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Unformatted text preview: t u +1 ; uj n ; 2 ( tt)n+ + ( n)
j uj au n x ; 2 ( xx)n+ ] = 0
x u : (2.2.2a)
Neglecting the second-order derivative terms in the local discretization errors, we obtain
the nite di erence equation
n Uj +1 ; U n
j t Solving for +( )
a Uj +1 ; Uj n Uj n x =0 : +1 , we obtain n Uj +1 n Uj n
= (1 + j ) n Uj ; nn
j Uj +1 (2.2.2b) 2.2. A Kinematic Wave Equation 9
n+1 n j j+1 Figure 2.2.1: Computational stencil of the forward time-forward space nite di erence
j ( n) a Uj t
x (2.2.3) : n
The parameter j is called the Courant number.
The nite di erence equation (2.2.2b) involves three points as indicated in the stencil
of Figure 2.2.1. It is easy to solve using the prescribed initial data (2.2.1b). Knowing
) for all , we calculate j1 for all of interest. Then, knowing j1 for all ,
we repeat the process to obtain j2 , etc.
Forward di erences are appropriate for approximating time derivatives in this type
of marching procedure, but it seems reasonable to also consider backward di erences or
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
- Spring '14