| The traditional finite difference method has an important limitation in practical applications, which is the requirement of a structured grid. The purpose of this thesis is to improve the finite difference scheme for application on complex domains. The analysis of the Finite Difference method is carried out for 1D model problems governed by the convection-diffusion equation. The Stencil Mapping method is developed for complex domains. One of the features of this new scheme is that the value at a node can be calculated by using only the neighbouring values on the 3-point stencil. This allows finite differencing for arbitrary nodal distribution in the mesh, and is developed for 2nd-order and 4th-order differencing schemes. The numerical solutions for typical boundary and initial value problems are compared with exact solutions. Local truncation error is introduced as an effective parameter to assess accuracy of the scheme. An adaptive meshing procedure is also presented. |
