Finite difference method

Abstract: Different numerical methods have been implemented to simulate internal natural convection heat transfer and also to identify the most accurate and efficient one. A laterally heated square enclosure, filled with air, was studied. A FORTRAN code based on the lattice Boltzmann method (LBM) was developed for this purpose. The finite difference method was applied to discretize the LBM equations. Furthermore, for comparison purpose, the commercially available CFD package FLUENT, which uses finite volume Method (FVM), was also used to simulate the same problem. Different discretization schemes, being the first order upwind, second order upwind, power law, and QUICK, were used with the finite volume solver where the SIMPLE and SIMPLEC algorithms linked the velocity-pressure terms. The results were also compared with existing experimental and numerical data. It was observed that the finite volume method requires less CPU usage time and yields more accurate results compared to the LBM. It has been noted that the 1st order upwind/SIMPLEC combination converges comparatively quickly with a very high accuracy especially at the boundaries. Interestingly, all variants of FVM discretization/pressure-velocity linking methods lead to almost the same number of iterations to converge but higher-order schemes ask for longer iterations.

Abstract: We present a parallel octree-based finite element method for large-scale earthquake ground motion simulation in realistic basins. The octree representation combines the low memory per node and good cache performance of finite difference methods with the spatial adaptivity to local seismic wavelengths characteristic of unstructured finite element methods. Several tests are provided to verify the numerical performance of the method against Green’s function solutions for homogeneous and piecewise homogeneous media, both with and without anelastic attenuation. A comparison is also provided against a finite difference code and an unstructured tetrahedral finite element code for a simulation of the 1994 Northridge Earthquake. The numerical tests all show very good agreement with analytical solutions and other codes. Finally, performance evaluation indicates excellent single-processor performance and parallel scalability over a range of 1 to 2048 processors for Northridge simulations with up to 300 million degrees of freedom. keyword: Earthquake ground motion modeling, octree, parallel computing, finite element method, elastic wave propagation

Abstract: The analogy between differential equations and difference equations has been recognized and exploited very frequently. Thus, shortly after the publication of PERRON'S paper on stability [5], there appeared a corresponding paper by TALI [3], a student of PERRON'S, with an almost identical title, in which similar methods were used to obtain analogous results for difference equations. In both papers a central concern is the relationship, for linear equations, between the condition that the non-homogeneous equation have some bounded solution for every bounded "second member" on the one hand, and a certain form of behaviour of the solutions of the homogeneous equation on the other. PERRON'S results for linear differential equations are the starting point of extensive later developments; some recent ones are found in a series of papers by MASSERA and SCH~FFER, later continued by SCH~,FFER, and brought together in a monograph [4] (see also [1], Chs. 12 and 13). The purpose of the present paper is to provide for linear difference equations the analogues of the most central results for linear differential equations in that theory.