Li R., Chen Z., Wu W. Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods

CRC, 2000. — 472 p.This text presents a comprehensive mathematical theory for elliptic, parabolic, and hyperbolic differential equations. It compares finite element and finite difference methods and illustrates applications of generalized difference methods to elastic bodies, electromagnetic fields, underground water pollution, and coupled sound-heat flows.Finite difference (on rectangular networks) and finite element methods are the two most important classes of numerical methods for partial differential equations. The finite difference method is particularly preferred for hyperbolic equations, especially quasi-linear ones which admit discontinuous solutions. The main defects of the difference method are: the considerable geometrical error of the approximation of curved domains by rectangular grids; the lack of a united and effective approach to deal with natural and internal boundary conditions; the difficulty to construct difference schemes with high accuracy, unless we allow the difference equation to relate more nodal points (which will in turn further increase the difficulty in dealing with boundary conditions).