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Unformatted text preview: 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 2Δ 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 2Δ 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)-10-8-6-4-2 2 4 6 8 10 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...
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This note was uploaded on 01/08/2012 for the course MPO 662 taught by Professor Iskandarani,m during the Spring '08 term at University of Miami.
- Spring '08