Set of equations

[jingluan]

Hi Rich,

1, Which dependent variables and which set of ODE's do you use for non-adiabatic modes please? For example, (24.1) to (24.6) in Unno's are one choice of dependent variables, and equations (24.7) to (24.12) in Unno's are one choice of ODE's. The reason I like to ask this question is explained in the following part.

2, Does the radiation diffusion make your ODE's a stiff system, especially when it is only a tiny little non-adiabatic please? I did a simple plane parallel model trying to check something, but I found the solution blows up (grows to very large amplitude and also oscillate crazily) due to the following equation,

i*omega*rho*kB*T*z0/F *\delta s = d(\delta F/F)/dz,

where z is the depth, and z0 is the length which I use to scale the depth, \delta s is the Lagrangian entropy perturbation, \delta F/F is the fractional radiative flux perturbation, I have converted partial derivative over time to i*omega. Because the coefficient, omega*rho*kB*T*z0/F, is very big, it brings in a branch of solutions which grows very fast with depth. The other branch declines very fast with depth, which is the physical branch I want. However, the fast growing branch blows up my solution. I tried some integration method for stiff system, but still failed.

So I am guessing maybe my choice of dependent variables and set of equations is ill-conditioned themselves??? That is why I ask about your choice.

Thank you so much for reading this long message. Any suggestions, comments, guesses are highly appreciated please. Thank you

Sincerely,
Jing

[rhtownsend]

Hi Rich,

Hi, Jing!

1, Which dependent variables and which set of ODE's do you use for non-adiabatic modes please? For example, (24.1) to (24.6) in Unno's are one choice of dependent variables, and equations (24.7) to (24.12) in Unno's are one choice of ODE's. The reason I like to ask this question is explained in the following part.

For the list of dependent variables and ODEs, see the file gyre/doc/equations.pdf included in the GYRE distribution. The ODEs are similar to those given in Unno et al, but instead of adopting equation (21.6) for the perturbed radiative heating term, I use (21.7).

2, Does the radiation diffusion make your ODE's a stiff system, especially when it is only a tiny little non-adiabatic please? I did a simple plane parallel model trying to check something, but I found the solution blows up (grows to very large amplitude and also oscillate crazily) due to the following equation,

i*omega*rho*kB*T*z0/F *\delta s = d(\delta F/F)/dz,

where z is the depth, and z0 is the length which I use to scale the depth, \delta s is the Lagrangian entropy perturbation, \delta F/F is the fractional radiative flux perturbation, I have converted partial derivative over time to i*omega. Because the coefficient, omega*rho*kB*T*z0/F, is very big, it brings in a branch of solutions which grows very fast with depth. The other branch declines very fast with depth, which is the physical branch I want. However, the fast growing branch blows up my solution. I tried some integration method for stiff system, but still failed.

So I am guessing maybe my choice of dependent variables and set of equations is ill-conditioned themselves??? That is why I ask about your choice.

You are correct -- when the solutions are only a little bit non-adiabatic, the ODEs are extremely stiff. This can make them very difficult to solve, especially when using shooting approaches which integrate from one boundary to the other. GYRE attempts to get around the stiffness issue by using multiple shooting (see Ascher, Mattheij & Russell, "Numerical Solution of Boundary Value Problems for Ordinary Differential Equations") and adopting implicit integration schemes.

I don't think this problem can be avoided by choosing a different set of dependent variables; the stiffness is intrinsic to the physics underlying the equations.

Thank you so much for reading this long message. Any suggestions, comments, guesses are highly appreciated please. Thank you

Happy to help, and also happy to talk a little about some of the challenges GYRE (and other oscillation codes) must overcome.

cheers,

Rich