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# Numerical Scheme - PowerPoint PPT Presentation

Numerical Methods. Marisa Villano, Tom Fagan, Dave Fairburn, Chris Savino, David Goldberg, Daniel Rave. An Overview. The Method of Finite Differences Error Approximations and Dangers Approxmations to Diffusions Crank Nicholson Scheme Stability Criterion. Finite Differences.

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### Numerical Methods

Marisa Villano, Tom Fagan,

Dave Fairburn, Chris Savino,

David Goldberg, Daniel Rave

• The Method of Finite Differences

• Error Approximations and Dangers

• Approxmations to Diffusions

• Crank Nicholson Scheme

• Stability Criterion

### Finite Differences

Best known numerical method of approximation

Marisa Villano

• Approximating the derivative with a difference quotient from the Taylor series

• Function of One Variable

• Choose mesh size Δx

• Then uj~ u(jΔx)

• Backward difference: (uj – uj-1) / Δx

• Forward difference: (uj+1 – uj) / Δx

• Centered difference: (uj+1 – uj-1) / 2Δx

2

• u(x + Δx) = u(x) + u΄(x)Δx + 1/2 u˝(x)(Δx)

+ 1/6 u˝΄(x)(Δx) + O(Δx)

• u(x – Δx) = u(x) – u΄(x)Δx + 1/2 u˝(x)(Δx)

- 1/6 u˝΄(x)(Δx) + O(Δx)

3

4

2

3

4

u΄(x) = u(x) – u(x – Δx) + O(Δx)

Δx

u΄(x) = u(x + Δx) – u(x) + O(Δx)

Δx

u΄(x) = u(x + Δx) – u(x – Δx) + O(Δx)

2Δx

2

2

• Centered difference: (uj+1 – 2uj + uj-1) / (Δx)

• Taylor Expansion

u˝(x) = u(x + Δx) – 2u(x) + u(x – Δx) + O(Δx)

(Δx)

2

2

n

u(jΔx, nΔt) ~ uj

• Backward difference for t and x

(jΔx, nΔt)~ (uj– uj ) / Δt

(jΔx, nΔt)~ (uj– uj ) / Δx

∂u

n

n-1

∂t

∂u

n

n-1

∂x

• Forward difference for t and x

(jΔx, nΔt)~ (uj – uj ) / Δt

(jΔx, nΔt)~ (uj – uj ) / Δx

∂u

n+1

n

∂t

∂u

n+1

n

∂x

• Centered difference for t and x

(jΔx, nΔt)~ (uj – uj ) / (2Δt)

(jΔx, nΔt)~ (uj – uj ) / (2Δx)

∂u

n+1

n-1

∂t

∂u

n+1

n-1

∂x

• Truncation Error: introduced in the solution by the approximation of the derivative

• Local Error: from each term of the equation

• Global Error: from theaccumulation of local error

• Roundoff Error: introduced in the computation by the finite number of digits used by the computer

### The Dangers of the Finite Difference Method

Evidence from an example in 8.1

Dave Fairburn

• 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

• ujn+1 - ujn = unj+1 - 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.

• 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:

ujn+1 = unj+1 - unj + unj-1

• Solving u with this scheme is now easy to do once we have the 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.

• 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 j’s.

• u1j = u0j-1 - u0j + u0j+1

• u15 = -1; u14 = 1; u16 = 1

• Now we can continue to increase the # of iterations, n, and create a table…

n-values

j values

• 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.

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

• 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.

### Approximations of Diffusions

Neumann Boundary Conditions and the Crank-Nicolson Scheme

Chris Savino

• Errors have accumulated from the approximations of the derivatives using the previous scheme

• The problem is the choice of the mesh Δt to the mesh Δx

• Let s=

can solve scheme

• 0 x l

• Simplest Approximations are

Crank-Nicolson Scheme derivatives on the boundary

• Can avoid any restrictions on stability conditions

• Unconditionally stable no matter what the value of s is.

### Stability Criterion for stability to hold

Approximations of the diffusion equation, ut=uxx

David Goldberg

Stability Criterion for stability to hold

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.”

Example for stability to hold

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

Example, continued for stability to hold

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

Example, continued for stability to hold

In attempting to use the method of finite differences, we are using a forward difference for utand 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.

Example, continued for stability to hold

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…

Example, continued for stability to hold

However, putting in different values can lead to the results being close to, or far from, that actual answer.

For instance, letting ∆x=π/20, and letting s=5/11, we get a relatively nice result. Letting s=5/9 does not get such a nice result.

So what, of significance, changes?

Example, Continued for stability to hold

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.

Example, continued for stability to hold

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.

Example, continued for stability to hold

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

Example, finished for stability to hold

Thus, to achieve stability, . This is why setting s=5/9 didn’t 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.

### Approximations of Diffusions for stability to hold

Example from 8.2

Daniel Rave

Summary for stability to hold

• Breif Review of Methods

• Wide Applicability

• Importance of Stability