Numerical Methods

1
Numerical Methods

• Marisa Villano, Tom Fagan,
• Dave Fairburn, Chris Savino,
• David Goldberg, Daniel Rave

2
An Overview

• The Method of Finite Differences
• Error Approximations and Dangers
• Approxmations to Diffusions
• Crank Nicholson Scheme
• Stability Criterion

3
Finite Differences

• Best known numerical method of approximation
• Marisa Villano

4
Finite Differences

• Approximating the derivative with a difference
  quotient from the Taylor series
• Function of One Variable
• Choose mesh size ?x
• Then uj u(j?x)

5
First Derivative Approximations

• Backward difference (uj uj-1) / ?x
• Forward difference (uj1 uj) / ?x
• Centered difference (uj1 uj-1) / 2?x

6
Taylor Expansion
2

• u(x ?x) u(x) u?(x)?x 1/2 u?(x)(?x)
• 1/6 u??(x)(?x) O(?x)
• u(x ?x) u(x) u?(x)?x 1/2 u?(x)(?x)
• - 1/6 u??(x)(?x) O(?x)

3
4
2
3
4
7
Taylor Expansion

• u?(x) u(x) u(x ?x) O(?x)
• ?x
• u?(x) u(x ?x) u(x) O(?x)
• ?x
• u?(x) u(x ?x) u(x ?x) O(?x)
• 2?x

2
8
Second Derivative Approximation
2

• Centered difference (uj1 2uj uj-1) / (?x)
• Taylor Expansion
• u?(x) u(x ?x) 2u(x) u(x ?x) O(?x)
• (?x)

2
2
9
Function of Two Variables
n

• u(j?x, n?t) uj
• Backward difference for t and x
• (j?x, n?t) (uj uj ) / ?t
• (j?x, n?t) (uj uj ) / ?x

?u
n
n-1
?t
?u
n
n-1
?x
10
Function of Two Variables

• Forward difference for t and x
• (j?x, n?t) (uj uj ) / ?t
• (j?x, n?t) (uj uj ) / ?x

?u
n1
n
?t
?u
n1
n
?x
11
Function of Two Variables

• Centered difference for t and x
• (j?x, n?t) (uj uj ) / (2?t)
• (j?x, n?t) (uj uj ) / (2?x)

?u
n1
n-1
?t
?u
n1
n-1
?x
12
Error

• Truncation Error introduced in the solution by
  the approximation of the derivative
• Local Error from each term of the equation
• Global Error from the accumulation of local
  error
• Roundoff Error introduced in the computation by
  the finite number of digits used by the computer

13
The Dangers of the Finite Difference Method

• Evidence from an example in 8.1
• Dave Fairburn

14
Example from 8.1

• Consider ut uxx u(x,0) h(x)
• We will use the finite difference method to
  approximate the solution
• Forward difference for ut
• Centered difference for uxx
• Re-write equation in terms of the finite
  difference approximations

15
Finite Difference Eqn.

• ujn1 - ujn unj1 - 2ujn unj-1

t
x
(
)
2
Error The local truncation error is O(
t)
from the left hand side and is O(
x)2 from
the right hand side.
16
Assumptions

• Assume that we choose a small change in x, and
  that the denominator on both sides of the
  equation are equal.
• We are now left with the scheme
• ujn1 unj1 - unj unj-1
• Solving u with this scheme is now easy to do once
  we have the initial data.

17
Initial Data

• Let u(x,0) h(x) a step function with the
  following properties
• h(x) 0 for all j except for j 5, so
• hj 0 0 0 0 1 0 0 0 0 0 0 .
• Initially, only a certain section, which is at j
  5 is equal to the value of 1.
• j serves as the counter for the x values.

18
How to solve?

• We know u0j 1 at j 5 and 0 at all other j
  initially (given by superscript 0).
• We can plug into our scheme to solve for u1j at
  all js.
• u1j u0j-1 - u0j u0j1
• u15 -1 u14 1 u16 1
• Now we can continue to increase the of
  iterations, n, and create a table

19
Solution for 4 iterations
4 1 -4 10 -16 19 -16 10 -4 1 0
3 0 1 -3 6 -7 6 -3 1 0 0
2 0 0 1 -2 3 -2 1 0 0 0
1 0 0 0 1 -1 1 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0
1 2 3 4 5 6 7 8 9 10
n-values
j values
20
Analysis of Solution

• Is this solution viable?
• Maximum principle states that the solution must
  be between 0 and 1 given our initial data
• At n 4, our solution has already ballooned to u
  19!
• Clearly, there are cases when the finite
  difference method can pose serious problems.

21
Charting the Error

• Assume the solution is constant and equal to 0.5
  (halfway between the possible 0 and 1)

22
Lessons Learned

• While the finite difference method is easy and
  convenient to use in many cases, there are some
  dangers associated with the method.
• We will investigate why the assumption that
  allowed us to simplify the scheme could have been
  a major contributor to the large error.

23
Approximations of Diffusions

• Neumann Boundary Conditions and the
  Crank-Nicolson Scheme
• Chris Savino

24
Approximations of Diffusions

• Errors have accumulated from the approximations
  of the derivatives using the previous scheme
• The problem is the choice of the mesh ?t to the
  mesh ?x
• Let s
• can solve scheme

25
Neumann Boundary Conditions

• 0 x l
• Simplest Approximations are

26

• To get smallest error, we use centered
  differences for the derivatives on the boundary
• Introduce ghost points
• Boundary Conditions become

27
Crank-Nicolson Scheme

• Can avoid any restrictions on stability
  conditions
• Unconditionally stable no matter what the value
  of s is.

28

• Centered Second Difference
• Pick a number theta between 0 and 1
• Theta scheme

29

• We analyze the scheme by plugging in a separated
  solution
• Therefore

30

• Must Check stability condition
• If then
• Therefore
• 
  is always true

31

• If then there is no restriction on
  the size of s for stability to hold
• The scheme is unconditionally stable
• When theta ½ it is called the Crank-Nicolson
  scheme
• If theta lt ½ then the scheme is stable if

32
Stability Criterion

• Approximations of the diffusion equation, utuxx
• David Goldberg

33
Stability Criterion

• The method of finite differences gives an answer,
  but it does not guarantee that this answer is
  meaningful.
• Values must be chosen appropriately, to ensure
  that the results make sense and are applicable to
  real world scenarios.
• This condition, that values must satisfy in order
  to be worthwhile, is called the stability
  criterion.

34
Example

• As per the book, take, for instance, the
  diffusion problem

35
Example, continued

• As can be easily shown, the graph of f(x) looks
  like this.

36
Example, continued

• In attempting to use the method of finite
  differences, we are using a forward difference
  for ut and a centered difference for uxx.
• This means that
• It is important to note here that the superscript
  n denotes a counter on the t variable, and the
  subscript j denotes a counter on the x variable.

37
Example, continued

• In order to make the calculations a bit cleaner,
  we are introducing a variable, s, which is
  defined by
• Rearranging, we have
• It would be nice if we could just plug in values
  and get a valid result

38
Example, continued

• However, putting in different values can lead to
  the results being close to, or far from, that
  actual answer.
• For instance, letting ?xp/20, and letting
  s5/11, we get a relatively nice result. Letting
  s5/9 does not get such a nice result.
• So what, of significance, changes?

39
Example, Continued

• As it turns out, changing the value of s can
  significantly change the validity of the
  solution. To see why, we return to our equation.

40
Example, continued

• Since the left hand side is a function of T and
  the right hand side is a function of X, they must
  be equal to a constant.

41
Example, continued

• This is a discrete version of an ODE, which when
  solved gives

42
Example, finished

• Thus, to achieve stability, . This
  is why setting s5/9 didnt give a valid result.
• It is to be noted that usually the necessary
  criterion is that
  , but that in this case it was irrelevant.
• So the stability criterion must be worked out
  before one can effectively use the method of
  finite differences.

43
Approximations of Diffusions

• Example from 8.2
• Daniel Rave

44
Summary

• Breif Review of Methods
• Wide Applicability
• Importance of Stability