JAVA语言的递归程序设计方法

递归程序设计方法分数值问题和非数值问题两种,对于数值问题从数学公式入手,对于非数值问题,应掌握方法,分步进行。

Solving the Sod Shock Tube Problem Using Localized Differential Quadrature (LDQ) Method

The localized differential quadrature (LDQ) method is a numerical technique with high accuracy for solving most kinds of nonlinear problems in engineering and can overcome the difficulties of other methods (such as difference method) to numerically evaluate the derivatives of the functions.Its high efficiency and accuracy attract many engineers to apply the method to solve most of the numerical problems in engineering.However,difficulties can still be found in some particular problems.In the following study,the LDQ was applied to solve the Sod shock tube problem.This problem is a very particular kind of problem,which challenges many common numerical methods.Three different examples were given for testing the robustness and accuracy of the LDQ.In the first example,in which common initial conditions and solving methods were given,the numerical oscillations could be found dramatically;in the second example,the initial conditions were adjusted appropriately and the numerical oscillations were less dramatic than that in the first example;in the third example,the momentum equation of the Sod shock tube problem was corrected by adding artificial viscosity,causing the numerical oscillations to nearly disappear in the process of calculation.The numerical results presented demonstrate the detailed difficulties encountered in the calculations,which need to be improved in future work.However,in summary,the localized differential quadrature is shown to be a trustworthy method for solving most of the nonlinear problems in engineering.

Numerical Simulation of Two-dimensional Nonlinear Sloshing Problems

Numerical simulation of a two-dimensional nonlinear sloshing problem is preceded by the finite element method. Two theories are used. One is fully nonlinear theory; the other is time domain second order theory. A liquid sloshing in a rectangular container subjected to a horizontal excitation is simulated using these two theories. Numerical results are obtained and comparisons are made. It is found that a good agreement is obtained for the case of small amplitude oscillation. For the situation of large amplitude excitation, although the differences between using the two theories are obvious the second order solution can still exhibit typical nonlinear features of nonlinear wave.

Numerical Simulation of Multicomponent Alloy Solidification

A non-equilibrium model of multicomponent melt solidification has been developed in which a Stefan problem with two boundaries is solved numerically, the boundaries being between the solid phase and the two-phase transition zone and between the two-phase transition zone and the liquid phase. The two-phase zone is represented as a porous medium with variable porosity. The additional force resisting the melt flow due to porosity and introduced by analogy with Darcy＇s law is taken into account. Computer simulation has been performed of the experiment on Sn-20 wt.%Pb binary alloy solidification by the method of downward-directed crystallization along the gravity vector. The paper shows the results of a quasi two-dimensional benchmark experiment on horizontal （i.e., at the right angle to the gravity vector） directional solidification of a binary Sn-3 wt.%Pb alloy. The calculations were done using two crystallization models： the equilibrium model and the non-equilibrium one. It is shown that the non-equilibrium model gives a better description of the thermal field evolution and solute distribution caused by natural convection.

On the values of representation functions

For a set A of nonnegative integers, the representation functions R2(A,n) and R3(A,n) are defined as the numbers of solutions to the equation n = a + a′ with a,a′∈ A, a < a′ and a a′, respectively. Let N be the set of nonnegative integers. Given n0 > 0, it is known that there exist A,A′■ N such that R2(A′,n) = R2(N \ A′,n) and R3(A,n) = R3(N \ A,n) for all n n0. We obtain several related results. For example, we prove that: If A ■ N such that R3(A,n) = R3(N \ A,n) for all n n0, then (1) for any n n0 we have R3(A,n) = R3(N \ A,n) > c1n - c2, where c1,c2 are two positive constants depending only on n0; (2) for any α < 116, the set of integers n with R3(A,n) > αn has the density one. The answers to the four problems in Chen-Tang (2009) are affirmative. We also pose two open problems for further research.

Numerical Approximation of a Reaction-Diffusion System with Fast Reversible Reaction

The authors consider the finite volume approximation of a reaction-diffusion system with fast reversible reaction.It is deduced from a priori estimates that the approximate solution converges to the weak solution of the reaction-diffusion problem and satisfies estimates which do not depend on the kinetic rate.It follows that the solution converges to the solution of a nonlinear diffusion problem,as the size of the volume elements and the time steps converge to zero while the kinetic rate tends to infinity.

On the Numerical Solution of Some Eikonal Equations:An Elliptic Solver Approach

The steady Eikonal equation is a prototypical first-order fully nonlinear equation. A numerical method based on elliptic solvers is presented here to solve two different kinds of steady Eikonal equations and compute solutions, which are maximal and minimal in the variational sense. The approach in this paper relies on a variational argument involving penalty, a biharmonic regularization, and an operator-splitting-based time-discretization scheme for the solution of an associated initial-value problem. This approach allows the decoupling of the nonlinearities and differential operators.Numerical experiments are performed to validate this approach and investigate its convergence properties from a numerical viewpoint.

Compressible Limit of the Nonlinear Schrdinger Equation with Different-Degree Small Parameter Nonlinearities

The authors study the compressible limit of the nonlinear Schrdinger equation with different-degree small parameter nonlinearities in small time for initial data with Sobolev regularity before the formation of singularities in the limit system.On the one hand,the existence and uniqueness of the classical solution are proved for the dispersive perturbation of the quasi-linear symmetric system corresponding to the initial value problem of the above nonlinear Schrdinger equation.On the other hand,in the limit system,it is shown that the density converges to the solution of the compressible Euler equation and the validity of the WKB expansion is justified.