Persamaan Diferensial Parsial...
Transcript of Persamaan Diferensial Parsial...
Tim Ilmu Komputasi
Coordinator contact:Dr. Putu Harry [email protected]
Persamaan Diferensial
Parsial CNH3C3Week 9: Finite Di�erenceMethod for ODE (Intro to
FDM for PDE)
Motivation
The decay problem
The so-called decay problemis chosen as a �rst example to
demonstrate what numerical modelling is. In mathematical terms,
this problem can be expressed as:
dC
dt= −κ · C (1.1)
where C is concentration of substance, t is time, and κ is a positive
constant parameter.
Motivation
The decay problem
The physical interpretation:
dC
dt(1.2)
refer to the temporal change of concentration per unit time. And
−κ · C (1.3)
speci�es the temporal change. For κ = 0, there will no change in
time. With κ 6= 0 is negative since concentration is always a
positive quantity. Accordingly, C will gradually decrease with time
at a rate in propotion to concentration itself.
Motivation
Example
Grandma used to keep a 10-litre cartoon of milk in a storage room.
Every night I take away 10% of the milk concentration on a daily
basis and replace it by a tab water. The following table shows the
result of this example. Notation I is the Milk content and II is the
Milk concentration.
Motivation
Example
Day I II (%) Day I II (%)
0 10 100 12 2.82 28.2
1 9.0 90 13 2.54 15.4
2 8.1 81 14 2.29 22.9
3 7.29 72.9 15 2.06 20.6
4 6.56 65.6 16 1.85 18.5
5 5.9 59.0 17 1.67 16.7
6 5.31 53.1 18 1.5 15.0
7 4.78 47.8 19 1.35 13.5
8 4.3 43.0 20 1.22 12.2
9 3.87 38.7 21 1.09 10.9
10 3.49 34.9 22 0.98 9.8
11 3.14 31.4
Explicit and Implicit scheme
The decay problem
Now we are interested with the discrete problem of
dC
dt= −κ · C (2.1)
The exact solution of this problem with initial concentration C0 is
C (t) = C0exp(−κ · t) (2.2)
In this case, three schemes are proposed which are Explicit, Implicit
and Crank-Nicholson Scheme.
Explicit and Implicit scheme
The decay problem
Now we are interested with the discrete problem of
dC
dt= −κ · C (2.1)
The exact solution of this problem with initial concentration C0 is
C (t) = C0exp(−κ · t) (2.2)
In this case, three schemes are proposed which are Explicit, Implicit
and Crank-Nicholson Scheme.
Explicit and Implicit scheme
The decay problem
Now we are interested with the discrete problem of
dC
dt= −κ · C (2.1)
The exact solution of this problem with initial concentration C0 is
C (t) = C0exp(−κ · t) (2.2)
In this case, three schemes are proposed which are Explicit, Implicit
and Crank-Nicholson Scheme.
Explicit and Implicit scheme
Discrete domain
The equation contains only one unknown variable which is t time
(C (t)), thus we only consider the discrete of time domain
(t ∈ [0,T ])
t=0- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -t=T
n-1 n n+1
Then we have a constant time step ∆t, thus we denote
C (t) ≈ Cn and C (t + ∆t) ≈ Cn+1 (2.3)
where n ∈ 0, 1, 2, · · ·N, with N = T/∆t.
Explicit and Implicit scheme
Explicit �nite di�erence
Thus we have �rst order �nite di�erence scheme
dC
dt≈ C (t + ∆t)− C (t)
∆t(2.4)
or can be written as
dC
dt≈ Cn+1 − Cn
∆t(2.5)
Explicit and Implicit scheme
Explicit �nite di�erence
Next the explicit FDM scheme of decay problem can be written as
Cn+1 − Cn
∆t= −κCn (2.6)
or
Cn+1 = (1− κ∆t)Cn (2.7)
Explicit and Implicit scheme
Algorithm Explicit
1. Start
2. De�ne �nal time T , ∆t, C0
3. Compute N = T/∆t, C 1 = C0
4. For n=1:N
5. Cn+1 = (1− κ∆t)Cn
6. End
7. plot(t, C)
Explicit and Implicit scheme
Algorithm Explicit
Problem of Grandma, using C0 = 100, ∆t = 3600, T = 15 hours.
Explicit and Implicit scheme
Implicit �nite di�erence
Next the explicit FDM scheme of decay problem can be written as
Cn+1 − Cn
∆t= −κCn+1 (2.8)
or
Cn+1 =Cn
(1 + κ∆t)(2.9)
Explicit and Implicit scheme
Algorithm Implicit
1. Start
2. De�ne �nal time T , ∆t, C0
3. Compute N = T/∆t, C 1 = C0
4. For n=1:N
5. Cn+1 = Cn/(1 + κ∆t)
6. End
7. plot(t, C)
Explicit and Implicit scheme
Algorithm Implicit
Problem of Grandma, using C0 = 100, ∆t = 3600, T = 15 hours.
Crank-Nicholson Scheme
Crank-Nicholson
As can be seen, the explicit scheme slightly underestimates the
correct concentration, whereas the implicit scheme slightly
overestimates concentration. A semi-implicit approach would
probably give the best solution, but this remains to be veri�ed by
the reader.
Crank-Nicholson Scheme
Crank-Nicholson
The semi-implicit scheme or Crank Nicholson of decay problem can
be written as
Cn+1 − Cn
∆t= −κ
(α · Cn+1 + (1− α) · Cn
)(3.1)
or
Cn+1 =(1− (1− α)κ∆t) · Cn
(1 + ακ∆t)(3.2)
where α = 0.5.
Crank-Nicholson Scheme
Algorithm Implicit
1. Start
2. De�ne �nal time T , ∆t, C0
3. Compute N = T/∆t, C 1 = C0
4. For n=1:N
5. Cn+1 = (1−(1−α)κ∆t)Cn
(1+ακ∆t)
6. End
7. plot(t, C)
Crank-Nicholson Scheme
Algorithm Implicit
Problem of Grandma, using C0 = 100, ∆t = 3600, T = 15 hours.
Homework
Homework
Repeat the exercise with
1. Explicit and Implicit scheme using ∆t = 1;
2. Semi-implicit using α = 0.25 and α = 0.75.
Next
Next
Next, the Finite Di�erence Method (FDM), explicit scheme will be
given in order to approximate the solution of 1D heat di�usion
problem.
Good Luck