Applied Numerical Analysis & Computational Mathematics最新文献

This work deals with the construction of difference schemes for the numerical solution of singularly perturbed boundary value problems, which appear while solving heat transfer equations with spherical symmetry. The projective version of integral interpolation (PVIIM) method is used. Derived schemes allow to approximate the solution of the problem and the derivatives of the solution at the same time. Moreover, they allow to approximate the boundary conditions of general form in the framework of the same method. New schemes are tested in order to compare them with well known difference schemes. Estimates for rates of classical and uniform convergence are carried out. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper a sufficient condition (Theorem 2.3) for the von Neumann stability of the Leapfrog-Euler scheme, which uses central spatial discretization of any order for 3D convection-diffusion equation, is derived in terms of the Courant and the diffusion numbers and the coefficients of approximation schemes. In the case of the second order differencing this condition becomes the necessary condition for the stability. Some particular sufficient conditions for the stability of the second and the fourth order schemes are also derived. A comparison of the results, which were obtained applying the derived stability conditions to compute the time step in the direct numerical simulations (DNS) of turbulent pipe flow with the help of the second and the fourth order schemes, is presented. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this work, the optimal control of an harmonic oscillator is considered. The external field is assumed to be weak and hence is represented by only dipole interaction. The dipole function is taken as a second degree polynomial in spatial coordinate. The objective term is composed of the expectation value of algebraic position operator. Penalty term related operator is taken as momentum. Some specific structures for spatial dependence is assumed and temporal equations are obtained for unknowns. The equations represent forward and backward evolution. The connection is provided by an algebraic equation coming from field amplitude related equation. The key unknown is the field amplitude since its determination leads us to evaluate all unknowns without remarkable difficulty. The purpose is not to determine field amplitude at the most general case but to construct an equation which involves this unknown only. The equation obtained via the linearisation of the field amplitude dependence is shown to be a linear integral equation. We do not attempt to solve it but discuss how to use appropriate methods for the solution. The equation is analytically solved at the zero interaction time limit. The real comprehensive implementation is left for future work. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The solution of the two-dimensional time-independent Schrödinger equation is considered by partial discretization. The discretized problem is treated as an ordinary differential equation problem and solved numerically by asymptotically symplectic methods. The problem is then transformed into an algebraic eigenvalue problem involving real, symmetric matrices. The eigenvalues of the two-dimensional harmonic oscillator and the twodimensional Henon-Heils potential are computed by the application of the methods developed. The results are compared with the results produced by full discretization. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The problem is considered of fitting circular arcs to discrete data using orthogonal distance regression. This can be formulated as a constrained nonlinear least squares problem, and an efficient Gauss-Newton method is developed. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Various iterative schemes have been proposed to solve the non-linear equations arising in the implementation of implicit Runge-Kutta methods. In one scheme, when applied to an s-stage Runge-Kutta method, each step of the iteration still requires s function evaluations but consists of r(>s) sub-steps. Improved convergence rate was obtained for the case r = s + 1 only. This scheme is investigated here for the case r = ks, k = 2, 3, …, and superlinear convergence is obtained in the limit k→∞. Some results are obtained for Gauss methods and numerical results are given. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)