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Unformatted text preview: bility restrictions, methods that use operator splitting become attractive. Let us consider splitting MacCormack's predictorcorrector scheme (6.4.2) for the conservation laws (7.1.2). In the x-direction, we neglect
g, predict using forward time-backward space di erencing, and correct using backward
time-forward space di erencing for one half time step to obtain tn
Un+1 = Un ; x (fjk ; fjn;1 k )
jk Un+1 =
jk ^ jk
Un + Un+1
2 ; t (^n+1 ; ^n+1)
2 x j+1 k jk (7.1.7)
(7.1.8) 6 Multidimensional Hyperbolic Problemss ^ jk
where ^jk+1 f (Un+1 ). The solution obtained at the conclusion of sweeping the mesh by
rows is denoted as Un+1. The sweep in the y-direction follows the same pattern with f
neglected and g included and the solution Un+1 used as initial data. Thus,
Un+1 = Un+1 ; y (gjk+1 ; gjn+1 1)
Un+1 + Un+1
jk (7.1.9) t (~ n+1 ; gn+1):
2 y gj k+1 jk
This split scheme merely needs to satisfy the one-dimensional stability conditions when
the boundary conditions between the x and y sweeps are implemented with care (cf.
Section 5.2). Un+1
jk = ; Problems 1. Analyze the stability of the Lax-Friedrichs scheme (7.1.2) for the kinematic wave
equation (7.1.4) using von Neumann's method....
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
- Spring '14