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I am getting a solution with NDSolve for studying the collapse of an interstellar isothermal cloud.

But I have two problems and questions:

  1. I am getting some warnings even if I am obtaining a good result : does that mean that something is wrong?
  2. I am doing a parametric study and I need to get a much faster solution (now about 1 minute). Is there some ideas to improve my code ?

Here is the code in condensed shape.

r0 = 10^-17.; rR = 6.8;
φK = 
  NDSolveValue[{+((2 Derivative[1][φ][r])/
       r) + Derivative[2][φ][r] == E^-φ[r], 
    Derivative[1][φ][r0] == 0, φ[r0] == 
     0}, φ, {r, r0, 1000}];
ϱK[r_] = E^-φK[r]; f[r_] = -D[φK[r], r];

With[{F = F[t, r], ρ = ρ[t, r], u = u[t, r]}, 
 eq = {r^2 D[ρ, t] + D[r^2 ρ u, r] == 
    0, ρ D[ u, t] + ρ u D[u, r] + D[ρ, r] + 
     Nfr F ρ == 0, ρ r^2 - D[r^2 F, r] == 0};
 ic = {F == f[r], ρ == ϱK[r], u == 0} /. t -> 0;
 bc = {{u == 1. 10^-4, F == 0} /. r -> r0, {u == 1 10^-4} /. r -> rR};]
 
mol[n : _Integer | {_Integer ..}, 
  o_: "Pseudospectral"] := {"MethodOfLines", 
  "DifferentiateBoundaryConditions" -> False, 
  "SpatialDiscretization" -> {"TensorProductGrid", "MaxPoints" -> n, 
    "MinPoints" -> n, "DifferenceOrder" -> o}, Method -> "Automatic"}

Using now NDSolve

mysol = ParametricNDSolveValue[{eq, ic, bc, 
   WhenEvent[ρ[t, r0] > 5. 10^4, tend = t; 
    "StopIntegration"]}, {ρ, F, u}, {t, 0, 30}, {r, r0, rR}, Nfr,
   Method -> mol[89], MaxStepFraction -> {1/10^4, Infinity}]

Looking for the density at the origin with Nfr=1.02

dens[t_, r_] = mysol[1.02][[1]][t, r]

Here I am getting some warnings

ParametricNDSolveValue::pdord: Some of the functions have zero differential order, so the equations will be solved as a system of differential-algebraic equations.

ParametricNDSolveValue::bcart: Warning: an insufficient number of boundary conditions have been specified for the direction of independent variable r. Artificial boundary effects may be present in the solution.

ParametricNDSolveValue::ivres: NDSolve has computed initial values that give a zero residual for the differential-algebraic system, but some components are different from those specified. If you need them to be satisfied, giving initial conditions for all dependent variables and their derivatives is recommended.

But I am getting some good results

LogPlot[dens[t, r0], {t, 0, tend}, PlotRange -> {0.2, 2 10^8}, 
 Frame -> True, PlotStyle -> {Blue}, 
 FrameLabel -> {"t réduit", "ρ réduit", 
   "Collapse et augmentation de la densité en r=r0"}]

Collapse and increase of density at r~0

Thanks a lot for any answers to my two questions.

PS :

Sorry , here are some precisions

  1. The system of equations comes from this link https://www.seas.harvard.edu/brenner/On_spherically.pdf

enter image description here

2 ) I used the acceleration F instead of the spatial derivative of the potential.

  1. Here are some examples of results giving the incresae of central density, different shapes of density for giving times, and different speed for different times. It seems to work but I am not sure at all....

enter image description here enter image description here enter image description here

*** EDIT 1 *****

Hera are two physical BC . But Mathematica stops .This is the reason why I used the acceleration F[t,r] instead of the derivative of the potential Phi : I was unable to enter a condition such like Derivative[0,1] [Phi[t,r0]==0 but it was Ok to enter F[t,r0]==0 !

Why does Mathematica rejects such BC ?? This is the main mystery for me.....

Derivative[0, 1][\[Rho][t, r0]] == 0; Derivative[1, 0][u[t, r0]] == 0

NDSolveValue::bdord: Boundary condition (\[Rho]^(0,1))[t,1.*10^-17] should have derivatives of order lower than the differential order of the partial differential equation.

EDIT 2

@ Alex Trounev from Hergé I am using Mathematica 12.0 and the code runs . I tried using viscosity but it doesnt'work. The model in reference is coherent except may be the factor N (Froude number) used by the authors. With N=1, equations are the same as ones in the very known following paper. https://ui.adsabs.harvard.edu/abs/1993ApJ...416..303F/abstract The isothermal sphere (or Bonnor Ebert sphere) should be stable or unstable and collapses depending on the mass of the cloud (Bonnor Ebert mass). My aim is to see if Mathematica 12.0 can solve this kind of Euler 1D problem but you are right : physic must be correct !

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    $\begingroup$ Please notice the bcart warning is a serious problem: mathematica.stackexchange.com/q/73961/1871 Where does this equation system come from? Can you add some more background info? $\endgroup$
    – xzczd
    Oct 5 '20 at 10:11
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    $\begingroup$ Model is not looks like collapse model. See for example my answer on mathematica.stackexchange.com/questions/205035/… $\endgroup$ Oct 5 '20 at 14:19
  • $\begingroup$ Thanks to xzczd and Alex Trounev. I hope my PS will answer to your first questions. $\endgroup$
    – Hergé
    Oct 7 '20 at 14:41
  • $\begingroup$ @Alex Trounev Equations are very different because I am working in a Newtonian frame, with very low speeds , and I am adressing the collapse of a simple isothermal (10°K) gas cloud. This is very far frome General RElativity ans the collapse of black hole with very huge relativistic potentials. $\endgroup$
    – Hergé
    Oct 7 '20 at 14:56
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    $\begingroup$ Yes it's possible, just adjust sub-option Coordinates of TensorProductGrid method. Here's an example: mathematica.stackexchange.com/a/204882/1871 For more info check the document of "Method of Lines". $\endgroup$
    – xzczd
    Oct 27 '20 at 14:41

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