pb2_11 - MPO 662 Problem Set 4 1 Consistency and stability...

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MPO 662 – Problem Set 4 1 Consistency and stability You have reverse-engineered a computer program and deciphered that it updates the variable u according to the following rule: u n +1 j - u n j Δ t + c 3 u n j - 4 u n j - 1 + u n j - 2 x = 0 (1) Determine the continuous PDE this finite difference equation is trying to approximate. Determine the truncation error for this scheme and derive the leading terms of the modified equations. Determine the stability characteristics of this scheme. 2 The Lax-Friedrich Scheme The forward Euler centered-space approximation to the advection equation, u t + cu x = 0, is unstable. Lax proposed a simple modification to stabilize it: replacing u n j in the time-derivative by its spatial average: u n +1 j - u n j +1 + u n j - 1 2 Δ t + c u n j +1 - u n j - 1 x = 0 (2) Show that this scheme is conditionally consistent and stable. 3 Modified equation analysis The analysis of the modified advection equation produces expressions of the form u t + cu x = a 2 u xx + a 3 u xxx + . . . + a n n u ∂x n (3) where a n are constants that depend on the grid spacing and time steps. Present heuristic arguments that the odd-derivative affect the phase properties of the scheme whereas the even derivatives affect its amplitude. 4 First-order conservative advection We will write a code to solve numerically the advection equation h t + uh x = 0 , on - 10 x 10 (4)
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-10 -8 -6 -4 -2 0 2 4 6 8 10 0 0.5 1 Figure 1: Initial conditions where h is the concentration of a pollutant, and u = 1 is the (constant) advective velocity, and with the initial condition: h ( x, 0) = max(0 , 1 - | x + 7 . 5
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