ATMS 581 / AMATH 586 — Spring 06
Homework 4
1. Suppose that the time derivative in the differential-difference equation
dφj
dt +cφj+1 −φj−1
2∆x= 0,
is approximated using a fourth-order Runge-Kutta Scheme. Determine the maxi-
mum value of c∆t/∆xfor which this scheme will be stable using the stability criteria
for the oscillation equation given in Table 2.2. Explain how this value can exceed
unity without violating the CFL condition.
2. Show the false 2D Lax-Wendroff scheme
δtφj+1
2+Uδ2xφj+V δ2yφj=U2∆t
2δ2
xφj+V2∆t
2δ2
yφj
for the approximation of the two-dimensional constant wind speed advection equa-
tion ∂ψ
∂t +U∂ψ
∂x +V∂ψ
∂y = 0,
is unstable for all ∆t. Here φj
m,n is the approximation to ψ(m∆x, n∆y, j∆t) and
the finite-difference operator notation is defined such that
δnxf(x) = f(x+n∆x/2) −f(x−n∆x/2)
n∆x.
(Hint: Do a Von Neumann analysis for an arbitrary 2D wave of the form φj
m,n =
Ajei(km∆x+ln∆y)and show there is at least one wave resolved on the mesh that is
unstable.)
3. Compute solutions to the one-dimensional diffusion equation
∂φ
∂t =D∂2φ
∂x2
on the periodic domain 0 ≤x≤1, subject to the initial condition
ψ(x, 0) = sin(2πx) + cos(6πx)/2 + sin(20πx)/5 + R,
where Ris randomly distributed number in the interval [0,4×10−7] and D= 0.01.
Download the matlab file from the website. This code uses the trapezoidal method
and a standard second-order centered approximation to the second derivative to
compute the approximate solution over the time interval 0 ≤t≤1/4.
(a) Derive an expression for the exact solution to this problem.
(b) Modify this code so that it also computes and plots (i) the exact solution
and (ii) a second approximate solution computed using forward time differenc-
ing
φn+1
j−φn
j
∆t=D φn
j+1 −2φn
j+φn
j−1
(∆x)2!.
