ABSTRACT
An introduction to issues encountered in creating discrete approximations to partial differential equations is presented. These issues include truncation error, round-off error, consistency, stability, convergence for marching problems, and relaxation for equilibrium problems. The concept of conservation form and the conservative property are considered in the context of the governing equation of fluid mechanic and heat transfer. Taylor series, polynomial fitting, integral, and other methods for creating discrete representations of these equations are examined in detail. An introduction to finite-element methods is presented where approximating and basis functions, residual, weak form, quadrature, and mass and stiffness matrices are discussed. The continuous and discontinuous Galerkin methods are introduced and examples of application to the advection–diffusion equation are given. Methods for creating discrete versions of the Laplace operator and treating diffusion terms in general are given including corrections schemes when a nonorthogonal mesh is encountered. Alterations necessary to these operators in irregular grids both interior at on boundaries are covered. The concluding material introduces the stability concept and the von Neumann analysis for single equations and stability analysis for systems of equations.
