Use the de nition of the central di erence operator

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Unformatted text preview: derivative. 22 x D uj = 2 (u x xx j ::: u : 8 Finite Di erence Methods 2.2 Simple Di erence Schemes for a Kinematic Wave Equation We have developed more than enough nite di erence formulas to begin solving some simple problems. Let us begin with the kinematic wave propagation problem ut ;1 + ( ) x=0 auu <x< ;1 ( 0) = ( ) ux x 1 t> <x< 0 (2.2.1a) 1 (2.2.1b) where the wave speed ( ) is a real function of . We have neglected boundary conditions for our initial study, so, in order to have a nite spatial domain, we will require that ( ) either have compact support au u x jj () 0 x x >X or be periodic ( + )= ( ) x X x where is a positive constant. Perhaps the simplest strategy for solving (2.2.1) is to approximate both time and spatial derivatives by rst-forward di erences. As in Section 2.1, let us cover the halfplane 0 by a uniform rectangular space-time mesh having cells of size (Figure 2.1.1), evaluate (2.2.1a) at ( ), and use the forward di erence approximations (2.1.3, 2.1.8) to obtain X t> x j ( ) + ( )( ) = n ut j n a uj n ux j xn +1 ; un n uj j t t t...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.

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