Functional integration

Abstract

This chapter provides in-depth coverage of the finite difference method (FDM) in the context of elliptic boundary value problems. The general procedure for deriving difference approximations to spatial derivatives using Taylor series expansions is first presented. The error incurred in the approximation – the truncation or discretization error – is thoroughly analyzed, and the procedure to develop higher-order approximations to derivatives is outlined. The implementation of the three canonical types of boundary conditions, i.e., Dirichlet, Neumann, and Robin, is discussed. Presentation of the matrix form of the discrete equations is finally followed by extension of the FDM to multidimensional geometries, including those described by the cylindrical coordinate system, and generalized curvilinear coordinates (body-fitted mesh).