13a 2 y l m1 l m1 applied to the four points

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Unformatted text preview: +1 l m;1 applied to the four points surrounding (j k) (Figure 7.1.1). The corrector step is the leap frog method t +1 t +1 Un+2 = Un ; x (fjn+1 k ; fjn;1 k ) ; y (gjn+1 ; gjn+1 1): (7.1.3b) jk jk k+1 k; The computational stencil of (7.1.3) is shown in Figure 7.1.1. The mesh spacing t, x, and y has been doubled relative to the one-dimensional scheme (6.4.1). This was done to simplify the writing of the scheme. As shown on the bottom of Figure 7.1.1, the Richtmyer two-step scheme can be regarded as a nine-point di erence formula on a staggered grid. This is possible because the physical ux vector and the divergence operator are rotationally invariant. Halving the mesh spacing to get a scheme that is more in line with our usual notation is much simpler with this staggered grid interpretation. The Courant, Friedrichs, Lewy Theorem is still available to restrict the domain of dependence of a di erence scheme to contain that of the partial di erential equation. For example, the solution of the model initial value problem ut + aux + buy = 0 ;1 < x y<1 t>0 (7.1.4a) 7.1. Split and Unsplit Di erence Methods 3 k j k j Figure 7.1.1: Computational stencil of the Richtmyer two-step method (7.1.3). Predicted solutions are shown with lled circles and corrected solutions are shown with blue circles and corrected solutions are shown in red. The Richtmyer two-step scheme can be regarded as a nine-point formula on a staggered grid (bottom). 4 Multidimensional Hyperbolic Problemss u(x y 0) = (x y) ;1 < x < 1 (7.1.4b) is u(x y...
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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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