Not exactly.@Cuchulain I was rereading your exponential fitting paper. As far as I understand, you produce a monotone scheme using a pertinent change of variable of the mesh itself. This change of variable ensures that the discretization of the advection terms are dominated by the diffusion ones. Technically, you end writing a M-matrix to conclude. Is that correct ?
Is it the diffusion term is denoted [$]D^+D^-[$] in your paper ? Isn't it a standard discretization of second order derivative on non uniform grid ?Not exactly.@Cuchulain I was rereading your exponential fitting paper. As far as I understand, you produce a monotone scheme using a pertinent change of variable of the mesh itself. This change of variable ensures that the discretization of the advection terms are dominated by the diffusion ones. Technically, you end writing a M-matrix to conclude. Is that correct ?
We use a modified diffusion term.
You're guessing! please do a bit of research.Is it the diffusion term is denoted [$]D^+D^-[$] in your paper ? Isn't it a standard discretization of second order derivative on non uniform grid ?Not exactly.@Cuchulain I was rereading your exponential fitting paper. As far as I understand, you produce a monotone scheme using a pertinent change of variable of the mesh itself. This change of variable ensures that the discretization of the advection terms are dominated by the diffusion ones. Technically, you end writing a M-matrix to conclude. Is that correct ?
We use a modified diffusion term.
I read quickly. You define the fitting factor in (31), the full scheme is at (33). (34) shows that the scheme is consistent with the equationYou're guessing! please do a bit of research.Is it the diffusion term is denoted [$]D^+D^-[$] in your paper ? Isn't it a standard discretization of second order derivative on non uniform grid ?
Not exactly.
We use a modified diffusion term.
Exponential fitting A-Z is described here, RTFM in bocca al lupo!
RTFMI read quickly. You define the fitting factor in (31), the full scheme is at (33). (34) shows that the scheme is consistent with the equationYou're guessing! please do a bit of research.
Is it the diffusion term is denoted [$]D^+D^-[$] in your paper ? Isn't it a standard discretization of second order derivative on non uniform grid ?
Exponential fitting A-Z is described here, RTFM in bocca al lupo!
[$]\sigma_h u'' + \mu u' = 0[$] (forget the others terms), [$]\sigma_h \sim \sigma + \frac{\mu^2 h^2}{\sigma}[$]
This might cause problems as [$]\sigma \mapsto 0[$] ?
You could also feel empathy for another mathematician that is interested by your work and is struggling to analyze itRTFMI read quickly. You define the fitting factor in (31), the full scheme is at (33). (34) shows that the scheme is consistent with the equation
You're guessing! please do a bit of research.
Exponential fitting A-Z is described here, RTFM in bocca al lupo!
[$]\sigma_h u'' + \mu u' = 0[$] (forget the others terms), [$]\sigma_h \sim \sigma + \frac{\mu^2 h^2}{\sigma}[$]
This might cause problems as [$]\sigma \mapsto 0[$] ?
Section 5.4 (Graceful degradation) ... upwinding QED.
I read quickly
Not a good idea. I really don' have time to explain that have already been documented. It's time wastage.
Another way is to split the equation : move the grid according to the advection equation. Then solve an auto-adjoint problem similar to the heat equation, for which monotoniticy is an easy task. Doing so one does not destroy the nice properties of CN and can interpret the diffusion as a Markov chain.That paper was from 1998, so it was corrected in my later books.
I am a trainer so I expect you to read what I post Especially stuff I invented.
One way to get monotonicity without fitting is to use use upwinding in convection.
Looks like overkill. What's the story about Markov??Another way is to split the equation : move the grid according to the advection equation. Then solve an auto-adjoint problem similar to the heat equation, for which monotoniticy is an easy task. Doing so one does not destroy the nice properties of CN and can interpret the diffusion as a Markov chain.That paper was from 1998, so it was corrected in my later books.
I am a trainer so I expect you to read what I post Especially stuff I invented.
One way to get monotonicity without fitting is to use use upwinding in convection.
Overkill is the only way at my disposal to tackle efficiently high dimensional problems.Looks like overkill. What's the story about Markov??Another way is to split the equation : move the grid according to the advection equation. Then solve an auto-adjoint problem similar to the heat equation, for which monotoniticy is an easy task. Doing so one does not destroy the nice properties of CN and can interpret the diffusion as a Markov chain.That paper was from 1998, so it was corrected in my later books.
I am a trainer so I expect you to read what I post Especially stuff I invented.
One way to get monotonicity without fitting is to use use upwinding in convection.
There are several methods >> CN around.
I would need a bit more convincing on that one.Overkill is the only way at my disposal to tackle efficiently high dimensional problems.Looks like overkill. What's the story about Markov??
Another way is to split the equation : move the grid according to the advection equation. Then solve an auto-adjoint problem similar to the heat equation, for which monotoniticy is an easy task. Doing so one does not destroy the nice properties of CN and can interpret the diffusion as a Markov chain.
There are several methods >> CN around.
For Markov: you can interpret a scheme as a Markov Chain provided u^{n+1}=Au^n, A stochastic matrix (a monotone scheme is necessary, but not sufficient. You need conservative properties). If you consider a heat equation with advection through exponential fitting or upwind, A is no longer stochastic. You can't anymore interpret your equation as a Markov chain. But, we really need a Markov interpretation for PDE pricing, for instance if one want to price Asian type options or Autocalls.
easy: consider simply the upwind scheme and try to deduce transition probabilities from it. [$]u_i^{n+1}=u_i^{n} -\frac{dt}{dx}(u_{i+1}^{n} - u_{i}^{n})[$] gives [$] A = (..0,1+\frac{dt}{dx},-\frac{dt}{dx},0...)[$]. You get one negative entries and the diagonal is greater than one, this is not a stochastic matrix. If you consider a backward difference, this is better, but boundary conditions will ruin it. If you move the mesh, you get [$]u_i^{n+1}=u_i^{n}, A = I_d[$], a beautiful stochastic matrix. Hence : if the exponential fitting degenerates as an upwind scheme, I can't use it straightforwardly - unless there are some trick to recover a stochastic matrix from it.I would need a bit more convincing on that one.Overkill is the only way at my disposal to tackle efficiently high dimensional problems.
Looks like overkill. What's the story about Markov??
There are several methods >> CN around.
For Markov: you can interpret a scheme as a Markov Chain provided u^{n+1}=Au^n, A stochastic matrix (a monotone scheme is necessary, but not sufficient. You need conservative properties). If you consider a heat equation with advection through exponential fitting or upwind, A is no longer stochastic. You can't anymore interpret your equation as a Markov chain. But, we really need a Markov interpretation for PDE pricing, for instance if one want to price Asian type options or Autocalls.
Yes you are right. So let us write this scheme properly:Your scheme above
[$]u_i^{n+1}=u_i^{n} -\frac{dt}{dx}(u_{i+1}^{n} - u_{i}^{n})[$]
is unconditionally unstable (use von Nemann discrete Fourier analysis) if we take (de facto standard) forward time [$]t > 0[$]. But maybe you mean something else (e.g. the confusing 'binomial' way [$]t < T[$] in this context).
Even if we get the FDM 'right' you need to satisfy CFL (Courant-Friedrichs-Lewy) constraint [$]\Delta t/\Delta x < 1[$].I have removed this constraint in my 2018 C++ book (p. 748) and it is unconditionally stable.