The solution of the di erence scheme must be trivial

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Unformatted text preview: stant. The scheme (2.2.4) only depends on the Courant n number j which, for Examples 2.2.1 and 2.2.2, is the constant . Is there a particular choice of the Courant number on (0 1] that produces more accurate solutions than others? Answer the same question for (2.2.2b) when 0. a a t= x a< n 2. What restrictions, if any, should be placed on the Courant number j for the forward time-centered space scheme (2.2.5) to satisfy the Courant, Friedrichs, Lewy theorem? We must, of course, study the behavior of (2.2.5) however, prior to doing this in Chapter 3, experiment by applying the method to the problems of Examples 2.2.1 and 2.2.2. Use the same mesh spacings as the earlier examples. 2.3 A Simple Di erence Scheme for the Heat Equation Let us determine whether similar or di erent phenomena occur when solving a simple initial-boundary value problem for the heat conduction equation. In particular, consider ut = 0 uxx ( 0) = ( ) ux 1 <x< 0 x t> x 0 (2.3.1a) 1 (2.3.1b) (0 ) = (1 ) = 0 u t u (2.3.1c) t where the di us...
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