Topics in Astrophysics

1. Topics in Astrophysics

2. Science is a Team Sport! Pictures on previous slide are all from computations run by Ken Chen, a postdoc at UC Santa Cruz The code he used was CASTRO, a compressible radiation hydrodynamics code developed at LBL. The rest of this talk is all from Prof. Michael Zingale of Stonybrook University.

3. Be Careful What You Simulate!

4. Astro 250 Numerical Methods for (Astro)Physics Computation on the Cosmological Scale • Small inhomogeneities in the early Universe seed structure • More than 10 billion particles • Self-gravity dominates the evolution Simulating the growth of structure and the formation of galaxies. (Springel et al. 2005)

5. Computation on the Stellar Scale Does a supernova begin as a single white dwarf or merging white dwarfs? Model both and see which looks more like nature. (Roepke and Hillebrandt 2005)

6. Astro 250 Numerical Methods for (Astro)Physics Computing A Star • Suppose you wanted to write down an equation of motion for every atom/ion/electron in a star • How much information do we need to store? • Number of particles in a C/O white dwarf: • For each particle, we store position and velocity, and composition (at a minimum) • Each number you store on a computer requires 8 bytes of memory • Memory = (3 + 3 + 1) * 1056 * 8 bytes ~ 5 x 1048 GB • A typical computer has ~ 10 GB of memory, so we would need 5 x 1047computers! That's just the nuclei

7. Astro 250 Numerical Methods for (Astro)Physics The Fluid Approximation (ref: Shu, Ch. 1) • Fluid approximation: average over the random atomic motions and focus on scales that are large compared to individual atoms but small compared to the star as a whole • Compressible Fluid: The density of the fluid varies with the pressure – liquids are mostly incompressible, gases are compressible … (Roger McLassus; Wikipedia)

8. Astro 250 Numerical Methods for (Astro)Physics The Fluid Approximation thermonuclear runaway on a neutron star(Zingale et al. 2001) ISS image of Cyclone Catarina (NASA) • Now we just need to worry about the bulk properties of the fluid—its density, velocity, pressure, etc.

9. Astro 250 Numerical Methods for (Astro)Physics Approximations In Astrophysics • N-body • Popular with dark matter simulations • Models only the 1/r2 gravitational interaction between masses • No continuous description of the system • Compressible Fluid • Mean free path is much smaller than the system we are modeling • Bulk fluid quantities provide a good description of the physics • Low Mach Number Fluid • How does the fluid velocity compare with the sound speed??

10. Astro 250 Numerical Methods for (Astro)Physics CFD Computational • Colorful Fluid Dynamics • We model our system of interest as a fluid • Evolution dictated by conservation laws: Conservation of mass Conservation of momentum Conservation of energy Now we just need to solve these …

11. Astro 250 Numerical Methods for (Astro)Physics Conservative System • This can be written in conservation law form: • With • To close this system, we need an equation of state • Simplest: gamma-law

12. Astro 250 Numerical Methods for (Astro)Physics Conservative System • To make it look more like advection, we write this in quasi-linear form. • Express flux vector in terms of u1=½,u2=½u,u3=½E • Compute the Jacobian: • Here sound speed is: • System becomes: • This matrix is complicated and hard to work with

13. Astro 250 Numerical Methods for (Astro)Physics Primitive Variable Formulation • We can instead cast things in terms of the primitive variables: density, velocity, and pressure • Our system becomes • Notice that the Jacobian for this formulation is much simpler

14. Astro 250 Numerical Methods for (Astro)Physics Eigensystem • Recall that a system is hyperbolic if the eigenvalues are real and finite. • For our system, the eigenvalues are: • These are the speeds at which information propagates in our system • Three distinct wave speeds for 3 equations • We'd get the same eigenvalues from the Jacobian of the conserved system • There is a rich mathematical description of the theory of hyperbolic systems of conservation laws. The book by LeVeque is an excellent introduction.

15. Astro 250 Numerical Methods for (Astro)Physics Eigensystem • We can also find the eigenvectors: • These are normalized such that

16. Astro 250 Numerical Methods for (Astro)Physics Characteristic Variables • A final form of the system is in terms of the characteristic variables • Construct matrices of the left and right eigenvectors • Satisfy: LR = RL = I • Define dw=Ldq • Our system can be written as:

17. Astro 250 Numerical Methods for (Astro)Physics Characteristic Variables • Here, w are the characteristic variables • The three equations are decoupled: • If our system were linear, we'd be done: • Transform into characteristic variables • Each characteristic variable advects at a given wave speed, without interacting with one-another • Solve and then transform back to primitive form (or conserved form) • We're non-linear: the wave-speeds and eigenvectors change with the solution

18. Astro 250 Numerical Methods for (Astro)Physics Jumps Across Waves • The characteristic system is telling us something interesting already. • Consider an initial discontinuity in the primitive variables • Each wave will carry a jump in their associate characteristic quantity away from the discontinuity at their speed • The corresponding jump in the primitive variable is just dq=L{1dw=Rdw

19. Astro 250 Numerical Methods for (Astro)Physics Jumps Across Waves All primitive quantities jump across the u-c wave Only density jumps across the middle (u) wave All primitive varaibles jump across the u+c wave

20. Astro 250 Numerical Methods for (Astro)Physics Jumps Across Waves

21. Astro 250 Numerical Methods for (Astro)Physics Solution Methodology • We will use the finite-volume discretization: Second-order in time requires time-centering the right hand side

22. Astro 250 Numerical Methods for (Astro)Physics Solution Methodology • We need to solve this in conservative form to get the shock speed correct • Basic idea: evaluate fluxes at interfaces by predicting the fluid state there, then solve Riemann problem:

23. Astro 250 Numerical Methods for (Astro)Physics Riemann Problem • No matter the method used to predict the interface states, we now have left and right states at each interface • Unlike the linear advection or Burger's equation, we rarely solve the Riemann problem for the Euler equations exactly. • We need to consider what is carried by each wave • Different types of waves are present depending on the behavior of the characteristics

24. Astro 250 Numerical Methods for (Astro)Physics Riemann Problem • If the flow is compressing, then we have a shock • Rankine-Hugoniot jump conditions provide the solution • If we are not compressing, then we are a rarefaction • It can be shown that entropy is constant across the left and right waves (consider the system of ½,u,s and look at the right eigenvectors) • We can integrate the characteristic variables along these waves and determine what must be constant • We find: Across left rarefaction Across right rarefaction (see E.g. LeVeque Eq. 14.48)

25. Astro 250 Numerical Methods for (Astro)Physics Structure of a Simulation Code • The structure of a compressible hydro code looks like: • Initialize • Evolve • Compute timestep • Fill ghost cells • Construct interface states • Solve Riemann problem • Do conservative update • Output

26. Astro 250 Numerical Methods for (Astro)Physics Bells and Whistles • We described the basics, but there are some other features typically implemented in hydro codes: • Flattening: shocks self-steepen (characteristics converge), and can become too sharp. Flattening adds some dissipation near shocks to smear them out a bit • Species advection: nuclear or chemical species can be advected with the flow: • Source terms: source terms (e.g. gravity) do not change the characteristic structure, but do participate in the prediction to the interfaces • Other geometries: extensions to cylindrical and spherical geometries involves volume and area factors throughout the code

27. Astro 250 Numerical Methods for (Astro)Physics Multidimensional Hydro • Unsplit methods generally preserve symmetries on the grid better Dimensionally split Unsplit This is a standard test problem where there are 4 states (2x2) initially, with the upper left and lower right state identical—the solution should be symmetric about the diagonal. These calculation use 256x256 zones. Aside from the splitting, the algorithm is identical.

28. git clone https://ccse.lbl.gov/pub/Downloads/BoxLib.git Details, details … And you’ll probably want to use adaptive mesh refinement to get enough resolution in the regions of interest:

29. Astro 250 Numerical Methods for (Astro)Physics git clone https://ccse.lbl.gov/pub/Downloads/BoxLib.git Now let’s try CASTRO itself First we need to download BoxLib, the software framework which CASTRO is based on: git clone https://ccse.lbl.gov/pub/Downloads/BoxLib.git Now let’s get CASTRO itself: git clone from https://github.com/BoxLib-Codes/Castro.git Then let’s try to build and run CASTRO … cd into Castro/Exec and see which run directory looks interesting.