Conservation Laws and the Model Equations

Abstract

We start out by casting our equations in the most general form, the integral conservation-law form, which is useful in understanding the concepts involved in finite-volume schemes. The equations are then recast into divergence form, which is natural for finite-difference schemes. The Euler and Navier—Stokes equations are briefly discussed in this chapter. The main focus, though, will be on representative model equations, in particular, the convection and diffusion equations. These equations contain many of the salient mathematical and physical features of the full Navier—Stokes equations. The concepts of convection and diffusion are prevalent in our development of numerical methods for computational fluid dynamics, and the recurring use of these model equations allows us to develop a consistent framework of analysis for consistency, accuracy, stability, and convergence. The model equations we study have two properties in common They are linear partial differential equations (PDE’s) with coefficients that are constant in both space and time, and they represent phenomena of importance to the analysis of certain aspects of fluid dynamics problems.