44 | | == The method == |
45 | | |
46 | | === Finding Pstar === |
47 | | |
48 | | Pstar is the solution for pressure in the "star region", which is set up at the intercell boundaries by the left and right cells involved in the local Riemann problems. This region is where the contact discontinuity lies, flanked on either side by a given combination of 2 shock or rarefaction waves. The first step at each intercell boundary is to solve for pstar. Pstar is found by Newton's iterative scheme, with the routine as it was in the Exact Riemann solver described before. |
49 | | |
50 | | === Sampling the solution === |
51 | | |
52 | | Once Pstar is found to within a certain degree of accuracy, ustar is given immediately by a simple formula in Toro, chapter 4. |
53 | | |
54 | | In conservative form, the Godunov method is written: |
| 44 | These fluxes are then fed into the Godunov scheme to update each (conserved) variable for each cell. In conservative form, the Godunov method is given by: |
66 | | is the numerical flux, which = the physical flux evaluated along the intercell boundary using the exact solution to the local Riemann Problem (LRP). |
| 56 | is the numerical flux (again, equal to the physical flux in the case of the God. method for the Euler equations) on the left boundary, and |
| 57 | |
| 58 | {{{#!latex |
| 59 | F{_{i+1/2}} |
| 60 | }}} |
| 61 | |
| 62 | is the flux along the right intercell boundary. |
| 63 | |
| 64 | == The method == |
| 65 | |
| 66 | === Finding Pstar === |
| 67 | |
| 68 | Pstar is the solution for pressure in the "star region", which is set up at the intercell boundaries by the left and right cells involved in the local Riemann problems. This region is where the contact discontinuity lies, flanked on either side by a given combination of 2 shock or rarefaction waves. The first step at each intercell boundary is to solve for pstar. Pstar is found by Newton's iterative scheme, with the routine as it was in the Exact Riemann solver described before. |
| 69 | |
| 70 | === Sampling the solution === |
| 71 | |
| 72 | Once Pstar is found to within a certain degree of accuracy, ustar is given immediately by a simple formula in Toro, chapter 4. |