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[...]... still be of little practical use if ine cient or inappropriate algorithms are used This motivates studying the concepts of sti ness, factorization, and algorithm development in general All of these concepts we hope to clarify in this book 1.2 Background The eld of computational uid dynamics has a broad range of applicability Independent of the speci c application under study, the following sequence of steps... class of problems of interest 1.2.4 Assessment and Interpretation of Results Finally, the results of the simulation must be assessed and interpreted This step can require post-processing of the data, for example calculation of forces and moments, and can be aided by sophisticated ow visualization tools and error estimation techniques It is critical that the magnitude of both numerical and physical-model... simulation of uid ows can involve a wide range of issues from grid generation to turbulence modelling to the applicability of various simpli ed forms of the Navier-Stokes equations Many of these issues are not addressed in this book Some of them are presented in the books by Anderson, Tannehill, and Pletcher 1] and Hirsch 2] Instead we focus on numerical methods, with emphasis on nite-di erence and nite-volume... collection of scalars in our presentation of hyperbolic systems Some of the abbreviations used throughout the text are listed and de ned below PDE ODE O E RHS P.S S.S k-D ~ bc O( ) Partial di erential equation Ordinary di erential equation Ordinary di erence equation Right-hand side Particular solution of an ODE or system of ODE's Fixed (time-invariant) steady-state solution k-dimensional space Boundary conditions,... an approach known as solution-adaptive gridding The numerical methods generally used in CFD can be classi ed as nite-di erence, nite-volume, nite-element, or spectral methods The choices of a numerical method and a gridding strategy are strongly interdependent For example, the use of nite-di erence methods is typically restricted to structured grids Here again, the success of a simulation can depend... referred to as the Navier-Stokes equations However, in the interest of e ciency, it is always prudent to consider solving simpli ed forms of the Navier-Stokes equations when the simpli cations retain the physics which are essential to the goals of the simulation Possible simpli ed governing equations include the potential- ow equations, the Euler equations, and the thin-layer Navier-Stokes equations These... consistent in terms of the well-posedness of a 1st-order PDE Hence the wave leaves the domain through the out ow boundary without distortion or re ection This type of phenomenon is referred to, simply, as the convection problem It represents most of the \usual" situations encountered in convecting systems Note that the left-hand boundary is the in ow boundary when a is positive, while the right-hand boundary... the nodes of a grid Otherwise, the use of a vector consisting of a collection of scalars should be apparent from the context and is not identi ed by any special notation For example, the variable u can denote a scalar Cartesian velocity component in the Euler and Navier-Stokes equations, a scalar quantity in the linear convection and di usion equations, and a vector consisting of a collection of scalars... Reynolds number and Mach number for the ow over an airfoil The requirements of the simulation include issues such as the level of accuracy needed, the turnaround time required, and the solution parameters of interest The rst two of these requirements are often in con ict and compromise is necessary As an example of solution parameters of interest in computing the ow eld about an airfoil, one may be interested... Navier-Stokes equations are brie y discussed in this Chapter The main focus, though, will be on representative model equations, in particular, the convection and di usion equations These equations contain many of the salient mathematical and physical features of the full Navier-Stokes equations The concepts of convection and di usion are prevalent in our development of numerical methods for computational