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ENGR90024 COMPUTATIONAL FLUID DYNAMICS
Lecture O01
Ordinary Differential Equations (ODEs):!
Introduction & Taylor series!
(Pages 1-4 of the Printed Lecture Notes)!
!
Mathematical equations that govern fluid flow
OpenFoam
Supercomputer
x PREFACE
(a) (b)
(c) (d)
(e) (f)
Figure 1: Some example applications in science and engineering which involve thesolution of ordinary and partial di↵erential equations.
x PREFACE
(a) (b)
(c) (d)
(e) (f)
Figure 1: Some example applications in science and engineering which involve thesolution of ordinary and partial di↵erential equations.
x PREFACE
(a) (b)
(c) (d)
(e) (f)
Figure 1: Some example applications in science and engineering which involve thesolution of ordinary and partial di↵erential equations.
Lecture & Workshop timetable3 MODULES!•ORDINARY DIFFERENTIAL EQUATIONS (ODE)!•PARTIAL DIFFERENTIAL EQUATIONS (PDE)!•OPENFOAM(HTTP://WWW.OPENFOAM.ORG)
Lecture & Workshop timetable
Week Starting Tue-morning Tues-afternoon Wed-Workshop Thu
1 Mar 3 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi
Workshop on Linux/C++/ODEs
Lecture on ODEs Prof Andrew Ooi
2 Mar 10 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on ODEs Lecture on OpenFOAM
Dr Stephen Moore
3 Mar 17 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
4 Mar 24 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on ODEs Lecture on OpenFOAM
Dr Stephen Moore
5 Mar 31 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
6 Apr 7 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson
Consultation for Assignment 1
Lecture on PDEs A/Prof Malcolm Davidson
7 Apr 14 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on OpenFOAM Lecture on PDEs
A/Prof Malcolm Davidson
Apr 21 Break
Break
Break
Break
8 Apr 28 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on PDEs Lecture on PDEs
A/Prof Malcolm Davidson
9 May 5 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on PDEs Lecture on PDEs
A/Prof Malcolm Davidson
10 May 12 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on PDEs Lecture on OpenFOAM
Dr Stephen Moore
11 May 19 Lecture on OpenFOAM Dr Stephen Moore
Lecture on OpenFOAM Dr Stephen Moore Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
12 May 26 Lecture on OpenFOAM Dr Stephen Moore
Lecture on OpenFOAM Dr Stephen Moore Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
Assignment 1 due!18th April
Assignment 2 due!30th May
Lecture & Workshop timetable
Week Starting Tue-morning Tues-afternoon Wed-Workshop Thu
1 Mar 3 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi
Workshop on Linux/C++/ODEs
Lecture on ODEs Prof Andrew Ooi
2 Mar 10 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on ODEs Lecture on OpenFOAM
Dr Stephen Moore
3 Mar 17 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
4 Mar 24 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on ODEs Lecture on OpenFOAM
Dr Stephen Moore
5 Mar 31 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
6 Apr 7 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson
Consultation for Assignment 1
Lecture on PDEs A/Prof Malcolm Davidson
7 Apr 14 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on OpenFOAM Lecture on PDEs
A/Prof Malcolm Davidson
Apr 21 Break
Break
Break
Break
8 Apr 28 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on PDEs Lecture on PDEs
A/Prof Malcolm Davidson
9 May 5 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on PDEs Lecture on PDEs
A/Prof Malcolm Davidson
10 May 12 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on PDEs Lecture on OpenFOAM
Dr Stephen Moore
11 May 19 Lecture on OpenFOAM Dr Stephen Moore
Lecture on OpenFOAM Dr Stephen Moore Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
12 May 26 Lecture on OpenFOAM Dr Stephen Moore
Lecture on OpenFOAM Dr Stephen Moore Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
Assignment 1 due!18th April
Assignment 2 due!30th May
Doug McDonell-309!11am-12pm
Lecture & Workshop timetable
Week Starting Tue-morning Tues-afternoon Wed-Workshop Thu
1 Mar 3 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi
Workshop on Linux/C++/ODEs
Lecture on ODEs Prof Andrew Ooi
2 Mar 10 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on ODEs Lecture on OpenFOAM
Dr Stephen Moore
3 Mar 17 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
4 Mar 24 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on ODEs Lecture on OpenFOAM
Dr Stephen Moore
5 Mar 31 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
6 Apr 7 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson
Consultation for Assignment 1
Lecture on PDEs A/Prof Malcolm Davidson
7 Apr 14 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on OpenFOAM Lecture on PDEs
A/Prof Malcolm Davidson
Apr 21 Break
Break
Break
Break
8 Apr 28 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on PDEs Lecture on PDEs
A/Prof Malcolm Davidson
9 May 5 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on PDEs Lecture on PDEs
A/Prof Malcolm Davidson
10 May 12 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on PDEs Lecture on OpenFOAM
Dr Stephen Moore
11 May 19 Lecture on OpenFOAM Dr Stephen Moore
Lecture on OpenFOAM Dr Stephen Moore Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
12 May 26 Lecture on OpenFOAM Dr Stephen Moore
Lecture on OpenFOAM Dr Stephen Moore Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
Assignment 1 due!18th April
Assignment 2 due!30th May
Alan Gilbert Theatre 2!5:15pm-6:15pm
Lecture & Workshop timetable
Week Starting Tue-morning Tues-afternoon Wed-Workshop Thu
1 Mar 3 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi
Workshop on Linux/C++/ODEs
Lecture on ODEs Prof Andrew Ooi
2 Mar 10 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on ODEs Lecture on OpenFOAM
Dr Stephen Moore
3 Mar 17 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
4 Mar 24 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on ODEs Lecture on OpenFOAM
Dr Stephen Moore
5 Mar 31 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
6 Apr 7 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson
Consultation for Assignment 1
Lecture on PDEs A/Prof Malcolm Davidson
7 Apr 14 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on OpenFOAM Lecture on PDEs
A/Prof Malcolm Davidson
Apr 21 Break
Break
Break
Break
8 Apr 28 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on PDEs Lecture on PDEs
A/Prof Malcolm Davidson
9 May 5 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on PDEs Lecture on PDEs
A/Prof Malcolm Davidson
10 May 12 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on PDEs Lecture on OpenFOAM
Dr Stephen Moore
11 May 19 Lecture on OpenFOAM Dr Stephen Moore
Lecture on OpenFOAM Dr Stephen Moore Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
12 May 26 Lecture on OpenFOAM Dr Stephen Moore
Lecture on OpenFOAM Dr Stephen Moore Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
Assignment 1 due!18th April
Assignment 2 due!30th May
Doug McDonell-503!2:15pm-3:15pm
Lecture & Workshop timetable
Week Starting Tue-morning Tues-afternoon Wed-Workshop Thu
1 Mar 3 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi
Workshop on Linux/C++/ODEs
Lecture on ODEs Prof Andrew Ooi
2 Mar 10 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on ODEs Lecture on OpenFOAM
Dr Stephen Moore
3 Mar 17 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
4 Mar 24 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on ODEs Lecture on OpenFOAM
Dr Stephen Moore
5 Mar 31 Lecture on ODEs Prof Andrew Ooi
Lecture on ODEs Prof Andrew Ooi Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
6 Apr 7 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson
Consultation for Assignment 1
Lecture on PDEs A/Prof Malcolm Davidson
7 Apr 14 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on OpenFOAM Lecture on PDEs
A/Prof Malcolm Davidson
Apr 21 Break
Break
Break
Break
8 Apr 28 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on PDEs Lecture on PDEs
A/Prof Malcolm Davidson
9 May 5 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on PDEs Lecture on PDEs
A/Prof Malcolm Davidson
10 May 12 Lecture on PDEs A/Prof Malcolm Davidson
Lecture on PDEs A/Prof Malcolm Davidson Workshop on PDEs Lecture on OpenFOAM
Dr Stephen Moore
11 May 19 Lecture on OpenFOAM Dr Stephen Moore
Lecture on OpenFOAM Dr Stephen Moore Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
12 May 26 Lecture on OpenFOAM Dr Stephen Moore
Lecture on OpenFOAM Dr Stephen Moore Workshop on OpenFOAM Lecture on OpenFOAM
Dr Stephen Moore
Assignment 1 due!18th April
Assignment 2 due!30th May
Old Engineering - EDS 1!12:00pm-2:00pm
Assessment
•10% Workshops (weekly)-1% satisfactory completion of every workshop-11 workshops in total so can afford to miss 1
•10% Assignment 1 (Due 18th April)-ODE & OpenFOAM
•20% Assignment 2 (Due 30th May)-PDE & OpenFOAM
•60% end of semester exam -ODE, PDE & OpenFOAM-Exam is a Hurdle. Need to pass the exam to pass this subject!!!!
Student comments“Open foam should be introduced in the workshops much earlier than they were”
• Last year, OpenFOAM workshops were not introduced until week 4• We will be introducing OpenFOAM workshop in week 3. There could also be
! OpenFOAM material in workshops in weeks 1 and 2.
“Assignments should have been released earlier”• Assignment 1 (due week 7) will be released week 1 or 2 of the semester• We will release assignment 2 in week 8 (last year was released in week 10).
Ordinary Differential EquationsIn the most generic form, ordinary differential equations (ODE) can bewritten as
g
✓dN�
dtN, · · · , d
3�
dt3,d2�
dt2,d�
dt,�, t
◆= 0
where N is the order of the ODE.
a(t)d3�
dt3+ b(t)
d2�
dt2+ c(t)
d�
dt+ d(t)�+ e(t) = 0
then the ODE is linear.
(O01.1)
(O01.2)
If every term in the ODE is linear, then the ODE is linear. If even one!term of the ODE is nonlinear, then the ODE is nonlinear.
If g is a linear function, e.g.
If g is a nonlinear function, then the ODE is nonlinear.
Example O01.1: For the ODEs shown below, state the order of the ODE and if the ODE is linear/nonlinear. a, b and c are constants.!
(a) t2d2�
dt2+ at
d�
dt+ b� = c
(b)d2�
dt2� a
�1� �2
� d�dt
+ � = 0
(a) t2d2�
dt2+ at
d�
dt+ b� = c
Let’s look at equation (a).
Linear Linear Linear
Linear
Since all terms in the above equations are linear, the ODE is a linear ODE
Let’s now look at (b)
(b)d2�
dt2� a
�1� �2
� d�dt
+ � = 0
d2�
dt2� a
d�
dt+ �2 d�
dt+ � = 0
LinearLinear
Linear
Nonlinear
Let’s now look at (b)
(b)d2�
dt2� a
�1� �2
� d�dt
+ � = 0
d2�
dt2� a
d�
dt+ �2 d�
dt+ � = 0
LinearLinear
Linear
Nonlinear
Since the third term in the above equation is nonlinear, the ODE is a nonlinear ODE!
End of Example O01.1
In many engineering problems, Eq. (O01.1) can be written as
Eq. (O01.3) is solved usually solved in the domain
with the initial condition
(O01.3)d�
dt= f(t,�)
tmin
t tmax
�(tmin) = �min
Initial Value Problem
d�
dt= f(t,�)
tmin
t tmax
�(tmin) = �min
Initial Value Problem
Mathematical MethodsExact (true) solution
Approximate solution
Numerical/Computer Methods
Compare
Example O01.2: !Solve the following ordinary differential equation (ODE)!!!!!with ɸ(t=0)=0. Use MATLAB to plot the solution for 0 < t < 8.
d�
dt= 1� � (O01.4)
d�
dt= f(t,�)
tmin
t tmax
�(tmin) = �min
Initial Value Problem
Mathematical MethodsExact (true) solution
Approximate solution
Numerical/Computer Methods
Compare
d�
dt= 1� �
d�
dt+ � = 1
Use integrating factor, multiply both sides by et
etd�
dt+ et� = et
d
dt
�et�
�= et
Integrating gives
1
et� = et +K
Where K is a constant. We are given that ɸ=0 when t=0. Substituting!into the above equation gives
1⇥ 0 = 1 +K
K = �1Thus the solution to Eq. (O01.4) is
� = 1� e�t
e0 ⇥ 0 = e0 +K
function MPO01p2() !!t=0:0.1:8; phi=1-exp(-t); !plot(t,phi); !xlabel('t'); ylabel('\phi(t)');
t=[0.0 0.1 0.2 0.3 0.4........8.0]
phi=[0.0 0.0952 0.1813 0.2592 ........ 1.0]
1-e-0.0 1-e-0.2 1-e-8.0
1-e-0.1 1-e-0.3
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
φ(t)
End of Example O01.2
d�
dt= f(t,�)
tmin
t tmax
�(tmin) = �min
Initial Value Problem
Mathematical MethodsExact solution
Approximate solution
Numerical/Computer Methods
Compare
� = 1� e�t
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
φ(t)
d�
dt= f(t,�)
tmin
t tmax
�(tmin) = �min
Initial Value Problem Approximate solution
Numerical/Computer Methods
Taylor series
TheTaylor series expansion of a function ɸ(t) about the pointtl can be written as
tl+1=tl+Δt.
�(tl+1) = �(tl) +�t
1!
d�
dt
����tl+
�t2
2!
d2�
dt2
����tl+
�t3
3!
d3�
dt3
����tl+ . . .
(O01.5)�(tl+1) =1X
n=0
(tl+1 � tl)n
n!dn�
dtn
����tl
(O01.6)
Equation (O01.5) can be expanded to look like
Example O01.3: Find the Taylor series for the function!!!!Plot the first few terms of this function and show that you can get closer to the original function if you use the more terms in the series.
�(t) = sin(t)
�(tl+1) = �(tl) +�t
1!
d�
dt
����tl+
�t2
2!
d2�
dt2
����tl+
�t3
3!
d3�
dt3
����tl+ . . .
If tl=0, then tl+1=Δt=t
�(t) = �(0) +t
1!
d�
dt
����0
+t2
2!
d2�
dt2
����0
+t3
3!
d3�
dt3
����0
+ . . .
You are given that ɸ(t)=sin(t), so
ɸ(0)=sin(0)=0!dɸ/dt(0)=cos(0)=1.0!
d2ɸ/dt2(0)=-sin(0)=0.0!d3ɸ/dt3(0)=-cos(0)=-1.0!d4ɸ/dt4(0)=sin(0)=0.0!d5ɸ/dt5(0)=cos(0)=1.0
From Eq. (O01.6), the Taylor’s series can be written as
�(tl+1) = �(tl) +�t
1!
d�
dt
����tl+
�t2
2!
d2�
dt2
����tl+
�t3
3!
d3�
dt3
����tl+ . . .
If tl=0, then tl+1=Δt=t
�(t) = �(0) +t
1!
d�
dt
����0
+t2
2!
d2�
dt2
����0
+t3
3!
d3�
dt3
����0
+ . . .
You are given that ɸ(t)=sin(t), so
ɸ(0)=sin(0)=0!dɸ/dt(0)=cos(0)=1.0!
d2ɸ/dt2(0)=-sin(0)=0.0!d3ɸ/dt3(0)=-cos(0)=-1.0!d4ɸ/dt4(0)=sin(0)=0.0!d5ɸ/dt5(0)=cos(0)=1.0
From Eq. (O01.6), the Taylor’s series can be written as
Substituting into the Taylor series Eq. (O01.6) gives!
�(t) = sin(t) = t� (1/6)t3
+(1/120)t5
�(1/5040)t7
+ . . .
�(t) = t
−4 −3 −2 −1 0 1 2 3 4−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t
tφ
�(t) = t�(1/6)t3
−4 −3 −2 −1 0 1 2 3 4−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t
t−(1/6) t3φ
�(t) = t�(1/6)t3+(1/120)t5
−4 −3 −2 −1 0 1 2 3 4−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t
t−(1/6) t3+(1/120) t5φ
�(t) = t�(1/6)t3+(1/120)t5�(1/5040)t7
−4 −3 −2 −1 0 1 2 3 4−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t
t−(1/6) t3+(1/120) t5−(1/5040) t7φ
End of Example O01.3
Sometimes it is convenient to group all higher order terms of theTaylorsseries into a single term. For example,
�(tl+1) = �(tl) +�t
1!
d�
dt(tl) +
�t2
2!
d2�
dt2(tl) +
�t3
3!
d3�
dt3(tl) +
�t4
4!
d4�
dt4(tl) + . . .
can be written as
or
�(tl+1) = �(tl) +�t
1!
d�
dt(⇠2)
where tl < ξ1 < tl+1 , tl < ξ2 < tl+1 and ξ1 ≠ ξ2
(O01.7)
(O01.8)
�(tl+1) = �(tl) +�t
1!
d�
dt(tl) +
�t2
2!
d2�
dt2(⇠1)
�(tl+1) = �(tl) +�t
1!
d�
dt(tl) +
�t2
2!
d2�
dt2(tl) +
�t3
3!
d3�
dt3(tl) +
�t4
4!
d4�
dt4(tl) + . . .
�(tl+1) = �(tl) +�t
1!
d�
dt(tl)+
�t2
2!
d2�
dt2(⇠)
where ξ is somewhere between tl and tl+1 i.e. tl < ξ < tl+1
�(tl+1) = �(tl) +�t
1!
d�
dt(tl) +
�t2
2!
d2�
dt2(tl) +
�t3
3!
d3�
dt3(tl) +
�t4
4!
d4�
dt4(tl) + . . .
�(tl+1) = �(tl) +�t
1!
d�
dt(tl)+
�t2
2!
d2�
dt2(⇠)
where ξ is somewhere between tl and tl+1 i.e. tl < ξ < tl+1
Exercise O01.4: For the function!!!!!!
�(t) = sin(t)show the validity of Eqs. (O01.7) & (O01.8) by finding !finding values of ξ1 and ξ2 for tl=0.0 and Δt =π/2.
Equation (O01.8), repeated here
tl+1 = tl +�t
tl ⇠2 tl+1
�(tl+1) = �(tl) +�td�
dt(⇠2)
tl+1 = tl +�t
tl ⇠2 tl+1
�(tl+1) = �(tl) +�td�
dt(⇠2)
For this example
tl = 0.0�t = ⇡/2
So
�(t) = sin(t)
tl+1 = 0 + ⇡/2 = ⇡/2
Hence, equation (O01.8), says that
�(tl+1) = sin(tl+1) = sin(⇡/2) = 1
=
�(tl) = sin(tl) = sin(0) = 0
+1 0 ⇡/2
d�
dt(⇠2) = cos (⇠2)
cos (⇠2)
1 = 0 + ⇡/2 cos (⇠2)
0 ⇠2 ⇡/2
Illustrate graphically
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t
φ
�t
sin(⇡/2)d�
dt(⇠2 = 0)
sin(0) ⇡/2
sin(0) +�td�
dt(⇠2 = 0)
� = sin(t)
sin(0) +�td�
dt(⇠2 = 0) > sin(⇡/2)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t
φ
sin(⇡/2)
sin(0)
� = sin(t)d�
dt(⇠2 = ⇡/2)
�tsin(0) +�t
d�
dt(⇠2 = ⇡/2)
sin(0) +�td�
dt(⇠2 = ⇡/2) < sin(⇡/2)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t
φ
� = sin(t)
sin(⇡/2)
sin(0)
d�
dt(⇠2 = 1.2)
�t
sin(0) +�td�
dt(⇠2 = 1.2)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t
φ
� = sin(t)
sin(⇡/2)
sin(0)
d�
dt(⇠2 = 1.2)
�t
sin(0) +�td�
dt(⇠2 = 1.2)
sin(0) +�td�
dt(⇠2 = 1.2) 6= sin(⇡/2)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t
φ
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t
φ
⇠2 = 0
⇠2 = ⇡/2
sin(0) +�td�
dt(⇠2 = 0)
sin(0) +�td�
dt(⇠2 = ⇡/2)
sin(⇡/2)
sin(⇡/2)
From the top graph, it is shown that for ξ2=0, the equation
evaluates to a value greater than sin(p/2). From the bottom graph, it is shown that for ξ2= p/2, the equation
evaluates to a value less than sin(p/2). Hence, in order for
sin(0) +�td�
dt(⇠2)
sin(0) +�td�
dt(⇠2)
sin(0) +�td�
dt(⇠2) = sin(⇡/2)
0<ξ2<p/2.
sin(⇡/2) = sin(0) +
⇡
2
cos(⇠2)
cos(⇠2) =2
⇡
⇠2 = 0.8807
Finding the value of ξ2 is easy
�t �t
�t�t
⇠2 = 0.0 ⇠2 = 0.5
⇠2 = 0.7 ⇠2 = 0.8807
End of Example O01.4