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Unformatted text preview: ure 2.3.3. r x : : As shown in Tables 2.3.1 and 2.3.3, the solution of (2.3.3a) with = 0 1 is producing a reasonable approximation of the exact solution. The larger errors at = 0 5 for r : x : 20 Finite Di erence Methods small times are due to the discontinuity in the initial data there. When = 1, the nite di erence solution bears little resemblance to the exact solution. As with the forward time-forward space scheme (2.2.2b) for the kinematic wave equation (2.2.1), it is oscillatory and increasing in amplitude. Apparently, some restriction must be placed on however, the explanation for this restriction is not as simple as it was with (2.2.1). We'll discuss it in the next chapter. r r 2.2. A Heat Equation 21 1 0.8 U 0.6 0.4 0.2 0 0.01 0.008 1 0.006 0.8 0.6 0.004 0.4 0.002 0.2 0 t 0 x 3 2 U 1 0 −1 −2 0.04 1 0.03 0.8 0.02 0.6 0.4 0.01 t 0.2 0 0 x Figure 2.3.3: Solutions of Example 2.3.1 obtained by the forward time-centered space scheme (2.3.3a) with = 0 1 (top) and = 1 (bottom). r : r 22 Finite Di erence Methods Bibliography 1] R. Courant, K.O. Friedrichs, and H. Lewy. On the partial di erential equations of mathematical physics. Technical Report NYO-7689, New York University, New York, 1956. Translated by P. Fox. 2] R. Courant, K.O. Friedrichs, and H. Lewy. Uber die partiellen di erenzenglechungen der mathematischen physik. Mathematische Annalen, 100:32{74, 1956. Also see 1]. 3] L. Lapidus and G.F. Pinder. Numerical Solution of Partial Di erential Equations in Science and Engineering. Wiley-Interscience, New York, 1982. 23...
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