Euler Equations

The Euler equations are a system of partial differential equations used in fluid dynamics which describe the flow of invicid fluids.

The parameter $\gamma$ is the heat capacity ratio also known as the adiabatic index. This system can be used to model airflow around a wing of an airplane for instance. Finding analytical solutions is pretty much impossible so we need to use numerical methods to approximate solutions.

Shu-Osher Shock Tube Problem

To compare different numerical methods there are benchmark problems such as the Shu-Osher shock tube problem. This problem is given by the following initial conditions

with $\varepsilon = 0.2$ and the adiabatic index $\gamma$ of $1.4$. Notice that the given initial conditions do not correspond to the variables that govern the Euler equations. You need to convert them to fit the conservation variables of the system.

Numerical Methods

Lax-Friedrichs Scheme

The Lax-Friedrichs scheme is based on finite differences and is used to approximate solutions to hyperbolic partial differential equations.

To get to a stable method for a linear hyperbolic differential equation a viscosity term had to be added making the solutions smoother. Discontinuities or shocks in the solution will be smoothed out. The same goes for non-linear hyperbolic differential equations.

Godunov Scheme

The Godunov schemes makes use of the exact solution to the Riemann problem at the boundary of two neighboring cells. This scheme is a lot better at preserving shocks in the solution.

Here $f^G(u_l,u_r)$ gives you the exact solution to the Riemann problem with flux at the boundary between the cells $i-1$ and $i$ or $i$ and $i+1$.

Animation

Here is an animation of the solution to the Shu-Osher shock tube problem using the Lax-Friedrichs and Godunov scheme.

Even with a reasonable resolution of 1000 cells the solutions are wildly different. Clearly the Lax-Friedrichs scheme produces a smoother solution but the Godunov scheme is a lot closer to the reference solution for the shock tube problem. Feel free to experiment with the code I have provided. The more cells you use the better the approximation becomes but beware of the added computational cost.