Homework_6_2011

Homework_6_2011 - CS 111 - Introduction to Computational...

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Unformatted text preview: CS 111 - Introduction to Computational Science Homework 6 Consider the diffusion equation in the interval [0, 1]: ∂u ∂2u = D 2, ∂t ∂x (1) where D is the diffusion constant. Here, we will take D = .5. An exact solution for this equation is u(x, t) = e−Dt cos(x) and a good approximation can be written as: un+1 − 2un+1 + un+1 un+1 − un i i−1 i i . = D i+1 ∆t ∆x2 (2) 1. Write down the linear system corresponding to (2). 2. Write a code that solve the diffusion equation (1) using the scheme (2) for a final time of ttarget = 1. You can assume that the exact solution is given at the domain’s boundary, i.e. at x = 0 and at x = 1. Take a time step of ∆t = .1 and discretize the interval [0, 1] with 100 points. 3. Use your code to make a movie showing the evolution of your numerical solution along with the exact solution. What to turn in : the linear system from question 1, your code as well as your movie. This can be emailed to the TAs. 1 ...
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This note was uploaded on 12/29/2011 for the course CS 111 taught by Professor Staff during the Fall '08 term at UCSB.

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