference schemes in one-dimensional gas dynamics

The linear implicit difference scheme is analyzed on specimen problems using numerical computations. Stability of some difference schemes is investigated, whose solvability is improved by introducing explicitness. A stability theorem is proved for the difference scheme of one-dimensional gas dynamics in the acoustic approximation near an arbitrary smooth solution.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 20–28, 1985.     

基于隐式离散极大值原理的聚合物驱最优注入策略

为了获得聚合物驱油的最大利润,建立了确定最佳聚合物注入浓度的最优控制模型.利用全隐式差分格式将连续模型离散化得到离散系统的状态方程.通过隐含离散系统的极大值原理获得了该最优控制问题的必要条件.给出了基于梯度的数值求解方法,在求解状态方程的过程中直接构造了伴随问题的系数矩阵.通过一个三维聚合物驱模型的计算实例表明了所提出方法的可行性和有效性.     

Numerical methods for Hamilton Jacobi functional differential equations

Initial and initial boundary value problems for first order partial functional differential equations are considered. Explicit difference schemes of the Euler type and implicit difference methods are investigated. The following theoretical aspects of the methods are presented. Sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that assumptions on the regularity of given functions are the same for both the methods. It is shown that conditions on the mesh for explicit difference schemes are more restrictive than suitable assumptions for implicit methods. There are implicit difference schemes which are convergent and corresponding explicit difference methods are not convergent. Error estimates for both the methods are construted.     

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.     

二维抛物型方程的高精度多重网格解法

提出了数值求解二维抛物型方程的一种新的高精度加权平均紧隐格式，利用Fourier分析方法证明了该格式是无条件稳定的，为了克服传统迭代法在求解隐格式是收敛速度慢的缺陷，利用了多重网格加速技术，大大加快了迭代收敛速度，提高了求解效率，数值实验结果验证了方法的精确性和可靠性。     

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.     

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.     

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.     

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.