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# 21 remark 2 well give a formal de nition of

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Unformatted text preview: Wave Equation 15 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 11 00 11 00 11 00 11 00 11 00 11 00 (j,n) 11 00 11 00 111 11 000 00 111 111 11 000 000 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 111 000 111 000 111 000 111 000 111 000 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 (j,n) 111 000 111 000 111 11 000 00 111 11 000 00 111 11 11 000 00 00 Figure 2.2.5: Domains of dependence of the solution of (2.2.1), the solution of (2.2.4) (left), and the solution of (2.2.2b) (right) for a point ( ). j xn t solution of the partial di erential equation as the space and time steps are reduced. Remark 3. Theorem 2.2.1 implies that (2.2.2b) is useless when is positive and that (2.2.4) is useless when is negative. Thus, we should maintain 0 for (2.2.4) and ; 0 for (2.2.2b). Stated in terms of the Courant number, we n n should maintain 0 j 1 for (2.2.4) and ;1 j 0 for (2.2.2b). Remark 4. The Courant, Friedrichs, Lewy Theorem is applicable under more general...
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