CE 30125 - Lecture 15
LECTURE 15
APPLICATIONS OF FD APPROXIMATIONS FOR SOLVING ORDINARY
DIFFERENTIAL EQUATIONS
Ordinary Differential Equations
Initial Value Problems
For Initial Value problems (IVPs), conditions are specified at only one value of the
ind
UNIVERSITY OF NOTRE DAME
Department of Civil Engineering
and Geological Sciences
CE 30125 Computational Methods
J.J. Westerink
Fall 2010
Sample Final Exams 2008 and 2009
NAME _
This exam is being conducted under the HONOR CODE. Please sign your exam to in
CE 30125 - Review 1
REVIEW NO. 1
TAYLOR SERIES
Find f x away from x = a given f a and the derivatives of f x evaluated at x = a
df
f x = f a + x a -dx
22
x a d f
+ - -2! dx 2
x=a
nn
x a d f
-+ + - -n! dx n
33
x=a
x a d f
+ - -3! dx 3
44
x=a
n+1
x=a
1
n+1
UNIVERSITY OF NOTRE DAME
Department of Civil Engineering
and Geological Sciences
CE 30125 Computational Methods
J.J. Westerink
Fall 2011
December 6th , 2011
Practice Exam #2
Name:
This exam is being conducted under the HONOR CODE. Please sign your exam to
UNIVERSITY OF NOTRE DAME
Department of Civil Engineering
and Geological Sciences
CE 30125 Computational Methods
J.J. Westerink
Fall 2011
December 6th , 2011
Practice Exam #2
Name:
This exam is being conducted under the HONOR CODE. Please sign your exam to
UNIVERSITY OF NOTRE DAME
Department of Civil and Environmental Engineering
and Earth Sciences
October 5th , 2012
Due: October 12th , 2012
CE 30125 Computational Methods
J.J. Westerink
Homework Set #5
Background: Since f (x) = g (x) + e(x) we can estimate
UNIVERSITY OF NOTRE DAME
Department of Civil and Environmental Engineering
and Earth Sciences
CE 30125 Computational Methods
J.J. Westerink
Homework Set #6 Solutions
Background: Initial-boundary value problems are typically solved from time level to the n
UNIVERSITY OF NOTRE DAME
Department of Civil and Environmental Engineering
and Earth Sciences
August 30th , 2012
Due: September 6th , 2012
CE 30125 Computational Methods
J.J. Westerink
Homework Set #1
Background: A T aylor Series provides a polynomial app
UNIVERSITY OF NOTRE DAME
Department of Civil and Environmental Engineering
and Earth Sciences
September 7th , 2012
Due: September 14th , 2012
CE 30125 Computational Methods
J.J. Westerink
Homework Set #2
Background: Newton Forward Interpolation reformulat
FINAL EXAM SOLUTIONS
Problem 1:
Solve the system of equations using LU dccampasitian. M'ake sure that you solve these equations such
that accuracy is ensured.
.711 ~ mg + 39:3 = 6
211:1 m $2 + $3 =- 4
3.11 + 432 ' $3 = 5
Rearranging for diagonal domin
CE 30125 - Lecture 16
LECTURE 16
NUMERICAL SOLUTION OF THE TRANSIENT DIFFUSION EQUATION USING THE FINITE DIFFERENCE (FD) METHOD
Solve the p.d.e.
2
u
u
- = D -2
t
x
Initial conditions (i.c.s)
u ( x, t = t o ) = u * o ( x )
Boundary conditions (b.c.s)
u
CE 30125 - Lecture 17
LECTURE 17
NUMERICAL INTEGRATION
Find
b
I=
f x dx
a
or
b
I=
vx
a
f x y dy dx
ux
Often integration is required. However the form of f x may be such that analytical
integration would be very difficult or impossible. Use numerical in
CE 30125 - Review 2
REVIEW NO. 2
NUMERICAL DIFFERENTIATION
Find a discrete approximation to differentiation
Use numerical differentiation to solve o.d.e.s and p.d.e.s on a computer
Recall that a computer doesnt do differential/integral mathematics and
CE 30125 - Review 3
REVIEW NO. 3
O.D.E. CLASSIFICATION
I.V.P.s
d2 y
dy
A - + B - + Cy = g t
dt
dt 2
y 0 = yo
dy
- 0 = V o
dt
d2y
dy
A - + B - + Cy = g x
dx
dx 2
y 0 = yo
y L = yL
B.V.P.s
st
Can always decompose an nth order i.v.p. into n simultaneous
CE 30125:
2008 TEST 1 SOLUTIONS
1
Problem 1
Solve the system of equations using a fast iterative solution method.
2
3
5
49
2
x1
8 4 x2 = 6
32
x3
7
Rearranging the matrix to make it diagonally dominant yields the following system:
2
942
x3
4 8 3 x2 = 6
UNIVERSITY OF NOTRE DAME
Department of Civil Engineering
and Geological Sciences
CE 30125 Computational Methods
J.J. Westerink
Fall 2008/2009
SAMPLE Test 1
NAME _
This exam is being conducted under the HONOR CODE. Please sign your exam to indicate that yo
2008 TEST 2 SOLUTIONS
December 5, 2008
Problem 1:
Derive the rst order accurate backward dierence approximation to the third derivative at node j. Use
interpolating polynomials to derive your result
OPTION 1: Newton forward method
The number of nodes need
UNIVERSITY OF NOTRE DAME
Department of Civil Engineering
and Geological Sciences
CE 30125 Computational Methods
J.J. Westerink
December 4, 2008
Test 2
NAME _
This exam is being conducted under the HONOR CODE. Please sign your exam to indicate that you agr
UNIVERSITY OF NOTRE DAME
Department of Civil Engineering
and Geological Sciences
CE 30125 Computational Methods
J.J. Westerink
Fall 2010
September 29th , 2010
Exam #1
Name:
This exam is being conducted under the HONOR CODE. Please sign your exam to indica
UNIVERSITY OF NOTRE DAME
Department of Civil Engineering
and Geological Sciences
CE 30125 Computational Methods
J.J. Westerink
Fall 2011
October 13th , 2011
Exam #1
Name:
This exam is being conducted under the HONOR CODE. Please sign your exam to indicate
Matlab Tutorial for
Computational Methods CE 30125
prepared by
Aaron S. Donahue
[email protected]
This document is a list of basic and key Matlab commands that will be helpful in the Computational Methods course. The document is broken up into sections, eac
CE 30125 - Lecture 2
LECTURE 2
INTRODUCTION TO INTERPOLATION
Interpolation function: a function that passes exactly through a set of data points.
Interpolating functions to interpolate values in tables
x
sin(x)
0.0
0.000000
0.5
0.479426
1.0
0.841471
1.5
Matlab Tutorial for
Computational Methods CE 30125
prepared by
Aaron S. Donahue
[email protected]
Disclaimer: Programming in Matlab is a very long and deep subject. The following is a synopsis
of statements that will help with what is done in this class, bu
CE 30125 - Lecture 1
LECTURE 1
INTRODUCTION
Formulating a Mathematical Model versus a Physical Model
Formulate the fundamental conservation laws to mathematically describe what is physically occurring. Also define the necessary constitutive relationships
CE30125 - Lecture 3
LECTURE 3
LAGRANGE INTERPOLATION
Fit N + 1 points with an N
th
degree polynomial
g(x)
f2
f0
x0
f1
f3
x1 x2 x3
f4
x4 .
f(x)
fN
xN
f x = exact function of which only N + 1 discrete values are known and used to establish an interpolatin