MAE 290B, Winter 2010
HOMEWORK 1
Due Mon 01182010 in class
Provide source codes used to solve all problems
PROBLEM 1
Consider the generalized family of Euler schemes
u
n
+1
=
u
n
+
αF
(
u
n
, t
n
) +
βF
(
u
n
+1
, t
n
+1
) Δ
t
with
α
+
β
= 1.
1. Show that the condition
α
+
β
= 1 makes these schemes at least firstorder accurate.
2. Show that the Implicit Euler method (
α
= 0
, β
= 1) is the most
∞
stable member of
this family while the CrankNicolson method (
α
=
β
= 1
/
2) is the most accurate one.
PROBLEM 2
Consider the ODE set
d
tt
u

u
+ 3
d
t
u
+ 5
w
=
t,
d
t
w

20
u
+ 10
w
= 0
,
(1)
with initial conditions
u
(0) = 0,
d
t
u
(0) =

1,
w
(0) = 1.
1. Find the exact analytical solution of this problem.
2. Determine the maximum value of the time step Δ
t
max
for which this ODE set can be
integrated using the Explicit Euler numerical scheme.
3. Integrate numerically this ODE set for 0
< t <
10 using Δ
t
= 2Δ
t
max
using the
Explicit Euler, Implicit Euler and Crank Nicolson schemes.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 MAE290B
 Numerical Analysis, Crank–Nicolson method, Numerical ordinary differential equations, exact analytical solution, John Crank

Click to edit the document details