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Differential Equations Lecture Work Solutions 326

# Differential Equations Lecture Work Solutions 326 - 1...

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1. Derive the modified equation for the Lax Wendroff method. u n +1 j u n j t + c u n j +1 u n j 1 2∆ x = c 2 t 2∆ x 2 u n j +1 2 u n j + u n j 1 (1) Expand each fraction in Taylor series (all terms on the right side of the following equations are evaluated at the point ( j, n ) u n +1 j u n j t = u t + t 2 u tt + t 2 6 u ttt + O (∆ t 3 ) c u n j +1 u n j 1 2∆ x = cu x + c x 2 6 u xxx + O (∆ x 4 ) c 2 2 t u n j +1 2 u n j + u n j 1 x 2 = c 2 2 tu xx + c 2 24 t x 2 u xxxx Now substitute these expansions in the Lax Wendroff scheme (1) u t + t 2 u tt + t 2 6 u ttt + cu x + c x 2 6 u xxx = c 2 2 tu xx + c 2 24 t x 2 u xxxx Reorganize u t = cu x + t 2 c 2 u xx u tt t 2 6 u ttt c x 2 6 u xxx + higher order terms (2) Differentiate this (2) with respect to t u tt = cu xt + t 2 c 2 u xxt u ttt + quadratic terms (3) Substitute in (2) u t = cu x + t 2 c 2 u xx + cu xt
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