ABSTRACT
Chapter 4 deals with the application of finite difference methods of various discretisation procedures of one- and two-dimensional diffusion equations, wave equations, complete transport equations, inviscid and viscid Burgers’ equations, Laplace equations with associated error and consistency and stability considerations using both explicit and implicit methods. The reason for explicit schemes failing to represent the physical behaviour of the parabolic equation is explained. First-order upwind scheme discretisation of convective terms is required but gives rise to numerical diffusion. For wave equations, explicit schemes provide a more natural finite difference approximation. Implicit methods are more suitable for diffusion equations than explicit methods. For one-dimensional problems, the Crank-Nicolson method is highly recommended. For two- or three-dimension cases, the Alternate Direction Implicit (ADI) scheme may be used, as this method allows the use of the Thomas algorithm. Discretisation of the conservative form of an equation guarantees the proper processing of any discontinuity present in the flow. Grid independence study is essential for any simulation. Solution of inviscid Burgers’ equation by explicit methods produces superior results. Godunov’s method is best for capturing shock. However, this method is computationally expensive when one is solving Euler equations.
