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Unformatted text preview: MPO 662 – Problem Set 1 Feel free to use symbolic computation software such as mathematica or the symbolic toolbox in matlab to carry out the algebra. See 1 to see an example with matlab. 1 Finite Difference Derivation Use Taylor series to derive explicit finite difference approximations according to the following specifications: • A 3 rd order approximation to the first derivatives using the points i- 2 ,i- 1 ,i and i +1. u x = u j- 2- 6 u j- 1 + 3 u j + 2 u j +1 6Δ x + O (Δ x 3 ) (1) • Use the same points as above and derive the highest possible approximation to the third derivative. u xxx =- u j- 2 + 3 u j- 1- 3 u j + u j +1 Δ x 3 + O (Δ x ) (2) • Derive a 4 th order centered approximations to the second derivative, using points i , i ± 1, and i ± 2. u xx =- ( u j- 2 + u j +2 ) + 16( u j- 1 + u j +1 )- 30 u j 12Δ x 2 + O (Δ x 4 ) (3) 2 Programming discretization of a grid Write a module (call it grid.f90) that contains a single subroutine that discretizes the interval x min ≤ x ≤ x max into N equally spaced cells with grid spacing Δ...
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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