Finite difference method

Abstract: In this paper a finite-difference-based lattice Boltzmann method for curvilinear coordinates is proposed in order to improve the computational efficiency and numerical stability of a recent method [R. Mei and W. Shyy, J. Comput. Phys. 143, 426 (1998)] in which the collision term of the Boltzmann Bhatnagar-Gross-Krook equation for discrete velocities is treated implicitly. In the present method, the implicitness of the numerical scheme is removed by introducing a distribution function different from that being used currently. As a result, an explicit finite-difference lattice Boltzmann method for curvilinear coordinates is obtained. The scheme is applied to a two-dimensional Poiseuille flow, an unsteady Couette flow, a lid-driven cavity flow, and a steady flow around a circular cylinder. The numerical results are in good agreement with the results of previous studies. Extensions to other lattice Boltzmann models based on nonuniform meshes are also discussed.

Abstract: The probability density evolution method (PDEM) for dynamic responses analysis of non-linear stochastic structures is proposed. In the method, the dynamic response of non-linear stochastic structures is firstly expressed in a formal solution, which is a function of the random parameters. In this sense, the dynamic responses are mutually uncoupled. A state equation is then constructed in the augmented state space. Based on the principle of preservation of probability, a one-dimensional partial differential equation in terms of the joint probability density function is set up. The numerical solving algorithm, where the Newmark-Beta time-integration algorithm and the finite difference method with Lax–Wendroff difference scheme are brought together, is studied. In the numerical examples, free vibration of a single-degree-of-freedom non-linear conservative system and dynamic responses of an 8-storey shear structure with bilinear hysteretic restoring forces, subjected to harmonic excitation and seismic excitation, respectively, are investigated. The investigations indicate that the probability density functions of dynamic responses of non-linear stochastic structures are usually irregular and far from the well-known distribution types. They exhibit obvious evolution characteristics. The comparisons with the analytical solution and Monte Carlo simulation method demonstrate that the proposed PDEM is of fair accuracy and efficiency. Copyright © 2005 John Wiley & Sons, Ltd.

Abstract: Partial differential equations are the chief means of providing mathematical models in physics, engineering and other fields. Generally these models must be solved numerically. This book provides a concise introduction to standard numerical techniques, ones chosen on the basis of their general utility for practical problems. The authors emphasize finite difference methods for simple examples of parabolic, hyperbolic and elliptic equations; finite element, finite volume and spectral methods are discussed briefly to see how they relate to the main theme. Stability is treated clearly and rigorously using maximum principles, energy methods, and discrete Fourier analysis. Methods are described in detail for simple problems, accompanied by typical graphical results. A key feature is the thorough analysis of the properties of these methods. Plenty of examples and exercises of varying difficulty are supplied.The book will be an excellent choice for students and teachers in mathematics, engineering and computer science departments seeking a concise introduction to the subject.