Programming assignment#3. (Due Nov 21.)
Math. 609
We consider applying explicit and implicit methods for solving a mildly stiff
system of ODE’s in this assignment. Specifically, we consider the system
Y=Y(t) : [0,1] →Rnwhere Yis the solution to
(0.1) Yt+(n+ 1)2
5AY =0
Y(0) = Y0≡(1,1,...,1)t.
Here Ais the n×nmatrix given by
Aij =
2 if i=j,
−1 if |i−j|= 1,
0 otherwise.
Problem 1. Show that the set of vectors {φ1,...,φn} ⊂ Rnwith compo-
nents given by
(φi)j= sin ijπ
n+ 1, i, j = 1,...,n
are eigenvectors for Aand compute the corresponding eigenvalues. Write
down the solution to (0.1) given the expansion of the initial data,
Y0=
n
X
i=1
ciφi.
You need not compute the coefficients {ci}above (and they will appear in
your expansion of the solution). Note that every component of the solution
decays exponentially as tincreases.
Problem 2. Given a time step size k, let Yidenote the sequence obtained
by applying the forward Euler method to (0.1). Give an expression for the
coefficients of Yiexpanded in the basis of eigenvectors. The forward Euler
method is considered stable for this problem provided that every component
of the solution decays. Find a “resonably sharp (see the next problem)”
condition on the time step size as a function of nwhich guarantees stability.
Problem 3. Code the forward Euler method applied to the ODE (0.1).
You can use CRS to store the matrix or simply store the diagonals in three
vectors. The forward Euler method only requires matrix evaluation and
is simple to code using either storage mode (note that the generation of
a full n×nmatrix is unacceptable). Apply your method to the cases of
1
