11 with the more stringent stability restrictions

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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 ^ jk Un+1 = Un ; x (fjk ; fjn;1 k ) jk Un+1 = jk ^ jk Un + Un+1 jk 2 ; t (^n+1 ; ^n+1) f f 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 fn rows is denoted as Un+1. The sweep in the y-direction follows the same pattern with f jk neglected and g included and the solution Un+1 used as initial data. Thus, jk tn ~ jk Un+1 = Un+1 ; y (gjk+1 ; gjn+1 1) jk k; ~ jk Un+1 + Un+1 jk (7.1.9) t (~ n+1 ; gn+1): ~ (7.1.10) 2 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.

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