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Unformatted text preview: The First-Order Wave Equation The first-order wave (advection) equation is ( c > 0) u t + c u x = 0 , u ( x, t = 0) = u ( x ) The solution propagates the initial data u to the right with speed c : u ( x, t ) = u ( x- ct ) The Riemann invariant u is constant along characteristics x = x + ct . Note that going backwards in time t - t simply propagates the initial data to the left with speed c. The upwind numerical method u n +1 i- u n i t =- c u n i- u n i- 1 x u n +1 i = u n i- c t x u n i- u n i- 1 is first-order accurate and stable for t x/c (CFL condition). Consistency: The LTE = t is given by ( h = x ) u ( x, t + t ) = u ( x, t )- c t h ( u ( x, t )- u ( x- h, t )) + t Taylor expanding, we get u + t u t + t 2 2 u tt + = u- c t u x- h 2 u xx + + t = t 2 u tt- ch 2 u xx + Stability: To analyze stability, we will derive the modified equation for the upwind method. In the Taylor expansion above with the LTE on the RHS, we delete the LTE and then do not use the wave equation: u + t u t + t 2 2 u tt +...
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- Fall '11