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Cuchulainn
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### Re: Monotone Schemes: what are they and why are they good?

Your scheme is still ambiguous at best!
You are approaching this problem incorrectly IMHO. A solution looking for a problem? It's all old hat at this stage.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl

JohnLeM
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Joined: September 16th, 2008, 7:15 pm

### Re: Monotone Schemes: what are they and why are they good?

Your scheme is still ambiguous at best!
You are approaching this problem incorrectly IMHO. A solution looking for a problem? It's all old hat at this stage.
What do you mean by ambiguous ? This is a perfect example of a very simple upwind scheme, and AFAIU it is exactly the limit case of the exponential fitting.

I think that this proves that one can't define a stochastic matrix using exponential fitting. Is that correct ?

Anyhow, again, without moving the mesh, it is not possible to define a stochastic matrix for the underlying PDE problem, unless massively twisting the boundary conditions.

JohnLeM
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Joined: September 16th, 2008, 7:15 pm

### Re: Monotone Schemes: what are they and why are they good?

A solution looking for a problem? It's all old hat at this stage.
In this thread, I thought that we were investigating properties of monotone schemes for Finance applications ?
A quite important requirement for pricing is also that the scheme defines a stochastic transition matrix. If I summarize the picture:

1) Such schemes exist in one or two dimensions: trees based methods. They are part of PDE schemes.
2) All classical PDE schemes based on finite differences on fixed grid should fail.
3) One can define such monotone PDE schemes, defining stochastic transition matrix, at the only condition of moving the grid. The minimum being moving the grid according to the advection terms. Note also that there exists schemes moving also the mesh accordingly to the diffusive part: these are the schemes I am using.

Cuchulainn
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### Re: Monotone Schemes: what are they and why are they good?

Your scheme is still ambiguous at best!
You are approaching this problem incorrectly IMHO. A solution looking for a problem? It's all old hat at this stage.
What do you mean by ambiguous ? This is a perfect example of a very simple upwind scheme, and AFAIU it is exactly the limit case of the exponential fitting.
It's unstable as I already said, You are mixing up forward and backward time. Please clear this up.
take initial value problem!
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl

JohnLeM
Posts: 515
Joined: September 16th, 2008, 7:15 pm

### Re: Monotone Schemes: what are they and why are they good?

Your scheme is still ambiguous at best!
You are approaching this problem incorrectly IMHO. A solution looking for a problem? It's all old hat at this stage.
What do you mean by ambiguous ? This is a perfect example of a very simple upwind scheme, and AFAIU it is exactly the limit case of the exponential fitting.
It's unstable as I already said, You are mixing up forward and backward time. Please clear this up.
take initial value problem!
It is not anymore, I already cleaned it up see above: $u_i^{n+1}=u_i^{n} +\frac{dt}{dx}(u_{i+1}^{n} - u_{i}^{n})$ gives $A = (..0,1-\frac{dt}{dx},+\frac{dt}{dx},0...)$, and of course $\frac{dt}{dx} \le 1$. It is a perfectly stable, monotone, first order, upwind scheme. Yet, you won't succeed defining a stochastic transition matrix with this scheme.

Cuchulainn
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### Re: Monotone Schemes: what are they and why are they good?

Ah, OK you had a typo.

Write it down 500 times.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl

Cuchulainn
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### Re: Monotone Schemes: what are they and why are they good?

Now we are on same wavelength, your correct scheme still suffers from $CFL \lt 1$. The next scheme has no such restriction (and it is monotone): using your notation

$u_i^{n+1}=u_i^{n} +\frac{dt}{dx}(u_{i+1}^{n} - u_{i}^{n+1})$

or
$u_i^{n+1} (1 + \frac{dt}{dx}) =u_i^{n} +\frac{dt}{dx}(u_{i+1}^{n})$

Monotone!

QED
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl

JohnLeM
Posts: 515
Joined: September 16th, 2008, 7:15 pm

### Re: Monotone Schemes: what are they and why are they good?

Now we are on same wavelength, your correct scheme still suffers from $CFL \lt 1$. The next scheme has no such restriction (and it is monotone): using your notation

$u_i^{n+1}=u_i^{n} +\frac{dt}{dx}(u_{i+1}^{n} - u_{i}^{n+1})$

or
$u_i^{n+1} (1 + \frac{dt}{dx}) =u_i^{n} +\frac{dt}{dx}(u_{i+1}^{n})$

Monotone!

QED
@Cuchullain, the question is not to know if these schemes are monotone or not. They are monotone, under a CFL condition for the explicit one. The question is: does this class of schemes define a stochastic transition matrix ? The answer is no.

Cuchulainn
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### Re: Monotone Schemes: what are they and why are they good?

The original thread title was "Monotone Methods", so what has changed? Start a new thread.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl

JohnLeM
Posts: 515
Joined: September 16th, 2008, 7:15 pm

### Re: Monotone Schemes: what are they and why are they good?

The original thread title was "Monotone Methods", so what has changed? Start a new thread.
@Cuchulainn, you are slightly of bad faith here I am expecting an answer : does exponential fitting produces a stochastic transition probability matrix or not ? If you prefer the same question but with another sight view: could I use exponential fitting for my framework or not (as an alternative to the schemes that I am using) ?

Cuchulainn
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### Re: Monotone Schemes: what are they and why are they good?

The original thread title was "Monotone Methods", so what has changed? Start a new thread.
@Cuchulainn, you are slightly of bad faith here I am expecting an answer : does exponential fitting produces a stochastic transition probability matrix or not ? If you prefer the same question but with another sight view: could I use exponential fitting for my framework or not (as an alternative to the schemes that I am using) ?
Your question is too vague.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl

JohnLeM
Posts: 515
Joined: September 16th, 2008, 7:15 pm

### Re: Monotone Schemes: what are they and why are they good?

The original thread title was "Monotone Methods", so what has changed? Start a new thread.
@Cuchulainn, you are slightly of bad faith here I am expecting an answer : does exponential fitting produces a stochastic transition probability matrix or not ? If you prefer the same question but with another sight view: could I use exponential fitting for my framework or not (as an alternative to the schemes that I am using) ?
Your question is too vague.
My question is very precise : does the upwind scheme defines a stochastic transition matrix ? I wrote you the scheme and the transition matrix just few lines above. We both know the answer, and I apologize for pushing this nail into.

What I would like to know is if we can give another answer than negative for the exponential fitting. I think that yes, with boundary conditions, but it will have a cost.

Cuchulainn
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### Re: Monotone Schemes: what are they and why are they good?

Do you know

1. Why exponential fitting?
2. Where can it be applied, and
3. Where not?
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl

JohnLeM
Posts: 515
Joined: September 16th, 2008, 7:15 pm

### Re: Monotone Schemes: what are they and why are they good?

Do you know

1. Why exponential fitting?
2. Where can it be applied, and
3. Where not?
Exponential fitting seems a good numerical scheme, hence I look forward to read your answers here.

Cuchulainn
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### Re: Monotone Schemes: what are they and why are they good?

It's OK. But not universal. No silver bullet.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl