Characterizing Lotka-Volterra Solutions, Stability Analysis, and Lax Method in Homework - , Assignments of Meteorology

Download Characterizing Lotka-Volterra Solutions, Stability Analysis, and Lax Method in Homework - and more Assignments Meteorology in PDF only on Docsity! 1 Tue. Sep. 16, 2008 ATMS 502 / CS 505 / CSE 566 Jewett Homework #1 – due in class Tuesday, Sep. 23, 2008 • As noted in class, you may work with others on this assignment, but what you hand in must be your own work. No copying allowed! • Any of the following questions requesting a non-mathematical answer can be completed in a few sentences. Keep your answers succinct! • In any derivation, show your work. 1. Characterize the Lotka-Volterra solutions we looked at early in the class. What went wrong with our approach - what kind of errors were apparent? Under what circumstances would the method converge to the correct solution? Why? 2. Name two restrictions on use of Von Neumann’s stability analysis – or, if you wish, state two circumstances under which it should not be applied (or would not strictly be correct). 3. From Taylor series, derive a second-order accurate 1-sided expression for the first derivative, i.e. determine the coefficients (a, b, c) such that € df dx = fx = af j + bf j−1 + cf j−2 4. Consider the Lax method discussed in today’s (Tue 9/16) class. € u j n+1 − u j−1 n + u j+1 n( ) /2 Δt = −c u j+1 n − u j−1 n 2Δx a. Derive the truncation error (not the full modified equation). Under what circumstances is the scheme consistent? What is the order of accuracy? b. Derive the amplification factor. c. What is the amplification factor if β=k∆x=0? d. What is the amplification factor if β=k∆x=π? What wavelength (n∆x, where n is an integer) is this? Describe the amplitude behavior vs. time for β=k∆x=π.