Timeline for Problems with solving PDEs
Current License: CC BY-SA 4.0
24 events
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| Sep 16, 2019 at 13:12 | comment | added | bbgodfrey | @user55777 "symbolic asymptotic solution, {(9 - 100 v0)/(-59 + 100 v0), v0}" | |
| Sep 16, 2019 at 4:23 | comment | added | bbgodfrey |
The first thing to check is whether the equilibrium described near the end of my answer is stable. Do this by linearizing the equations about this equilibrium. Then, Fourier-transform the linearized equations in x Laplace-transform them in t, and determine whether the resulting eigensystem has exponentially growing modes in t. If so, the equilibrium is unstable, and numerical solutions always will blow up in time. If the equilibrium is stable, on the other hand, then the numerical method used by NDSolve must be unstable, and attention should be given to finding a stable algorithm.
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| Sep 15, 2019 at 13:25 | comment | added | user55777 | @bbgodfrey thank you for the suggestions! What is the relation between the 1D and 2D solutions? Does the steady state of 1D solution imply that 2D solution should also reach a steady state under instabilities governed by the x-dependent terms? In other words, if those terms are destabilizing/stabilizing effects the 2D solution will reach a steady state more slow/quickly compared to that of 1D solution. So the existing numerical instability is due to the not-that-good spatial discretization? | |
| Sep 13, 2019 at 12:40 | comment | added | bbgodfrey |
@user55777 I chose 2/5, because it was somewhat larger than the value of v for the initial plateau but still much less than 1. By asymptotic, I meant steady state. The oscillations could be from a physical instability, but they still have to start from some irregularity in x. Because they grow even with c = 0, the only source left is numerical noise. Thanks for checking the spatial derivative terms.
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| Sep 13, 2019 at 8:05 | comment | added | user55777 | @bbgodfrey Very impressive. By eliminating x-dependency, you obtained the "asymptotic solution". What did you mean by asymptotic solution here? Do you think that if the 2D eqns have a steady state, it should be (close to) the asymptotic soln after the same order of transition time as in the 1D soln? | |
| Sep 13, 2019 at 7:25 | comment | added | user55777 | @bbgodfrey for the 1D solution from the constant initial conditions, you said that "these oscillations are growing from truncation errors". I guess they could grow from the instabilities described by the equation. Am I right? | |
| Sep 13, 2019 at 5:14 | comment | added | user55777 |
@bbgodfrey many thanks for your valuable time. I understand that you stop the integration by monitoring the max among 7 x-positions of v. Well, is there any special reason for choosing the criterion 2/5 in the event Max@Table[v[L (i - 1/2)/7, t], {i, 7}] > 2/5?
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| Sep 13, 2019 at 4:01 | comment | added | bbgodfrey |
@user55777 Yes, I did. However, if WhenEvent is used to terminate the computation before it reaches v == 1, as I did for the fourth and fifth plots, then using Max is unnecessary. By the way, I have some new ideas for investigating the issues here, but I will not have time to do so for a few days.
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| Sep 13, 2019 at 3:52 | comment | added | user55777 |
@bbgodfrey "replacing (1 - v[x, t]) by Max[1 - v[x, t], 10^-8] in the right side of eqv"... did you replace both two terms on rhs?
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| Sep 11, 2019 at 13:27 | comment | added | bbgodfrey | @user55777 I was careless in my choice of words. I did mean "too big". I also added the code that created the first plot. | |
| Sep 11, 2019 at 13:23 | history | edited | bbgodfrey | CC BY-SA 4.0 |
corrected comment on size of MaxtepSize and provided code for first figure.
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| Sep 11, 2019 at 9:48 | comment | added | user55777 |
@bbgodfrey "...MaxStepFraction is too small to resolve sharp variations in the solution." I always supposed that it is a too big MaxStepFraction that fails to resolve sharp variations in the solution. Please give me some advice.
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| Sep 11, 2019 at 9:42 | comment | added | user55777 |
@bbgodfrey sorry for the late reply. I need time to go through all the reply. Btw, fig.1 is a plot of the problem term, which term did you refer to? Did you mean 1 - v[x, t]?
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| Sep 11, 2019 at 3:35 | history | edited | bbgodfrey | CC BY-SA 4.0 |
fixed typo
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| Sep 10, 2019 at 21:08 | history | edited | bbgodfrey | CC BY-SA 4.0 |
added Addendum
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| Sep 10, 2019 at 15:12 | comment | added | bbgodfrey |
@user55777 I was mistaken. The 2D solution does not come close to reaching steady state, as can be seen by solving the 1D problem numerically. (The symbolic staty state in 1D is u == 11/39, v == 1/5.) Are you sure that the spatial derivative terms in the 2D equations are correct?
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| Sep 10, 2019 at 13:37 | comment | added | user55777 | @bbgodfrey could you show by plotting that it reaches such a steady state? | |
| Sep 10, 2019 at 13:32 | comment | added | bbgodfrey |
@user55777 It reaches a steady state at around t == 50 but goes unstable at around t == 100.
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| Sep 10, 2019 at 13:18 | comment | added | user55777 |
@bbgodfrey when observing u we can see its peaks increase in time near tmax, why did you say the solution reaches a steady state rapidly? Could you update by adding some descriptions?
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| Sep 10, 2019 at 12:52 | comment | added | bbgodfrey | @user55777 The solution rapidly reaches a steady state but apparently becomes numerically unstable at much later time. | |
| Sep 10, 2019 at 12:15 | comment | added | bbgodfrey |
@xzczd I am aware of this capability, but the variation in x and in t both are extreme.
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| Sep 10, 2019 at 7:05 | comment | added | xzczd♦ |
Perhaps you can adjust temperal grid and spatial grid separately using e.g. MaxStepFraction -> {1/30, 1/1000}? (More discussion can be found here. )
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| Sep 10, 2019 at 5:52 | history | edited | bbgodfrey | CC BY-SA 4.0 |
added plot
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| Sep 10, 2019 at 5:37 | history | answered | bbgodfrey | CC BY-SA 4.0 |