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Unformatted text preview: Chapter 14 Finite Volume Method and Time Discretization 14.1 Introduction The previous chapter focussed primarily on the issues of turning the partial dif- ferential equation into a finite volume formulation, and on the various ways of reconstructing the function at cell edges from cell values. The result is a spatially discretized ordinary differential equation that we need to integrate in time. Here we focus on identifying suitable time-discretization that are both accurate and sta- ble. The stability analysis is however, more complicated now since we are dealing with systems of ordinary differential equations. Hence instead of a scalar amplifi- cation factor we have an amplification matrix whose impact on the growth of the numerical solution must be investigated. This requires the analysis of the structure of the matrix and its eigenvalues. The Von-Neumann stability analysis provide a simpler approach to study the time-stability of the finite difference scheme. 14.2 The Forward Euler ODE system For the sake of simplicity (and later on necessity) we assume that the flow field is constant and positive ( u > 0). Furthermore we need an initial condition which we assume to be T j ( t = 0) = T j , and an inlet boundary condition which take as as T n 1 2 = T n i where T j and T n i are known quantities. The piecewise constant FV formulation becomes then: d T j d t =- u T j- T i Δ x , j = 1 (14.1) d T j d t =- u T j- T j- 1 Δ x , j = 2 , 3 , . . . , N (14.2) 141 142 CHAPTER 14. FINITE VOLUME METHOD AND TIME DISCRETIZATION Note that j = 1 is special because of the need to apply the inlet boundary condition. The above is a system of ODE that must be advanced in time with some suitable time-stepping scheme. The Forward Euler scheme, even though it is only first order in time, is one of the simplest scheme which allow the unknowns to be “marched” in time explicitly. So the fully discrete system takes the form:in time explicitly....
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This note was uploaded on 01/08/2012 for the course MSC 321 taught by Professor Staff during the Fall '08 term at University of Miami.
- Fall '08