Download - PDEs - Slides (3)
-
8/3/2019 PDEs - Slides (3)
1/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Part III
20401
Tony Shardlow
1/120
-
8/3/2019 PDEs - Slides (3)
2/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Outline
1 Uniqueness for reaction diffusion model
2 Stability for diffusion model
3 Finite differences
4 Numerical analysis
5 Finite differences for convection diffusion model
2/120
-
8/3/2019 PDEs - Slides (3)
3/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Outline
1 Uniqueness for reaction diffusion model
2 Stability for diffusion model
3 Finite differences
4 Numerical analysis
5 Finite differences for convection diffusion model
3/120
-
8/3/2019 PDEs - Slides (3)
4/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
A model reaction diffusionproblem
Model reaction diffusion problem
For a constant r 0,
u + ru = f for 0 < x < 1
u(0) = , u(1) = .
This is a two point Boundary Value Problem (BVP).
It is a simple model of a system with reaction and diffusion in
equilibrium.
4/120
-
8/3/2019 PDEs - Slides (3)
5/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Example soln 1
u = 1 for 0 < x < 1, (1.1)
u(0) = 0; u(1) = 0.
The solution is u(x) =1
2(x x2) .
0 0.5 10
1/8
x
temperature
Problem 1.1
0 0.5 10
1/(w^2)
Problem 1.2
defle
ction
x5/120
-
8/3/2019 PDEs - Slides (3)
6/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Example soln 2
u + w2u = 1 for 0 < x < 1 (1.2)
u(0) = 0; u(1) = 0.
The solution is
u(x) =1
w2
(exp(wx) + exp(w(1 x)))
w2(1 + exp(w)).
0 0.5 10
1/8
temperature
Problem 1.1
0 0.5 10
1/(w^2)
Problem 1.2
deflection
6/120
-
8/3/2019 PDEs - Slides (3)
7/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Uniqueness for reaction diffusionmodel
u + ru = f for 0 < x < 1
u(0) = , u(1) = .
Definition (well posed)
A boundary value problem is well posed ifexistence a solution exists
uniqueness the solution is unique
stability the solution depends continuously on the data.
We know solutions exists (using SoV) .
We now look at uniqueness .
HOMEWORK
You can now try Problem 17/120
-
8/3/2019 PDEs - Slides (3)
8/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Uniqueness for reaction diffusionmodel
u + ru = f for 0 < x < 1
u(0) = , u(1) = .
Definition (well posed)
A boundary value problem is well posed ifexistence a solution exists
uniqueness the solution is unique
stability the solution depends continuously on the data.
We know solutions exists (using SoV) .
We now look at uniqueness .
HOMEWORK
You can now try Problem 18/120
-
8/3/2019 PDEs - Slides (3)
9/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Theorem (uniqueness of soln)
There is at most one solution u to
u + ru = f, u(0) = , u(1) = ,
where r 0.
Proof.Suppose that u, v are solns.As the PDE is linear,
the difference w(x) = u(x) v(x) satisfies
w + r w = 0, w(0) = 0, w(1) = 0.
9/120
-
8/3/2019 PDEs - Slides (3)
10/120
-
8/3/2019 PDEs - Slides (3)
11/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Proof ctd.
10
(w)2 dx 0
+r1
0w2 dx 0
= 0,
with r 0.
The first term implies that w(x) = 0 for x (0, 1).Hence w is constant in (0, 1).
As w(0) = 0 and is constant, we conclude that w 0.As w = u v, we see u v and the two solutions are thesame.
We have proved uniqueness of solutions.
HOMEWORK
You can now try Problem 2 and 3
11/120
-
8/3/2019 PDEs - Slides (3)
12/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Proof ctd.
10
(w)2 dx 0
+r1
0w2 dx 0
= 0,
with r 0.
The first term implies that w(x) = 0 for x (0, 1).Hence w is constant in (0, 1).
As w(0) = 0 and is constant, we conclude that w 0.As w = u v, we see u v and the two solutions are thesame.
We have proved uniqueness of solutions.
HOMEWORK
You can now try Problem 2 and 3
12/120
-
8/3/2019 PDEs - Slides (3)
13/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Proof ctd.
10
(w)2 dx 0
+r1
0w2 dx 0
= 0,
with r 0.
The first term implies that w(x) = 0 for x (0, 1).Hence w is constant in (0, 1).
As w(0) = 0 and is constant, we conclude that w 0.As w = u v, we see u v and the two solutions are thesame.
We have proved uniqueness of solutions.
HOMEWORK
You can now try Problem 2 and 3
13/120
-
8/3/2019 PDEs - Slides (3)
14/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Proof ctd.
10
(w)2 dx 0
+r1
0w2 dx 0
= 0,
with r 0.
The first term implies that w(x) = 0 for x (0, 1).Hence w is constant in (0, 1).
As w(0) = 0 and is constant, we conclude that w 0.As w = u v, we see u v and the two solutions are thesame.
We have proved uniqueness of solutions.
HOMEWORK
You can now try Problem 2 and 3
14/120
-
8/3/2019 PDEs - Slides (3)
15/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Proof ctd.
10
(w)2 dx 0
+r1
0w2 dx 0
= 0,
with r 0.
The first term implies that w(x) = 0 for x (0, 1).Hence w is constant in (0, 1).
As w(0) = 0 and is constant, we conclude that w 0.As w = u v, we see u v and the two solutions are thesame.
We have proved uniqueness of solutions.
HOMEWORK
You can now try Problem 2 and 3
15/120
-
8/3/2019 PDEs - Slides (3)
16/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Outline
1 Uniqueness for reaction diffusion model
2 Stability for diffusion model
3 Finite differences
4 Numerical analysis
5 Finite differences for convection diffusion model
16/120
-
8/3/2019 PDEs - Slides (3)
17/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Diffusion problem
The third condition for a well posed problem is stability or
continuous dependence on the problem data.For simplicity, we investigate stability for the diffusion problem(case r = 0 of the reaction diffusion model).
Model diffusion problem
u = f for 0 < x < 1
u(0) = ; u(1) = .
To establish stability, we show that u(x)
depends continuously on boundary data ( and ).
17/120
-
8/3/2019 PDEs - Slides (3)
18/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Diffusion problem
The third condition for a well posed problem is stability or
continuous dependence on the problem data.For simplicity, we investigate stability for the diffusion problem(case r = 0 of the reaction diffusion model).
Model diffusion problem
u = f for 0 < x < 1
u(0) = ; u(1) = .
To establish stability, we show that u(x)
depends continuously on boundary data ( and ).
18/120
-
8/3/2019 PDEs - Slides (3)
19/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Diffusion problem
The third condition for a well posed problem is stability or
continuous dependence on the problem data.For simplicity, we investigate stability for the diffusion problem(case r = 0 of the reaction diffusion model).
Model diffusion problem
u = f for 0 < x < 1
u(0) = ; u(1) = .
To establish stability, we show that u(x)
depends continuously on boundary data ( and ).
19/120
-
8/3/2019 PDEs - Slides (3)
20/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Maximum principle
Our main tool is the maximum principle.
Lemma (maximum principle)
Suppose that f (x) < 0 for all x (0, 1).Ifu = f , then u(x) attains its maximum value at one of thetwo end points x = 0, 1.
Proof.
Suppose for a contradiction that (0, 1) is a local maximum.
From calculus, u() 0 and u() = 0.Hence u() 0.But we assumed u = f < 0 for all x (0, 1).Proved by contradiction.
20/120
-
8/3/2019 PDEs - Slides (3)
21/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Maximum principle
Our main tool is the maximum principle.
Lemma (maximum principle)
Suppose that f (x) < 0 for all x (0, 1).Ifu = f , then u(x) attains its maximum value at one of thetwo end points x = 0, 1.
Proof.
Suppose for a contradiction that (0, 1) is a local maximum.
From calculus, u() 0 and u() = 0.Hence u() 0.But we assumed u = f < 0 for all x (0, 1).Proved by contradiction.
21/120
( )
-
8/3/2019 PDEs - Slides (3)
22/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Upper bound on soln u(x)
Lemma
Ifu(x) = 0 for all x (0, 1),then u(x) M for x [0, 1] and M = max{u(0), u(1)}.
Proof.
For > 0, letv(x) = u(x) + x
2.
Then v = u 2 = 2 < 0 for x (0, 1).
By the maximum principle, v(x) max{v(0), v(1)} .
Now u(x) = v(x) x2 v(x). Hence,
u(x) v(x) max{v(0), v(1)} = max{u(0), u(1) + }.
As this holds for any > 0, we are done
22/120
U b d l ( )
-
8/3/2019 PDEs - Slides (3)
23/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Upper bound on soln u(x)
Lemma
Ifu(x) = 0 for all x (0, 1),then u(x) M for x [0, 1] and M = max{u(0), u(1)}.
Proof.
For > 0, letv(x) = u(x) + x
2.
Then v = u 2 = 2 < 0 for x (0, 1).
By the maximum principle, v(x) max{v(0), v(1)} .
Now u(x) = v(x) x2 v(x). Hence,
u(x) v(x) max{v(0), v(1)} = max{u(0), u(1) + }.
As this holds for any > 0, we are done
23/120
U b d l ( )
-
8/3/2019 PDEs - Slides (3)
24/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Upper bound on soln u(x)
Lemma
Ifu(x) = 0 for all x (0, 1),then u(x) M for x [0, 1] and M = max{u(0), u(1)}.
Proof.
For > 0, letv(x) = u(x) + x
2.
Then v = u 2 = 2 < 0 for x (0, 1).
By the maximum principle, v(x) max{v(0), v(1)} .
Now u(x) = v(x) x2 v(x). Hence,
u(x) v(x) max{v(0), v(1)} = max{u(0), u(1) + }.
As this holds for any > 0, we are done
24/120
U b d l ( )
-
8/3/2019 PDEs - Slides (3)
25/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Upper bound on soln u(x)
Lemma
Ifu(x) = 0 for all x (0, 1),then u(x) M for x [0, 1] and M = max{u(0), u(1)}.
Proof.
For > 0, letv(x) = u(x) + x
2.
Then v = u 2 = 2 < 0 for x (0, 1).
By the maximum principle, v(x) max{v(0), v(1)} .
Now u(x) = v(x) x2 v(x). Hence,
u(x) v(x) max{v(0), v(1)} = max{u(0), u(1) + }.
As this holds for any > 0, we are done
25/120
L b d l ( )
-
8/3/2019 PDEs - Slides (3)
26/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Lower bound on soln u(x)
Lemma
Ifu(x) = 0 for all x (0, 1)then m u(x) for x [0, 1] and m = min{u(0), u(1)}.
Proof.
Define w(x) = u(x).Then w(x) = 0 for x (0, 1).By previous lemma,
w(x) max{w(0), w(1)}.
and using w = u
u(x) max{u(0), u(1)} = min{u(0), u(1)}
and so u(x) min{u(0), u(1)} 26/120
L b d l ( )
-
8/3/2019 PDEs - Slides (3)
27/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Lower bound on soln u(x)
Lemma
Ifu(x) = 0 for all x (0, 1)then m u(x) for x [0, 1] and m = min{u(0), u(1)}.
Proof.
Define w(x) = u(x).Then w(x) = 0 for x (0, 1).By previous lemma,
w(x) max{w(0), w(1)}.
and using w = u
u(x) max{u(0), u(1)} = min{u(0), u(1)}
and so u(x) min{u(0), u(1)} 27/120
Summary
-
8/3/2019 PDEs - Slides (3)
28/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Summary
Theorem
Let u be a smooth solution of
u = 0 for 0 < x < 1
u(0) = ; u(1) = ;
For all x (0, 1),
min{, } u(x) max{, }.
HOMEWORK
You can now try Problem 4
We now use this to show stability with respect to changes inthe boundary data , .
28/120
Summary
-
8/3/2019 PDEs - Slides (3)
29/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Summary
Theorem
Let u be a smooth solution of
u = 0 for 0 < x < 1
u(0) = ; u(1) = ;
For all x (0, 1),
min{, } u(x) max{, }.
HOMEWORK
You can now try Problem 4
We now use this to show stability with respect to changes inthe boundary data , .
29/120
Continuity with respect to
-
8/3/2019 PDEs - Slides (3)
30/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
boundary data
Theorem
Suppose that
u = f for 0 < x < 1
u(0) = ; u(1) = .
(D)
u = f for 0 < x < 1
u(0) = + 0; u(1) = + 1.
(D)
Then
supx(0,1)
|u(x) u(x)| max{|0|, |1|}.
Small changes (0, 1) to boundary data
cause small changes to the solution u
30/120
Proof
-
8/3/2019 PDEs - Slides (3)
31/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Proof
Let e = u u. Subtracting (D) from (D) gives
e = 0 for 0 < x < 1
e(0) = 0; e(1) = 1.
As homogeneous, previous theorem implies the stability bound:
min{0, 1} e(x) max{0, 1}
This implies|e(x)| max{|0|, |1|}.
HOMEWORK
You can now try Problem 5
31/120
Proof
-
8/3/2019 PDEs - Slides (3)
32/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Proof
Let e = u u. Subtracting (D) from (D) gives
e = 0 for 0 < x < 1
e(0) = 0; e(1) = 1.
As homogeneous, previous theorem implies the stability bound:
min{0, 1} e(x) max{0, 1}
This implies|e(x)| max{|0|, |1|}.
HOMEWORK
You can now try Problem 5
32/120
Proof
-
8/3/2019 PDEs - Slides (3)
33/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Proof
Let e = u u. Subtracting (D) from (D) gives
e = 0 for 0 < x < 1
e(0) = 0; e(1) = 1.
As homogeneous, previous theorem implies the stability bound:
min{0, 1} e(x) max{0, 1}
This implies|e(x)| max{|0|, |1|}.
HOMEWORK
You can now try Problem 5
33/120
Outline
-
8/3/2019 PDEs - Slides (3)
34/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Outline
1 Uniqueness for reaction diffusion model
2 Stability for diffusion model
3 Finite differences
4 Numerical analysis
5 Finite differences for convection diffusion model
34/120
Scientific computing
-
8/3/2019 PDEs - Slides (3)
35/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Scientific computing
Scientific computing involves constructing numerical
solution techniques for mathematical models and usingcomputers to analyse and solve models that arise inscience and engineering.
Without computing, we could not find or even approximatesolutions to most mathematical models and PDEs.
Numerical analysis is the mathematical theoryunderpinning the techniques used in computationalscience. It aims to show existing algorithms are efficientand accurate and develop better algorithms.
We introduce one numerical solution technique for PDEs,known as finite differences.
35/120
Scientific computing
-
8/3/2019 PDEs - Slides (3)
36/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Scientific computing
Scientific computing involves constructing numerical
solution techniques for mathematical models and usingcomputers to analyse and solve models that arise inscience and engineering.
Without computing, we could not find or even approximatesolutions to most mathematical models and PDEs.
Numerical analysis is the mathematical theoryunderpinning the techniques used in computationalscience. It aims to show existing algorithms are efficientand accurate and develop better algorithms.
We introduce one numerical solution technique for PDEs,known as finite differences.
36/120
Scientific computing
-
8/3/2019 PDEs - Slides (3)
37/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Sc e t c co put g
Scientific computing involves constructing numerical
solution techniques for mathematical models and usingcomputers to analyse and solve models that arise inscience and engineering.
Without computing, we could not find or even approximatesolutions to most mathematical models and PDEs.
Numerical analysis is the mathematical theoryunderpinning the techniques used in computationalscience. It aims to show existing algorithms are efficientand accurate and develop better algorithms.
We introduce one numerical solution technique for PDEs,known as finite differences.
37/120
Scientific computing
-
8/3/2019 PDEs - Slides (3)
38/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
p g
Scientific computing involves constructing numerical
solution techniques for mathematical models and usingcomputers to analyse and solve models that arise inscience and engineering.
Without computing, we could not find or even approximatesolutions to most mathematical models and PDEs.
Numerical analysis is the mathematical theoryunderpinning the techniques used in computationalscience. It aims to show existing algorithms are efficientand accurate and develop better algorithms.
We introduce one numerical solution technique for PDEs,known as finite differences.
38/120
Finite difference approximation
-
8/3/2019 PDEs - Slides (3)
39/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
pp
The idea of the finite difference method is to approximate
u(x) by uj at x = xj for xj on a grid
Definition (grid)
A uniform grid on [0, 1] is defined by
xj = jh, j = 0, . . . , n, h = 1/n.
0 = x0 x1 x2 x3 xn = 1
h =1
n
h h
39/120
Approximating derivatives on aid
-
8/3/2019 PDEs - Slides (3)
40/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability for
diffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
grid
Definition (centered differencing)
To approximate derivatives u
(xj) at a grid point x = xj, write
u(xj) 1
hu(xj) ,
where u(x) = u(x + h/2) u(x h/2).
xj h
2
xj +h
2
u(xj h)
u(xj)
u(xj +h)
40/120
Error
-
8/3/2019 PDEs - Slides (3)
41/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Theorem (Taylors theorem)
u(x + h) = u(x) + hu(x) +1
2h2u(x) + +
1
n!hnu(n)()
for some (x, x + h).
Then
u(x + h) u(x) = hu(x) +1
2h2u()
= hu
(x) + O(h2
) ,
where we use the notation O(h2) for short. It means anyquantity bounded by Kh2 for some constant K.
HOMEWORK
You can now try Problem 7 and 8 41/120
Error
-
8/3/2019 PDEs - Slides (3)
42/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Theorem (Taylors theorem)
u(x + h) = u(x) + hu(x) +1
2h2u(x) + +
1
n!hnu(n)()
for some (x, x + h).
Then
u(x + h) u(x) = hu(x) +1
2h2u()
= hu
(x) + O(h2
) ,where we use the notation O(h2) for short. It means anyquantity bounded by Kh2 for some constant K.
HOMEWORK
You can now try Problem 7 and 8 42/120
Error
-
8/3/2019 PDEs - Slides (3)
43/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Theorem (Taylors theorem)
u(x + h) = u(x) + hu(x) +1
2h2u(x) + +
1
n!hnu(n)()
for some (x, x + h).
Then
u(x + h) u(x) = hu(x) +1
2h2u()
= hu
(x) + O(h2
) ,where we use the notation O(h2) for short. It means anyquantity bounded by Kh2 for some constant K.
HOMEWORK
You can now try Problem 7 and 8 43/120
Error
-
8/3/2019 PDEs - Slides (3)
44/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Theorem (Taylors theorem)
u(x + h) = u(x) + hu(x) +1
2h2u(x) + +
1
n!hnu(n)()
for some (x, x + h).
Then
u(x + h) u(x) = hu(x) +1
2h2u()
= hu
(x) + O(h2
) ,where we use the notation O(h2) for short. It means anyquantity bounded by Kh2 for some constant K.
HOMEWORK
You can now try Problem 7 and 8 44/120
Lemma
-
8/3/2019 PDEs - Slides (3)
45/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Lemma
The error in approximating u(xj) by u(xj)/h is
ej = u(xj) 1h
u(xj) = O(h2).
Proof.
By Taylors theorem,
u(x + h/2) =u(x) +h
2u(x) +
h2
8u(x) + O(h3)
u(x h/2) =u(x) h
2u(x) +
h2
8u(x) + O(h3)
so u(xj)
h=
(u(xj + h/2) u(xj h/2))
h
=hu(xj) + O(h
3)
h= u(xj) + O(h
2).
45/120
Lemma
-
8/3/2019 PDEs - Slides (3)
46/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Lemma
The error in approximating u(xj) by u(xj)/h is
ej = u(xj) 1h
u(xj) = O(h2).
Proof.
By Taylors theorem,
u(x + h/2) =u(x) +h
2u(x) +
h2
8u(x) + O(h3)
u(x h/2) =u(x) h
2u(x) +
h2
8u(x) + O(h3)
so u(xj)
h=
(u(xj + h/2) u(xj h/2))
h
=hu(xj) + O(h
3)
h= u(xj) + O(h
2).
46/120
Lemma
-
8/3/2019 PDEs - Slides (3)
47/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Lemma
The error in approximating u(xj) by u(xj)/h is
ej = u(xj) 1h
u(xj) = O(h2).
Proof.
By Taylors theorem,
u(x + h/2) =u(x) +h
2u(x) +
h2
8u(x) + O(h3)
u(x h/2) =u(x) h
2u(x) +
h2
8u(x) + O(h3)
so u(xj)
h=
(u(xj + h/2) u(xj h/2))
h
=hu(xj) + O(h
3)
h= u(xj) + O(h
2).
47/120
Approximate u(xj)
-
8/3/2019 PDEs - Slides (3)
48/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Definition (second centered difference)
To approximate the second derivative, write
1
h22u(xj) u
(xj)
where
2u(xj) =[u(xj)]
=u(xj +h
2) u(xj
h
2)
= u(xj + h) u(xj) (u(xj) u(xj h))
= u(xj + h) 2u(xj) + u(xj h) .
48/120
Approximate u(xj)
-
8/3/2019 PDEs - Slides (3)
49/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
Definition (second centered difference)
To approximate the second derivative, write
1
h22u(xj) u
(xj)
where
2u(xj) =[u(xj)]
=u(xj +h
2) u(xj
h
2)
= u(xj + h) u(xj) (u(xj) u(xj h))
= u(xj + h) 2u(xj) + u(xj h) .
49/120
Finite differences for modelreaction diffusion PDE
-
8/3/2019 PDEs - Slides (3)
50/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
reaction diffusion PDE
Consider the reaction diffusion equation
u + ru = f, u(0) = u(1) = 0.
At the grid points xj = jh where h = 1/n, the PDE is
u(xj) + ru(xj) = f(xj) j = 1, 2, . . . , n 1.
Let fj = f(xj) and make approximations
u(xj) uj , u(xj)
2uj/h2.
Finite difference approximationuj is soln of
1
h22uj + ruj = fj j = 1, 2, . . . , n 1.
with boundary conditions u0 = un = 050/120
Finite differences for modelreaction diffusion PDE
-
8/3/2019 PDEs - Slides (3)
51/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
reaction diffusion PDE
Consider the reaction diffusion equation
u + ru = f, u(0) = u(1) = 0.
At the grid points xj = jh where h = 1/n, the PDE is
u(xj) + ru(xj) = f(xj) j = 1, 2, . . . , n 1.
Let fj = f(xj) and make approximations
u(xj) uj , u(xj)
2uj/h2.
Finite difference approximation uj
is soln of
1
h22uj + ruj = fj j = 1, 2, . . . , n 1.
with boundary conditions u0 = un = 051/120
Finite differences for modelreaction diffusion PDE
-
8/3/2019 PDEs - Slides (3)
52/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finite
differences forconvectiondiffusionmodel
reaction diffusion PD
Consider the reaction diffusion equation
u + ru = f, u(0) = u(1) = 0.
At the grid points xj = jh where h = 1/n, the PDE is
u(xj) + ru(xj) = f(xj) j = 1, 2, . . . , n 1.
Let fj = f(xj) and make approximations
u(xj) uj , u(xj)
2uj/h2.
Finite difference approximation uj
is soln of
1
h22uj + ruj = fj j = 1, 2, . . . , n 1.
with boundary conditions u0 = un = 052/120
Finite differences for modelreaction diffusion problem
-
8/3/2019 PDEs - Slides (3)
53/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
p
Definition (finite difference method)
The finite difference method is to find uj u(xj) such that
1
h22uj + ruj = fj j = 1, 2, . . . , n 1, ()
and u0 = un = 0.
As2uj = uj1 2uj + uj+1,
the finite difference method is
1
h2(uj1 2uj + uj+1) + r uj = fj j = 1, 2, . . . , n 1.
53/120
Finite differences for modelreaction diffusion problem
-
8/3/2019 PDEs - Slides (3)
54/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Definition (finite difference method)
The finite difference method is to find uj u(xj) such that
1
h22uj + ruj = fj j = 1, 2, . . . , n 1, ()
and u0 = un = 0.
As2uj = uj1 2uj + uj+1,
the finite difference method is
1
h2(uj1 2uj + uj+1) + r uj = fj j = 1, 2, . . . , n 1.
54/120
Linear system of eqns
-
8/3/2019 PDEs - Slides (3)
55/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
1h2
(uj1 2uj + uj+1) + ruj = fj j = 1, 2, . . . , n 1.
and rearranging
1
h
2uj1 + (
2
h
2+ r)uj
1
h
2uj+1 = fj.
The BCs u(0) = u(1) = 0 give u0 = 0 and un = 0 . Writeas a system of linear equations:
( 2h2
+ r) 1h2
0
. . . . . . 1
h2( 2h2
+ r) 1h2
. . .. . .
0 1h2
( 2h2
+ r)
u1...
uj...
un1
=
f1...fj...
fn1
55/120
Example (r = 0, f(x) = 1)
-
8/3/2019 PDEs - Slides (3)
56/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Take n = 6, i.e. h = 1/6. The system is
72 36 0 0 036 72 36 0 0
0 36 72 36 00 0 36 72 36
0 0 0 36 72
u1u2u3u4
u5
=
1111
1
Solve the system of equations (e.g., with MATLAB):
u1 = u5 = 5/72;
u2 = u4 = 1/9 = 8/72;u3 = 1/8 = 9/72.
Note that uj = u(xj) =12 (xj x
2j ). That is, the finite
difference solution is exact in this example.
56/120
Example (r = 0, f(x) = 1)
-
8/3/2019 PDEs - Slides (3)
57/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
Take n = 6, i.e. h = 1/6. The system is
72 36 0 0 036 72 36 0 0
0 36 72 36 00 0 36 72 36
0 0 0 36 72
u1u2u3u4
u5
=
1111
1
Solve the system of equations (e.g., with MATLAB):
u1 = u5 = 5/72;
u2 = u4 = 1/9 = 8/72;u3 = 1/8 = 9/72.
Note that uj = u(xj) =12 (xj x
2j ). That is, the finite
difference solution is exact in this example.
57/120
Example
-
8/3/2019 PDEs - Slides (3)
58/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
u
+ 16u
= 1for
0 0 if the local error satisfies
|Tj| = O(hk) j = 1, 2, . . . , n 1.
Theorem (2nd order consistent)
The method is 2nd order consistent if the local truncation errorsatisfies
|Tj| = O(h2), j = 1, 2, . . . , n 1.
67/120
-
8/3/2019 PDEs - Slides (3)
68/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
Definition (kth order consistent)
A finite difference scheme is said to be kth order consistent fork > 0 if the local error satisfies
|Tj| = O(hk) j = 1, 2, . . . , n 1.
Theorem (2nd order consistent)
The method is 2nd order consistent if the local truncation errorsatisfies
|Tj| = O(h2), j = 1, 2, . . . , n 1.
68/120
Proof.
-
8/3/2019 PDEs - Slides (3)
69/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
By Taylors theorem,
u(x h) =u(x) hu(x) + h22
u(x) h36
u(x) + O(h4)
u(x + h) =u(x) + hu(x) +h2
2u(x) +
h3
6u(x) + O(h4).
andTj = u
(xj) 1
h2(u(xj h) 2u(xj) + u(xj + h)).
We can add the expansions for u(x h) and u(x + h), to find
u(xj h) + u(xj + h) = 2u(xj) + h2u(xj) + O(h
4).
Rearranging givesTj = O(h
2).
69/120
Global error
-
8/3/2019 PDEs - Slides (3)
70/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
HOMEWORK
You can now try Problem 9 and 10
We relate the global error ej at the grid point xj to the localtruncation error Tj
Tj =
1
h2 2
u(xj) + ru(xj) fj j = 1, 2, . . . , n 1,
0 = 1
h22uj + ruj fj j = 1, 2, . . . , n 1.
Subtracting these equations and letting ej = u(xj) uj gives
Tj = 1
h22ej + rej j = 1, 2, . . . , n 1.
We have a linear system of eqns for ej.70/120
Global error
-
8/3/2019 PDEs - Slides (3)
71/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
HOMEWORK
You can now try Problem 9 and 10
We relate the global error ej at the grid point xj to the localtruncation error Tj
Tj =
1
h2 2
u(xj) + ru(xj) fj j = 1, 2, . . . , n 1,
0 = 1
h22uj + ruj fj j = 1, 2, . . . , n 1.
Subtracting these equations and letting ej = u(xj) uj gives
Tj = 1
h22ej + rej j = 1, 2, . . . , n 1.
We have a linear system of eqns for ej.
71/120
Global error
-
8/3/2019 PDEs - Slides (3)
72/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
HOMEWORK
You can now try Problem 9 and 10
We relate the global error ej at the grid point xj to the localtruncation error Tj
Tj =
1
h2 2
u(xj) + ru(xj) fj j = 1, 2, . . . , n 1,
0 = 1
h22uj + ruj fj j = 1, 2, . . . , n 1.
Subtracting these equations and letting ej = u(xj) uj gives
Tj = 1
h22ej + rej j = 1, 2, . . . , n 1.
We have a linear system of eqns for ej.
72/120
Global error
-
8/3/2019 PDEs - Slides (3)
73/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
HOMEWORK
You can now try Problem 9 and 10We relate the global error ej at the grid point xj to the localtruncation error Tj
Tj =
1
h2 2
u(xj) + ru(xj) fj j = 1, 2, . . . , n 1,
0 = 1
h22uj + ruj fj j = 1, 2, . . . , n 1.
Subtracting these equations and letting ej = u(xj) uj gives
Tj = 1
h22ej + rej j = 1, 2, . . . , n 1.
We have a linear system of eqns for ej.
73/120
Stability theorem for finitedifferences
Theorem (Stability theorem)
-
8/3/2019 PDEs - Slides (3)
74/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
Theorem (Stability theorem)
Suppose that {uj}nj=0 satisfy the tridiagonal system of
equations
auj1 + buj cuj+1 0 j = 1, 2, . . . , n 1.
Let a, b, c denote real coefficients with
a 0c 0
b a + c > 0
(S)
then
uj max{0, u0, un} for all j = 0, 1, 2, . . . , n.
74/120
Proof.
Suppose for a contradiction there exists u 0 for
-
8/3/2019 PDEs - Slides (3)
75/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
Suppose for a contradiction, there exists uk 0 fork = 1, . . . , n 1 such that
uk =max{u0, u1, u2, . . . , un}
min{uk1, uk+1} 0, this is a contradiction.
75/120
Proof.
Suppose for a contradiction there exists uk 0 for
-
8/3/2019 PDEs - Slides (3)
76/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
Suppose for a contradiction, there exists uk 0 fork = 1, . . . , n 1 such that
uk =max{u0, u1, u2, . . . , un}
min{uk1, uk+1} 0, this is a contradiction.
76/120
Proof ctd.
W l d h i h
-
8/3/2019 PDEs - Slides (3)
77/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
We conclude that either
1
there is a maximum, which is negative (uk 0) or2 there is a maximum at the boundary uk max{u0, un}.
.
HOMEWORK
You can now try Problem 11
77/120
Proof ctd.
W l d th t ith
-
8/3/2019 PDEs - Slides (3)
78/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
We conclude that either
1
there is a maximum, which is negative (uk 0) or2 there is a maximum at the boundary uk max{u0, un}.
.
HOMEWORK
You can now try Problem 11
78/120
Stability+consisitency convergence
1 F PDE th i i i l ti
-
8/3/2019 PDEs - Slides (3)
79/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
1 For PDE, the maximum principle gave us continuousdependence on data and hence well posedness.
2 For numerical method, the discrete maximum principlegives us continuous dependence of global error ontruncation error.
3 This type of result for numerical methods is usually
referred to as stability and
Stability+consistency convergence
Recall, consistency means local truncation error Tj is
O(hk), some k > 1.convergence means global error u(xj) uj is small.
4 No proofs given, but Theorem given next.
79/120
Stability+consisitency convergence
1 For PDE the maximum principle gave us continuous
-
8/3/2019 PDEs - Slides (3)
80/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
1 For PDE, the maximum principle gave us continuousdependence on data and hence well posedness.
2 For numerical method, the discrete maximum principlegives us continuous dependence of global error ontruncation error.
3 This type of result for numerical methods is usually
referred to as stability and
Stability+consistency convergence
Recall, consistency means local truncation error Tj is
O(hk), some k > 1.convergence means global error u(xj) uj is small.
4 No proofs given, but Theorem given next.
80/120
Stability+consisitency convergence
1 For PDE the maximum principle gave us continuous
-
8/3/2019 PDEs - Slides (3)
81/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
1 For PDE, the maximum principle gave us continuousdependence on data and hence well posedness.
2 For numerical method, the discrete maximum principlegives us continuous dependence of global error ontruncation error.
3 This type of result for numerical methods is usually
referred to as stability and
Stability+consistency convergence
Recall, consistency means local truncation error Tj is
O(hk), some k > 1.convergence means global error u(xj) uj is small.
4 No proofs given, but Theorem given next.
81/120
P III
Stability+consisitency convergence
1 For PDE the maximum principle gave us continuous
-
8/3/2019 PDEs - Slides (3)
82/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
1 For PDE, the maximum principle gave us continuousdependence on data and hence well posedness.
2 For numerical method, the discrete maximum principlegives us continuous dependence of global error ontruncation error.
3 This type of result for numerical methods is usually
referred to as stability and
Stability+consistency convergence
Recall, consistency means local truncation error Tj is
O(hk), some k > 1.convergence means global error u(xj) uj is small.
4 No proofs given, but Theorem given next.
82/120
P t III
Theorem (Stability+constistency=convergence)
-
8/3/2019 PDEs - Slides (3)
83/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusion
model
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
(S y+ y g )
Suppose that
consistent: the finite difference method is kth order consistent.
stable the method has the form
auj1 + buj cuj+1 = fj
with a, c 0 and b a + c > 0.Then, the numerical approximation converges to the exact
solution,
ej = |uj u(xj)| = O(hk) as h 0.
83/120
Part III
Stability of finite differenceapprox
Recall the finite difference approximation
-
8/3/2019 PDEs - Slides (3)
84/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
Recall the finite difference approximation
1h2
uj1 + ( 2h2
+ r)uj 1h2
uj+1 = fj j = 1, 2, . . . , n 1,
To show that the approximation is stable,we apply the Stability Theorem and look at condition (S).
a 0 1h2
0
c 0 1
h2 0
b a + c 2
h2+ r
2
h2
The centred approximation method is stable as r 0.
84/120
Part III
Stability of finite differenceapprox
Recall the finite difference approximation
-
8/3/2019 PDEs - Slides (3)
85/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
Recall the finite difference approximation
1h2
uj1 + ( 2h2
+ r)uj 1h2
uj+1 = fj j = 1, 2, . . . , n 1,
To show that the approximation is stable,we apply the Stability Theorem and look at condition (S).
a 0 1h2
0
c 0 1
h2 0
b a + c 2
h2+ r
2
h2
The centred approximation method is stable as r 0.
85/120
Part III
Stability of finite differenceapprox
Recall the finite difference approximation
-
8/3/2019 PDEs - Slides (3)
86/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
Recall the finite difference approximation
1h2
uj1 + ( 2h2
+ r)uj 1h2
uj+1 = fj j = 1, 2, . . . , n 1,
To show that the approximation is stable,we apply the Stability Theorem and look at condition (S).
a 0 1h2
0
c 0 1
h2 0
b a + c 2
h2+ r
2
h2
The centred approximation method is stable as r 0.
86/120
Part III
Stability of finite differenceapprox
Recall the finite difference approximation
-
8/3/2019 PDEs - Slides (3)
87/120
Part III
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
pp
1h2
uj1 + ( 2h2
+ r)uj 1h2
uj+1 = fj j = 1, 2, . . . , n 1,
To show that the approximation is stable,we apply the Stability Theorem and look at condition (S).
a 0 1h2
0
c 0 1
h2 0
b a + c 2
h2+ r
2
h2
The centred approximation method is stable as r 0.
87/120
Part III
1 Proof of theorem not given See written notes
-
8/3/2019 PDEs - Slides (3)
88/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
1 Proof of theorem not given. See written notes.
2 It is similar to the proof of stability for the diffusionproblem.
3 We conclude that the global error for the centereddifference approximation of the diffusion problem satisfies
global error = |uj u(xj)| = O(h2).
88/120
Part III
Outline
1 Uniqueness for reaction diffusion model
-
8/3/2019 PDEs - Slides (3)
89/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
2 Stability for diffusion model
3 Finite differences
4 Numerical analysis
5 Finite differences for convection diffusion model
89/120
Part III
Convection diffusion problem
The problem
-
8/3/2019 PDEs - Slides (3)
90/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
u + ru = f
is known as a reaction diffusion equation, as the term u
models diffusion and the term u models reaction.We now look at a convection diffusion equation, replacing uwith u.
Convection-diffusionu + wu = f for 0 < x < 1
u(0) = ; u(1) =
for some f : (0, 1) R
, a scalar w R
, and boundary values, R.The scalar w is known as the wind.
It controls the strength and direction of the convection.
90/120
Part III
Problem 1.3
u + wu = 0 for 0 < x < 1 (1.3)
(0) 1 (1) 0
-
8/3/2019 PDEs - Slides (3)
91/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
u(0) = 1; u(1) = 0.
The solution is
u(x) =exp(w) exp(wx)
exp(w) 1
0 0.50
1
x
temperature
Problem 1.3
w=0
w=5
w=20
0 0.5 10
1/8
Problem 1.4
w=20
w=5
w=0
91/120
Part III
Problem
u + wu = 1 for 0 < x < 1 (1.4)
(0) 0 (1) 0
-
8/3/2019 PDEs - Slides (3)
92/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
u(0) = 0; u(1) = 0.
The solution is
u(x) =x
w
1 exp(wx)
w(exp(w) 1)
0 0.50
1
x
temperature
Problem 1.3
w=0
w=5
w=20
0 0.5 10
1/8
Problem 1.4
w=20
w=5
w=0
92/120
Part III
Centered finite differences
There are two derivatives to approximate.
-
8/3/2019 PDEs - Slides (3)
93/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
u(xj)
u(xj) 2u(xj)
h2=
u(xj + h) 2u(xj) + u(xj h)
h2.
u
(xj)
We cannot use u(xj) = u(xj + h/2) u(xj h/2) as we canonly take values on the grid.We use the averaged centered difference:
u(xj) u(xj)
h= 1
2h
u(xj + h
2) + u(xj h
2)
=1
2h(u(xj + h) u(xj h)) .
93/120
Part III
Numerical approximation
The exact solution at the grid points u(xj) satisfies
-
8/3/2019 PDEs - Slides (3)
94/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
u(xj) + wu(xj) = fj j = 1, 2, . . . , n 1.
Replace derivatives by centered finite differences
Centered finite difference method
Find uj such that
1
h22uj +
w
huj = fj j = 1, 2, . . . , n 1.
That is,
1
h2(uj+1 2uj + uj1) +
w
2h(uj+1 uj1) = fj.
94/120
Part III
Linear system of equations
For j = 1, . . . , n 1,
-
8/3/2019 PDEs - Slides (3)
95/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences for
convectiondiffusionmodel
1
h2 (uj+1 2uj + uj1) +w
2h (uj+1 uj1) = fj
1
h2
w
2h
a
uj1 +
2
h2
b
uj +
1
h2+
w
2h
c
uj+1 = fj
To take care of the boundary conditions The BC u(0) = gives u0 = and
a + bu1 cu2 = f0 bu1 cu2 = f0 + a.
The BC u(1) = gives un = and
aun2 + bun1 = fn1 + c.
95/120
Part III
Linear system of equations
For j = 1, . . . , n 1,
-
8/3/2019 PDEs - Slides (3)
96/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
1
h2 (uj+1 2uj + uj1) +w
2h (uj+1 uj1) = fj
1
h2
w
2h
a
uj1 +
2
h2
b
uj +
1
h2+
w
2h
c
uj+1 = fj
To take care of the boundary conditions The BC u(0) = gives u0 = and
a + bu1 cu2 = f0 bu1 cu2 = f0 + a.
The BC u(1) = gives un = and
aun2 + bun1 = fn1 + c.
96/120
Part III
Linear system of equations
For j = 1, . . . , n 1,
-
8/3/2019 PDEs - Slides (3)
97/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
1
h2 (uj+1 2uj + uj1) +w
2h (uj+1 uj1) = fj
1
h2
w
2h
a
uj1 +
2
h2
b
uj +
1
h2+
w
2h
c
uj+1 = fj
To take care of the boundary conditions The BC u(0) = gives u0 = and
a + bu1 cu2 = f0 bu1 cu2 = f0 + a.
The BC u(1) = gives un = and
aun2 + bun1 = fn1 + c.
97/120
Part III
Linear system of equations
Collect all the equations as a linear system.
-
8/3/2019 PDEs - Slides (3)
98/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
b c 0. . .
. . .
a b c. . .
. . .
0 a b
u1...
uj...
un1
=
f1 + a...fj...
fn1 + c
.
Top row comes from left hand boundary condition and thebottom rows comes from the right boundary condition.
Also note the boundary condition affects the right hand sidevector in top/bottom entry.
In contrast to reaction diffusion equation,the matrix is non-symmetric as a = c.
98/120
Part III
Numerical analysis
HOMEWORK
Y t P bl 12
-
8/3/2019 PDEs - Slides (3)
99/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
You can now try Problem 12
1 Is the approximation consistent? If u is smooth,
|Tj| Ch2, j = 1, 2, . . . , n 1.
In other words, Tj = O(h2).
Proof by Taylors theorem.
2 Is the approximation stable?
99/120
Part III
Numerical analysis
HOMEWORK
Y t P bl 12
-
8/3/2019 PDEs - Slides (3)
100/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
You can now try Problem 12
1 Is the approximation consistent? If u is smooth,
|Tj| Ch2, j = 1, 2, . . . , n 1.
In other words, Tj = O(h2).
Proof by Taylors theorem.
2 Is the approximation stable?
100/120
Part III
Numerical analysis
HOMEWORK
You can now try Problem 12
-
8/3/2019 PDEs - Slides (3)
101/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
You can now try Problem 12
1 Is the approximation consistent? If u is smooth,
|Tj| Ch2, j = 1, 2, . . . , n 1.
In other words, Tj = O(h2).
Proof by Taylors theorem.
2 Is the approximation stable?
101/120
Part III
Apply stability theorem
Recall the tridiagonal matrix entries ..
-
8/3/2019 PDEs - Slides (3)
102/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
1h2 w2h a
uj1 + 2h2 b
uj + 1h2 + w2h c
uj+1 = fj
To show that the approximation is stable we simply need tocheck that the tridiagonal coefficients satisfy (S).
a 0 1
h2+
w
2h 0 (?)
c 0
1
h2
w
2h 0
b a + c 2
h2
2
h2()
102/120
Part III
Apply stability theorem
Recall the tridiagonal matrix entries ..
1 2 1
-
8/3/2019 PDEs - Slides (3)
103/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
1h2 w2h a
uj1 + 2h2 b
uj + 1h2 + w2h c
uj+1 = fj
To show that the approximation is stable we simply need tocheck that the tridiagonal coefficients satisfy (S).
a 0 1
h2+
w
2h 0 (?)
c 0
1
h2
w
2h 0
b a + c 2
h2
2
h2()
103/120
Part III
20401
Apply stability theorem
Recall the tridiagonal matrix entries ..
1 2 1
-
8/3/2019 PDEs - Slides (3)
104/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvectiondiffusionmodel
1h2 w2h a
uj1 + 2h2 b
uj + 1h2 + w2h c
uj+1 = fj
To show that the approximation is stable we simply need tocheck that the tridiagonal coefficients satisfy (S).
a 0 1
h2+
w
2h 0 (?)
c 0
1
h2
w
2h 0
b a + c 2
h2
2
h2()
104/120
Part III
20401
Apply stability theorem
Recall the tridiagonal matrix entries ..
1 2 1
-
8/3/2019 PDEs - Slides (3)
105/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvection
diffusionmodel
1h2 w2h a
uj1 + 2h2 b
uj + 1h2 + w2h c
uj+1 = fj
To show that the approximation is stable we simply need tocheck that the tridiagonal coefficients satisfy (S).
a 0 1
h2+
w
2h 0 (?)
c 0
1
h2
w
2h 0
b a + c 2
h2
2
h2()
105/120
Part III
20401
Apply stability theorem
Recall the tridiagonal matrix entries ..
1 2 1
-
8/3/2019 PDEs - Slides (3)
106/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvection
diffusionmodel
1h2 w2h a
uj1 + 2h2 b
uj + 1h2 + w2h c
uj+1 = fj
To show that the approximation is stable we simply need tocheck that the tridiagonal coefficients satisfy (S).
a 0 1
h2+
w
2h 0 (?)
c 0
1
h2
w
2h 0
b a + c 2
h2
2
h2()
106/120
Part III
20401
Apply stability theorem
Recall the tridiagonal matrix entries ..
1 w 2 1 w
-
8/3/2019 PDEs - Slides (3)
107/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvection
diffusionmodel
1h2 w2h a
uj1 + 2h2 b
uj + 1h2 + w2h c
uj+1 = fj
To show that the approximation is stable we simply need tocheck that the tridiagonal coefficients satisfy (S).
a 0 1
h2+
w
2h 0 (?)
c 0
1
h2
w
2h 0
b a + c 2
h2
2
h2()
107/120
Part III
20401
Condition for stability
Assume that w > 0 then
1 w
-
8/3/2019 PDEs - Slides (3)
108/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvection
diffusionmodel
a 0 1
h2 +w
2h 0 ()
c 0 1
h2
w
2h 0 (?)
The centred approximation method is stable when
1
h2
w
2h 0
w h
2 1 h
2
w.
108/120
Part III
20401
Stability
Assume that w < 0 then
1 w
-
8/3/2019 PDEs - Slides (3)
109/120
20401
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvection
diffusionmodel
a 0
1
h2 +
w
2h 0 (?)
c 0 1
h2
w
2h 0 ()
Thus, the centred approximation method is stable whenever
1
h2+
w
2h 0
wh
2 1 h
2
w.
We deduce that a sufficient condition for stability is that
|w|h2 1.
The ratio |w|h2 is called the mesh Peclet number.
Computationally, if wh2 > 1 the centred difference solution109/120
Part III
20401
Stability
Assume that w < 0 then
1 w
-
8/3/2019 PDEs - Slides (3)
110/120
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvection
diffusionmodel
a 0
1
h2 +
w
2h 0 (?)
c 0 1
h2
w
2h 0 ()
Thus, the centred approximation method is stable whenever
1
h2+
w
2h 0
wh
2 1 h
2
w.
We deduce that a sufficient condition for stability is that
|w|h2 1.
The ratio |w|h2 is called the mesh Peclet number.
Computationally, if wh2 > 1 the centred difference solutionh 110/120
Part III
20401
Stability
Assume that w < 0 then
1 w
-
8/3/2019 PDEs - Slides (3)
111/120
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvection
diffusionmodel
a 0
1
h2 +
w
2h 0 (?)
c 0 1
h2
w
2h 0 ()
Thus, the centred approximation method is stable whenever
1
h2+
w
2h 0
wh
2 1 h
2
w.
We deduce that a sufficient condition for stability is that
|w|h2 1.
The ratio |w|h2 is called the mesh Peclet number.
Computationally, if wh2 > 1 the centred difference solutionh 111/120
Part III
20401
Solution of convection diffusionmodel with w = 100;
Central differencing h = 1/n with n = 20
-
8/3/2019 PDEs - Slides (3)
112/120
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvection
diffusionmodel
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
12
14
16
x
u
Note the oscillation: the exact solution has no oscillation
112/120
Part III
20401
Alternative method: upwindfinite difference
Solution of convection with w = 100; h = 1/n with n = 20
12
-
8/3/2019 PDEs - Slides (3)
113/120
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvection
diffusionmodel
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
x
u
There is no oscillation.The truncation error for this upwind finite differecing is O(h)
(first order) compared to the second order central differencingmethod.Even though truncation error bigger, the solution is better.
113/120
Part III
20401
Alternative method: Upwindfinite difference method
One sided finite difference approximation:
u(xj +h)
-
8/3/2019 PDEs - Slides (3)
114/120
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvection
diffusionmodel
xj h xj xj +h
u(xj h)
u(xj)
u(xj) +u(xj)
1
hu(xj) u
(xj),1
h+u(xj) u
(xj),
where
u(xj) =u(xj) u(xj h)
+u(xj) =u(xj + h) u(xj).
114/120
Part III
20401
Upwind method
1 For u, use the centered difference: u(xj) 1h2
2u(xj)
2
-
8/3/2019 PDEs - Slides (3)
115/120
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvection
diffusionmodel
2
u(xj) =[u(xj)]=u(xj + h) 2u(xj) + u(xj h)
2 For u, if w > 0, approximate u(xj) 1
h
u(xj)
whereu(xj) = u(xj) u(xj h).
or, if w < 0, approximate u(xj) 1h
+u(xj) where
+u(xj) = u(xj + h) u(xj).
Called the upwind difference approximation to u(x).
115/120
Part III
20401
Upwind method
1 For u, use the centered difference: u(xj) 1h2
2u(xj)
2
-
8/3/2019 PDEs - Slides (3)
116/120
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvection
diffusionmodel
2
u(xj) =[u(xj)]=u(xj + h) 2u(xj) + u(xj h)
2 For u, if w > 0, approximate u(xj) 1
h
u(xj)
whereu(xj) = u(xj) u(xj h).
or, if w < 0, approximate u(xj) 1h
+u(xj) where
+u(xj) = u(xj + h) u(xj).
Called the upwind difference approximation to u(x).
116/120
Part III
20401
Upwind method
1 For u, use the centered difference: u(xj) 1h2
2u(xj)
2
-
8/3/2019 PDEs - Slides (3)
117/120
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvection
diffusionmodel
2
u(xj) =[u(xj)]=u(xj + h) 2u(xj) + u(xj h)
2 For u, if w > 0, approximate u(xj) 1
h
u(xj)
whereu(xj) = u(xj) u(xj h).
or, if w < 0, approximate u(xj) 1h
+u(xj) where
+u(xj) = u(xj + h) u(xj).
Called the upwind difference approximation to u(x).
117/120
Part III
20401
Linear system of eqns
When w > 0 we obtain
1
(u 2u + u ) +w
(u u ) = f
-
8/3/2019 PDEs - Slides (3)
118/120
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvection
diffusionmodel
h2(
j+1 j+
j1) +
h(
j j1)
j
1
h2
w
h
auj1 +
2
h2+
w
h
buj +
1
h2
cuj+1 = fj
Apply BCs u(0) = and u(1) = , this the following linearsystem
b c 0. . .
. . .
a b c. . .
. . .
0 a b
u1...
uj...
un1
=
f1 + a...fj...
fn1 + c
.
118/120
Part III
20401
Consistency and Stability forupwind
HOMEWORK
You can now try Problem 13
-
8/3/2019 PDEs - Slides (3)
119/120
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvection
diffusionmodel
1 You show upwind method is 1st order consistentTaylorexpansions.
2 And the method is stable independent of h.
We knowconsistent + stable convergent
Hence the global erroror the error in approximating true solution u(x) is
|u(xj) uj| = O(h).
119/120
Part III
20401
Consistency and Stability forupwind
HOMEWORK
You can now try Problem 13
-
8/3/2019 PDEs - Slides (3)
120/120
Uniquenessfor reactiondiffusionmodel
Stability fordiffusionmodel
Finitedifferences
Numericalanalysis
Finitedifferences forconvection
diffusionmodel
1 You show upwind method is 1st order consistentTaylorexpansions.
2 And the method is stable independent of h.
We knowconsistent + stable convergent
Hence the global erroror the error in approximating true solution u(x) is
|u(xj) uj| = O(h).
120/120