By Clive A. J. Fletcher
The aim of this textbook is to supply senior undergraduate and postgraduate engineers, scientists and utilized mathematicians with the particular strategies, and the framework to improve abilities in utilizing the suggestions, that experience confirmed potent within the quite a few brances of computational fluid dynamics.
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Additional info for Computational Techniques for Fluid Dynamics: Volume 2: Specific Techniques for Different Flow Categories
The numerical solution is defined throughout the entire spatial domain at every time step, but in time-dependent problems the approximate solution is typically available at only a few time levels at any given step of the numerical simulation. As a consequence, the use of series expansions is generally restricted to the representation of functional variations along spatial coordinates. Time derivatives are almost always approximated by finite differences. 30 1 Introduction s0 1 0 s1 s2 1 0 1 0 s3 s4 s5 1 0 1 0 1 0 x 0 2π Fig.
It is often helpful to consider the algebraic equations used to generate the approximate solution as arising from one of two approaches. 1) is replaced with a finite difference. 2) tn and the algebraic equations that constitute the numerical method can be most easily interpreted as providing an approximation to the integral of F . 1) is replaced by finite differences. 2). Some of the simplest schemes can be easily interpreted using either approach. 1 Stability, Consistency, and Convergence The basic goal when computing a numerical approximation to the solution of a differential equation is to obtain a result that indeed approximates the true solution.
A numerical method is A-stable if it is absolutely stable for all t Ä 0. Forward differencing is not A-stable but, as will be discussed in Sect. 3, the other methods whose absolute stability regions are shown in Fig. 1 are A-stable. 18) reduces to the oscillation equation d D i! 19) dt which serves as an important model for many nondissipative dynamical systems. 19) by setting and ! D f . 20) Here the last relation defines an “exact amplification factor” Ae , which in the case of the oscillation equation, is a complex number of modulus 1.