Homework 4: Space-Time Discretization and Stability
Due Thursday, November 28th at 1 am.
In this problem, we will work with the 1D wave equation
ut = κuxx, t ∈ [0, 1], x ∈ [0, 1], κ = 0.1.
Use periodic boundary conditions1 u(t, 0) = u(t, 1), and initial heat distribution
u(0, x) = sin(2pix).
Problem 1:
Implement the forward Euler method to solve the problem. We can write this method in shorthand as
Dtu
n
j = κD2xunj
and explicitly as
un+1j − unj
∆t
= κ
unj+1 − 2unj + unj−1
∆2x
We suggest you use linear algebra to implement this iteration. That is, define the vector
Un =

un1
un2
...
unm

and implement the forward Euler iteration as
Un+1 = Un + κ∆t∆2x
AUn
where A is a discretized operator.
Use dt = 0.05 and dx = 0.1. Save the solution u(1, x), a column vector of size 10, into A1.dat.
What happens if you try to take dt = dx = 0.05, or dt = dx = 0.01?
Problem 2:
Implement the backward Euler method to solve the problem. We can write this method in shorthand
as
Dtu
n
j = κD2xun+1j
and explicitly as
un+1j − unj
∆t
= κ
un+1j+1 − 2un+1j + un+1j−1
∆2x
Here, as in the previous problem, use linear algebra to set up the iteration. You will need to solve a
linear system in this case, since backward Euler is an implicit method.
Use dt = 0.05 and dx = 0.1. Save the solution u(1, x), a column vector of size 10, into A2.dat.
Use dt = 0.01 and dx = 0.01. Save the solution u(1, x), a column vector of size 100, into A3.dat.
1Periodic boundary conditions force simple modifications to discretized spatial operators. Last weeks’ homework will
1
Problem 3:
Implement the Crank-Nicolson method to solve the problem:
Dtu
n
j =
κ
2
(
D2xu
n
j +D2xuPj n+ 1
)
and explicitly as
un+1j − unj
∆t
= κ2∆2x
(
unj+1 − 2unj + unj−1 + un+1j+1 − 2un+1j + un+1j−1
)
Here, as in the previous problem, use linear algebra to set up the iteration.
Use dt = 0.05 and dx = 0.1. Save the solution u(1, x), a column vector of size 10, into A4.dat.
Use dt = 0.01 and dx = 0.01. Save the solution u(1, x), a column vector of size 100, into A5.dat.
Make a movie of the time evolution of your heat distributions. Play with how large a step size you can
get away with when working with the implicit methods. You don’t have to turn anything in for this
problem.
2  Email:51zuoyejun

@gmail.com