Building a 1D Hydro Code

Introduction

I've created this page to document the important aspects of building a hydro code. It contains general information that I have learned from reading Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction by E.F. Toro. In the near future, it will contain useful information from my own personal experience as I build my first hydro code.

The Equations of Fluid Dynamics

The first time I saw the fluid equations was in the normal conservative-law form. These are conceptually simple and make intuitive sense.

Name of Law	Formula
Conservation of Mass	\frac{\partial}{\partial t}\rho + \frac{\partial}{\partial x} (\rho v) = 0
Conservation of Momentum	\frac{\partial}{\partial t} (\rho v) + \frac{\partial}{\partial x} (\rho v^2 + p) = 0
Conservation of Energy	\frac{\partial}{\partial t} E + \frac{\partial}{\partial x} (v(E+p)) = 0

Where \rho is mass density, v is velocity (only in x-direction for 1D), p is pressure, and E is total energy per unit volume. E is further defined as E = \rho(\frac{1}{2}v^2 + e) where e is the specific internal energy. Basically, these laws state that in a given volume, the change in a conserved quantity must be equal to the flux through the boundaries of that volume.

Solving the Riemann Problem

Discretisation

Godunov's First Order Upwind Scheme