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Unformatted text preview: Chapter 17 Finite Volume Method . This chapter focusses on introducing finite volume method for the solution of partial differential equations. These methods have gained wide-spread acceptance in recent years for their robustness, their intuitive formulation, and their compu- tational advantages. We will take the opportunity to look at issues pertaining to method formulation, discretization, and analysis. 17.1 The partial differential equation The partial differential equation we will focus on is a scalar equation that represents the transport of a substance under the influence of advection by the air flow and mixing. The transport equation is frequently written in the advective form: ∂T ∂t + vectoru · ∇ T = ∇ · ( α ∇ T ) (17.1) where T is the subtance transported, e.g. temperature, humidity or a pollutant concentration, vectoru is the velocity field presumed known, and α is the diffusion coef- ficient and which can represent either molecular diffusion or eddy mixing. The velocity field cannot be arbitrary and must satisfy some sort of mass con- servation equation. Here we will assume the flow to be incompressible so that its mass conservation equation reduces to: ∇ · vectoru = 0 (17.2) The advective form can be interpreted as the time evolution of the T field along characteristic lines given by d x d t = vectoru . It is thus closest to a Lagrangian description of the flow where one follows individual particles. In the Eulerian frame, however, another important issues is the conservation of the tracer T for long period of times. This stems not only from physical considerations but also for the need to account for the sources and sinks of T in long calculations. It 151 152 CHAPTER 17. FINITE VOLUME METHOD is imperative that the discretization does not introduce spurios sources (this the prime imperative in climate models for example). A slightly different form of the equation called the conservative form can be derived and forms the starting point for the derivation of finite volume methods. Multiplying the continuity equation by T , adding it to the resultant equations to the advective form, and recalling that vectoru · ∇ T + T ∇ · vectoru = ∇ · ( vectoruT ) we can derive the conservative form of the transport- diffusion equation: ∂T ∂t + ∇ · ( vectoruT ) = ∇ · ( α ∇ T ) . (17.3) 17.2 Integral Form of Conservation Law a38a37 a39a36 δV δS a0 a0 a18 vectoru a54 vectorn Figure 17.1: Skectch of the volume δV and its bounding surface δS . The partial differential equation is valid at all points in the domain which we could consider as infitesimal volumes. Anticipating that infinitesimal discrete volumes are unaffordable and would have to be ”inflated” to a finite size, we proceed to derive the conservative form for a finite volume δV bounded by a surface δS as shown in figure 17.1; the integral yields: integraldisplay δV ∂T ∂t d V + integraldisplay δV ∇ · ( vectoruT ) d V = integraldisplay δV ∇ ·...
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- Fall '08