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Meeting update

Updates from last week:

Pretty close with the Jeans Instability module. I wrote a page on the theory of the instability here- https://astrobear.pas.rochester.edu/trac/astrobear/wiki/u/erica/JeansInstability and a page on the module I wrote here- https://astrobear.pas.rochester.edu/trac/astrobear/wiki/u/erica/JeansTest. There is still a needed modification to the analytical function that I am working on. Given the initial perturbation is sinusoidal in x and exponential in t, I think what I need to have my module do is: 1. at t = 0 assign rho(x,0) = rho0 + 0.1*rho0*cos(kx) (i.e. prescribe a small amplitude perturbation, 2. for later times t=t', have the density distribution continually perturbed by an exponential function of t'. This I believe should take the form: rho(x,t) ~ rho(x,0)*exp(4*pi*G*rho0)t.In this way, the amplitude of the initial perturbation should grow in time and have a growth rate ~ 1/tff. The next thing I need to do then in the problem module is to define and store an array of initial rho. To store this array, I added it to the Info structure. However, I am having difficulty populating this array with Info%q(i, :, :, 1) in the problem initiation subroutine in problem.f90. It seems that this is a fortran language error I am making, but I should ask, if one would like to store information on fluid variables from a given time level, is this a reasonable place to store (inside of the info structure)?

Plot of xi(t) for the intermediate 1/30 case- . Also, after thinking more of the critical pressure, I have convinced myself that it does change with time over the course of the sim, as the parameters of the BE sphere change.

Read chapters 15 and 16 of Toro. These are 'splitting schemes for PDEs with Source Terms', and 'Methods for multi-dimensional PDEs', respectively. I think the general method for multi-dimension schemes is a VERY straight-forward extension of the 1D schemes. However, I still need to delve a little more into 1. how the additional (shear) waves are handled numerically — this involves reading up on how to mathematically/numerically separate the different dimensions in the eigen system, 2. How the geometric source terms can reduce the dimension of the problem, while maintaining mathematical equivalence to the higher dimension Cartesian counterparts.

Upcoming week:

Perform a test with it (thinking the 2D cylindrical explosion in first section of Ch. 17). If I do this test, I need to work through the math of how the 2 sets of equations are equivalent (the 1D + source term of cylindrical and the 2D Cartesian version), and I need to make sure I can solve the 1D "numerical" test of this problem (again, this is devising a scheme that can solve for cells in r using a split scheme to handle the geometric source term) to compare with the 2D results.