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Unformatted text preview: Homework 4 PHZ 5156 Due Thursday, October 1 1. Consider the diffusion equation in one spatial dimension, u ( x,t ) t = D 2 u ( x,t ) x 2 with boundary conditions u ( x = 0 ,t ) = u ( x = L,t ) = 0. Write a code that uses the Crank-Nicholson method, as discussed in class, to integrate the diffusion equation with the given boundary and initial conditions. First, convince yourself that we can write the algorithm as, 1 2 ( u n +1 j + u n j )- ( u n +1 j +1 + u n j +1 ) + 2 ( u n +1 j + u n j )- ( u n +1 j- 1 + u n j- 1 ) = u n j where = D t 2 x 2 , the j represent the discrete spatial step, and n represent the discrete time step. Take x j = j x for the spatial discretization, with x = L N +1 . Then the boundary conditions are u n j =0 = 0 and u n j = N +1 = 0. Write the above N linear equations as a matrix equation, Q n = U n where Q is an N N square matrix and n and U n are both column vectors with length N . Show that Q is a symmetric, tridiagonal matrix, with diagonal elements...
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This note was uploaded on 08/08/2011 for the course PHZ 5156 taught by Professor Johnson,m during the Fall '08 term at University of Central Florida.
- Fall '08