Introduction

The method I chose for practice writing a higher dimension code is the 1st order accurate, split scheme, Godunov + Exact Riemann Solver to solve the 2D Euler equations for a cylindrical explosion. Given the higher dimension of the problem, new types of waves are present in the solution, namely shears. Shears are passively advected with the flow, as can be shown by combining the continuity equation and the corresponding momenta equations.

Initialization

This method begins by initializing a 2D Cartesian grid, with a circle in the center. The primitive fluid variables (rho, x-velocity, y-velocity, pressure) are set inside and outside of this circle.

Upsides / Downsides

The good aspects of this method are: 1) accurate resolution of shear waves, 2) simple to construct.

The downside is: 1) not higher order (if wanted to add this - code could become quite complex to take into account the addition of shears into the TVD algorithms)