| 5 | | 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. This means that the velocity of the shear (i.e. velocity in the tangential direction relative to wave propagation) does not change across either shock or rarefaction. This means that the value of the tangential velocity only changes discontinuously across the middle wave of the Riemann problem, i.e. the contact discontinuity. The value of the shear velocity on the left side of the contact is just the tangential velocity of the left state, the value of the shear velocity on the right side of the contact is of the right state. This makes for an extremely straightforward/easy addition to the exact Riemann solver. |
| | 5 | 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. This means that the velocity of the shear (i.e. velocity in the tangential direction relative to wave propagation) does not change across either shock or rarefaction. This means that the value of the tangential velocity only changes discontinuously across the middle wave of the Riemann problem, i.e. the contact discontinuity. The value of the shear velocity on the left side of the contact is just the tangential velocity of the left state, and vice versa. This makes for an extremely straightforward/easy modification to the exact Riemann solver. |
