14b is ux y t x at y bt 714c thus the domain

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Unformatted text preview: t) = (x ; at y ; bt): (7.1.4c) Thus, the domain of dependence of a point (x0 y0 t0 ) is the single point (x0 ; at0 y0 ; bt0 ). Let us consider a ve- or nine-point explicit di erence scheme of the form n Ujk+1 X XC = j +1 k+1 l=j ;1 m=k;1 n lm Ulm The domain of dependence of the mesh point (j k n + 1) for the nine-point scheme is the rectangle D = f(x y)jxj;1 x xj+1 yk;1 y yk+1g: (7.1.5a) A ve-point scheme would have Cj 1 k 1 = 0. Its domain of dependence would be less clear however, for simplicity, we'll de ne it as jx ; xj j jy ; yk j D = f(x y )j 1g: (7.1.5b) x+ y The domain of dependence of the point (j k n + 1) for the partial di erential equation is the single point (xj ; a t yk ; b t). The various domains of dependence are shown in Figure 7.1.2. Thus, the Courant, Friedrichs, Lewy condition is max( jaj x t jbj y t ) 1: (7.1.6a) for the nine-point scheme and jaj t jbj t (7.1.6b) x + y) 1 for the ve-point scheme. The Lax-Friedrichs scheme applied to (7.1.4a) yields 1 n Ujk+1 = 1 (1 ; 2 )Ujn+1 k + 1 (1 + 2 )Ujn;1 k + 4 (1 ; 2 )Ujnk+1 + 1 (1 + 2 )Ujnk;1 4 4 4 where = b yt : = a xt 7.1. Split and Unsplit Di erence Methods 5 k j Figure 7.1.2: Domain of dependence of nine-point (solid line) and ve-point (dashed line) di erence schemes and of the partial di erential equation ( ). Using the Maximum Principle, we nd a su cient condition for stability in the maximum norm as max(j j j j) 1 2 which is more restrictive than in one dimension. Using the von Neumann method (cf. Problem 1 at the end of this section), we may show that the Lax-Friedrichs scheme is stable in L2 when 2 + 2 1: 2 Example 7.1.1. With the more stringent sta...
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