18.303 Problem Set
Due Wednesday, 10 November 2010.
Problem 1: Stability
(a) For the 1d diffusion equation
the following discretization:
∂2u
∂x2
=
∂u
∂t
on x ∈ [−∞, ∞], use Von Neumann analysis to analyze the stability of
un+1
− 12 (unm+1 + unm−1 )
un
− 2un+1
+ unm−1
m
m
.
= m+1
∆t
∆x2
That is, assume that the solution on the n-th timestep is un = g n u for an eigenvalue (“growth factor”) g and
eigenvector u, and solve for g under the assumption that um = eikx . Then find conditions on ∆t (if any) such
that it is stable (no exponential growth, i.e. |g| ≤ 1). What is the order of accuracy, in time and space, of this
discretization? If you get |g| = 1 anywhere, what does this mean about the solutions (especially as ∆x and ∆t
→ 0)?
∗
(b) Suppose we are discretizing a wave equation ∂w
∂t = D̂w for some D̂ = −D̂. First, we discretize D̂ in space,
T
replacing it by some matrix D to obtain an ODE du
dt = Du; assume D is real and D = −D (i.e, we have chosen
the discretization to mimic the properties of D̂), and thus D has purely imaginary eigenvalues λ = iω. Now
consider the Crank–Nicolson discretization:
un+1 − un
un+1 + un
=D
.
∆t
2
(As shown in class, this is 2nd-order accurate in time and space.) Show that this is unconditionally stable
(different from the proof in class: in class, we did it for a matrix A that was negative-definite). Do the solutions
of this discretized equation grow, decay, or...? How does this compare to the exact equation ∂w
∂t = D̂w?
Problem 2: Waves, boundary conditions, and conservation laws
2
In class (and notes), we showed that we can turn the scalar wave equation b∇ · (a∇u) = ∂∂t2u (a > 0 and b > 0) into
∂v
two coupled first-derivative equations: ∂u
∂t = b∇ · v, ∂t = a∇u by introducing a new (vector) unknown v(x, t). By
T
defining w = (u, v) , we obtained the form
∂w
b∇·
=
w = D̂w,
a∇
∂t
where D̂ was anti-Hermitian under the inner product hw, w0 i =
tions (e.g. u|dΩ = 0).
R
Ω
1
0
b ūu
+ a1 v̄ · v0 , for appropriate boundary condi-
(a) Suppose u|dΩ = 0. What boundary condition does this imply for v?
(b) Suppose v · n̂|dΩ = 0, where n̂ is a unit vector perpendicular to the dΩ boundary. What boundary conditions
does this imply for u?
(c) In class, we showed that D̂∗ = −D̂ implies “conservation of energy:” hw, wi must be constant in time if ∂w
∂t = D̂w.
Here, you will give a different conservation law: just as we saw in class for the in the diffusion equation, for
any function n(x) in the left nullspace D̂∗ n = 0 = −D̂n, show that hn, wi is constant in time if ∂w
∂t = D̂w.
Describe the left nullspace of the operator D̂ above, for the boundary conditions in part (b), and give the resulting
conservation law(s). [Tip 1: the nullspace of ∇· in 3d is the curl of any vector field. Tip 2: for any vector fields
F and G, ∇ · (F × G) = G · (∇ × F) − F · (∇ × G), which allows you to “integrate by parts” on F · (∇ × G).]
1