| 198 | | It is interesting to see the Godunov method performs quite well for the pressure and velocity, but falls short with density (and hence internal energy, which is a function of density). As I have shown above, we see the development of the structure that is 'trying to mimic' the discontinuous jump in density within the star region of the exact solution. By solving the Riemann problem with the Godunov method over successive time steps, we have smoothed out this region in our discretization of the problem. This must happen when we are inside of the star region, and there is a discontinuity that occurs in a fluid variable. I can imagine this smoothing happening only within the (global) star region , as outside of that region, the waves have not yet penetrated the surrounding medium. Given the only variable with the potential of not being continuous in this region is density, I imagine the only smoothing we see of sharp peaks in the solution is with density, and any quantities that depend on density. This then supports why we see close agreement with pressure and velocity between methods. |
| | 198 | It is interesting to see the Godunov method performs quite well for the pressure and velocity, but falls short with density (and hence internal energy, which is a function of density). As I have shown above, we see the development of the structure that is approaching the discontinuous jump in density within the star region of the exact solution. By solving the Riemann problem with the Godunov method over successive time steps, we have smoothed out this region in our discretization of the problem. This must happen when we are inside of the star region, and there is a discontinuity that occurs in a fluid variable. I can imagine this smoothing happening only within the (global) star region , as outside of that region, the waves have not yet penetrated the surrounding medium. Given the only variable with the potential of not being continuous in this region is density, I imagine the only smoothing we see of sharp peaks in the solution is with density, and any quantities that depend on density. This then supports why we see close agreement with pressure and velocity between methods. |
