<div dir="ltr"><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">Dear Flash users,</p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><br></p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">I have a numerical problem and I was wondering if someone knows this kind of issue.</p>
<p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"> </p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">In my astrophysical problem, I have a strong radiative cooling layer separated by a </p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">shock and a contact discontinuity.</p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><br></p>
<p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">In my simulation when the cooling is really strong, the gradients of density, pressure and temperature become really strong too. </p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">At some point there is in this layer two region, one is the cooling layer and another is an extremely thin spike of cooled matter.</p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><br></p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">When I
try a better refinement I find that the density spike of high compression is 2 times greater if I multiply the refinement by 2 </p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">and
actually, <span style="font-size:11pt">the density would diverge if I do not stop to increase the refinement
level at some point for the new simulations.</span></p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:11pt">So I chose a refinement of the
space to resolve the cooling layer very well except for this spike of density
near the contact discontinuity. </span></p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:11pt"><br></span></p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:11pt">Then, I try to decrease the CFL condition
coefficient to see if the solution is sensitive to the decrease in time step. </span></p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:11pt">The problem is that for each different simulations with
different levels of refinement, </span></p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:11pt">the decrease in the CFL condition changes the
numerical solution (and not even in the same way for every different simulation
with different spatial refinement). </span></p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:11pt">For example, in some simulations with a given refinement, </span><span style="font-size:11pt">I find the formation of secondary shocks propagating</span></p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:11pt">in the layer with high velocity when I decrease the CFL and in some others I do not find them or I find really weaker ones.</span></p>
<p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"> </p>
<p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">I do not understand why because the time step dt
calculated in the simulation is </p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">for every simulation at least 3 times of
magnitude lower than the cooling time step dt_cool during the simulation (I
checked for all simulations).</p>
<p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"> </p>
<p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">The cooling time step in the simulation dt_cool is
calculated as dt_cool=0.5*pressure/(gamma-1)/Cooling (Cooling in erg/s/cm^3 )</p>
<p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">And dt << dt_cool for every simulation (see below)</p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"> t dt dt_hydro dt_cool CFL</p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"> ******* 4.2186E+09 3.6423E-01 ( 1.151E+16, -1.000E+16, 0.00 ) | 3.642E-01 1.073E+05 0.0400000<br> ******* 4.2186E+09 3.6423E-01 ( 1.151E+16, -1.000E+16, 0.00 ) | 3.642E-01 1.073E+05 0.0400000<br> ******* 4.2186E+09 3.6423E-01 ( 1.151E+16, -1.000E+16, 0.00 ) | 3.642E-01 1.073E+05 0.0400000<br> ******* 4.2186E+09 3.6423E-01 ( 1.151E+16, -1.000E+16, 0.00 ) | 3.642E-01 1.073E+05 0.0400000<br> ******* 4.2186E+09 3.6423E-01 ( 1.151E+16, -1.000E+16, 0.00 ) | 3.642E-01 1.073E+05 0.0400000<br> ******* 4.2186E+09 3.6423E-01 ( 1.151E+16, -1.000E+16, 0.00 ) | 3.642E-01 1.073E+05 0.0400000<br> ******* 4.2186E+09 3.6423E-01 ( 1.151E+16, -1.000E+16, 0.00 ) | 3.642E-01 1.073E+05 0.0400000<br> ******* 4.2186E+09 3.6423E-01 ( 1.151E+16, -1.000E+16, 0.00 ) | 3.642E-01 1.073E+05 0.0400000<br> ******* 4.2186E+09 3.6423E-01 ( 1.151E+16, -1.000E+16, 0.00 ) | 3.642E-01 1.073E+05 0.0400000<br></p>
<p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"> </p>
<p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">I would expect that the decrease in the CFL condition
does not change so much the solution in this case but actually changes </p><p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">a lot the numerical
solution after some times in the strong radiative regime and my numerical solution seems to not be converged.</p>
<p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"> </p>
<p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">Thank you very much for any suggestions,</p>
<p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"> </p>
<p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">Best regards,</p>
<p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"> </p>
<p class="gmail-MsoPlainText" style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">Antoine.</p></div>