UNIVERSITY OF NOTRE DAME
Department of Civil Engineering
and Geological Sciences
CE 30125
November 18, 2010
J.J. Westerink
Due: November 30, 2010
Homework Set # 6
Corrections Nov. 23, 2010
Background:
Partial differentiation in a discrete context is implemented simply by holding the indices in the
directions other than those being differentiated constant. We can then apply all the one-dimensional discrete
differentiation formulae. Note that for a three dimensional spatial coordinate system (
x,y,z
), the discrete indices
would be (
i,j,k
).
Problem 1
Write down a second order accurate approximation to the following three dimensional partial differential equa-
tion at node (
i,j,k
)
Background:
Initial value problems are solved one step at a time (going from one time level to the next time
level is called
time stepping
). It involves writing a discrete algebraic form of the differential equation to
connect two time levels. Starting with the functional value specified as the initial condition, you advance from
one level to the next solving for one unknown at a time using the algebraic equation that you have developed.
Problem 2
Solve the problem
Using a forward Euler method for
with time steps equal to
,
, and
.
Compare, by plotting the three solutions and comment. Develop your own MATLAB or Fortran code to do this
problem.
∂
2
f
∂
x
2
-------
∂
2
f
∂
y
2
-------
∂
2
f
∂
z
2
-------
∂
2
f
∂
x
∂
y
-----------
∂
2
f
∂
y
∂
z
-----------
+
+
+
+
B x y z
,
,
(
)
=
dy
dt
-----
y
2
t
5
y
–
(
)
5
---------------
=
y
0
(
)
1
=
0
t
4
≤
≤
Δ
t
0.2
=
Δ
t
0.02
=
Δ
t
0.002
=

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