open-astrophysics-bookshelf/numerical_exercises

An introduction to numerical methods for astrophysical hydrodynamics

Computational Hydrodynamics for Astrophysics

part of the Open Astrophysics Bookshelf

Notes on numerical methods for computational astrophysical hydrodynamics.

These notes describe the way I think about the numerical methods commonly used with grid-based codes in astrophysical hydrodynamics. The notes are written in LaTeX, and should build by typing 'make' in the main directory.

Working implementations for all of the solvers are contained either in the main pyro code or in the set of hydro examples, both referenced below.

Chapters

The following chapters are mostly written:

• Simulation Overview
• Finite-Volume Grids
• Advection
• Burgers' Equation
• Euler Equations: Theory
• Euler Equations: Numerical Methods
• Elliptic Equations and Multigrid
• Diffusion
• Model Multiphysics Problems
• Reactive Flow
• Planning a Simulation
• Incompressible Flow and Projection Methods
• Low Mach Number Methods

The following are things I'd like to add in the next 1-2 years:

• Fluid Instabilities
• Rotation and Self-gravity
• Radiation Hydrodynamics
• MHD
• AMR
• Mapped Grids

The following are things hopefully will eventually get written:

• Relativisitc Flows
• Higher-Order Methods
• Implicit Hydrodynamics

Getting PDFs and Source

A PDF version of these notes is available at:

http://open-astrophysics-bookshelf.github.io/numerical_exercises/

(this PDF is automatically generated by a Github action each time changes are pushed to main.)

There are two sets of companion codes that go along with these notes:

• hydro_examples: https://github.com/zingale/hydro_examples

  simple, standalone, 1-d solvers that illustrate the basic ideas.

• pyro: https://github.com/python-hydro/pyro2

  pyro is a 2-d full simulation code designed with simplicity in mind that implements the core solvers described in these notes along with various test problems.

numerical_exercises's Issues

About Eq. (1.26) in section "Implicit methods"

I believe that Eq. (1.26) at here does not want expansions on time. I think the superscript of "t" on the right hand side should be changed from "n" to "n+1".
PS: I think the subscript of the Jacobin matrix, "0", may be a little confusing since the matrix uses digits (1-n) as the subscriptions of its elements. Maybe a few more explanations will make it clearer. Thanks.

A small typo in the book

On page 31 in section 3.4, I think ilo=ng (assuming counting starts from 0)

ilo = ng #not nx
ihi = nx + ng - 1
state = np. zeros (( nx + 2*ng , nvar ))

Links to hydro_examples repo point to old version

The links (that I've checked, in the Advection chapter in particular) to the hydro_examples repo all point to zingale/hydro_examples rather than python-hydro/hydro_examples. They do correctly redirect, but should be updated.

Found these by searching for the \hydroexdoit macro.

Author acronym BCH is not defined

Appears first on page 240 but is not defined previously. Presumably Bell, Colella, and Howell?

Also, the url in the Bell, Colella, and Howell reference is broken.

Abuse of notation leads to confusion

Some abuse of notation in the Multidimensional Advection section leads to lack of clarity.

In particular, the definition of the interface fluxes in Eq. 3.45 or in code here uses the same symbol (e.g. (ua)_{i+1/2,j}) on both sides of the equation. This is also confusing given the definition of a_{i,j} in Eq. 3.41, and the fact that you are assuming u (and v) to be constant.