HW4[1] - MAE 290B Winter 2010 HOMEWORK 3 Due Wed in class...

MAE 290B, Winter 2010
HOMEWORK 3
Due Wed 02-24-2010 in class
Provide source codes used to solve all problems
PROBLEM 1
Consider the advection - diffusion equation
∂
t
u + c∂
x
u = ν∂
xx
u,
1. Discretize this equation spatially using 2nd-order centered finite difference formulae
for the convective terms and the diffusive terms.
2. Discretize the resulting set of ODEs temporally using a low-storage, mixed RK3- θ
method. Express this discretization as a function of the two non-dimensional parate-
mers CN = c Δ t/ Δ x (Courant Number) and V N = ν Δ t/ (Δ x )
2
.
3. Apply Von-Neumann stability analysis to determine the stability of the resulting spatio-
temporal scheme. For this purpose,
(a) Find the amplification factor σ = u
n +1
j
/u n
j
as a function of CN , V N and k Δ x .
(b) Plot the contour | σ | = 1 as a function of k Δ x and CN for V N = 0 , 5 , 10 , 20.
Use this plot to discuss the stability of the scheme
(c) Consider now the case V N = 0. Find the maximum value of CN for which the
scheme is stable. Relate this result to the stability region of the RK3 scheme.