Finite difference method

Abstract: This paper describes the applications of the method of fundamental solutions (MFS) for 1-, 2- and 3-D diffusion equations. The time-dependent fundamental solutions for diffusion equations are used directly to obtain the solution as a linear combination of the fundamental solution of the diffusion operator. The proposed scheme is free from the conventionally used Laplace transform or the finite difference scheme to deal with the time derivative of the governing equation. By properly placing the field points and the source points at a given time level, the solution is advanced in time until steady state solutions are reached. Test results obtained for 1-, 2- and 3-D diffusion problems show good comparisons with the analytical solutions and some with the MFS based on the modified Helmholtz fundamental solutions, thus the demonstration present numerical scheme of MFS with the space–time unification has been demonstrated as a promising mesh-free numerical tool to solve homogeneous diffusion problem.

Abstract: : Problems relating to the computation of viscous compressible flows based on numerical solutions of the Navier-Stokes equations are reviewed. A general introduction to the Navier-Stokes equations and a discussion of their interest in aerodynamic problems are first presented. Then the following aspects of numerical methods are considered: limitation of the computational domain and boundary conditions on the outer boundary; various approaches in finite difference methods and description of some representative schemes; treatment of boundary conditions at a solid wall; treatment of shock waves, and general considerations on accuracy and computing times. Finally reported computations of two-dimensional or three-dimensional flows are presented in table form with summary indications on the problems treated and the methods used.

Abstract: The heat transfer arising due to the movement of a continuous heated plate in processes such as hot rolling and hot extrusion has been studied. Of particular interest were the resulting temperature distribution in the solid and the proper imposition of the boundary conditions at the location where the material emerges from a furnace or die. These considerations are important in the simulation and design of practical systems. A numerical study of the thermal transport process has been carried out, assuming a two-dimensional steady circumstance. The boundary layer equations, as well as full governing equations including buoyancey effects, are solved employing finite difference techniques. The effect of various physical parameters, which determine the temperature and flow fields, is studied in detail. The significance of these results in actual manufacturing processes is discussed.