On suggestion from JohnLeM.
Always good to start with a definition,thus avoiding other alternate mental models
https://www.cfd-online.com/Wiki/Monotone_scheme
Excellent !On suggestion from JohnLeM.
Always good to start with a definition,thus avoiding other alternate mental models
https://www.cfd-online.com/Wiki/Monotone_scheme
Some incentives towards monotone schemes that comes to my mind:On suggestion from JohnLeM.
Always good to start with a definition,thus avoiding other alternate mental models
https://www.cfd-online.com/Wiki/Monotone_scheme
Both. For instance consider the one-dimensional heat equation, and the notations above ([$]u_i^{n+1} \sim u(t^n,x^i)[$]). To be precise, there exists a finite-difference scheme [$]u^{n+1} = H u^{n}[$], H being a [$]I \times I[$] matrix having REAL eigenvalues, satisfying BOTH propertiesunconditionally stables,
yes and no (A or L?)
Yes you are right in a general context. However this stability property is equivalent to monotonicity for finite-difference schemes for the heat equations. Anyhow, I just meant monotone in the sense above: H has real eigenvalues.Point 1) is stability in some norm, it has nothing to do with monotone scheme which is one that satisfies maximum principle.
Ok ok, I'll send the math to your datasim address to have your feedback on it.Can you write down the precise 'ehancements'? e.g. for heat equation and then BS?
You are right ! Might it be a side-effect of artificial quantum intelligence and economical slump, dragging out young brains to twilight zone ?Thanks!
BTW have you noticed how quiet it is here! Where have all the flowers gone?
www.youtube.com/watch?v=Zy9aB4WrgqM
The Crank-Nicolson scheme becomes monotone for small enough time-steps (in relation to the space-steps ^2). Either the limit corresponds to the explicit scheme stability limit or to twice the latter, I don't remember exactly.
Hélas non :/ Crank Nicolson approach always produces non monotone schemes. To experience them: take the Crank Nicolson scheme for the heat equation, with a nasty initial condition, as heavyside (u_0(x) = 0, x<0, +1, x >0}. You should see oscillations whatever the time step is. But it is true that oscillations are damped out with smaller time step.The Crank-Nicolson scheme becomes monotone for small enough time-steps (in relation to the space-steps ^2). Either the limit corresponds to the explicit scheme stability limit or to twice the latter, I don't remember exactly.
I started to work on it. I need a half day more to finish a first readable draft. I hope you will receive it at end of this week !Can you write down the precise 'ehancements'? e.g. for heat equation and then BS?