7 | | Given the above properties of the Riemann problem, an algebraic expression can be derived which gives p in the central "star-region" (denoted by '*'). The overall structure of the Riemann problem then is to solve this algebraic equation for p*. Once p* is known, u* follows immediately. The remaining rho*_L and rho*_R follow from expressions valid for the specific L- or R- wave present. |
8 | | |
9 | | Specifically, this means that the code will determine at every point (x,t), that point's relative position to the different waves present. Once the position is determined, the solution is given by analytic expressions for the following five possibilities: pre- or post- shock, ahead of rarefaction head, behind rarefaction tail, or within the rarefaction fan. The position of each sampling point (x,t) is determined by a characteristic speed in the grid given by s = dx/t, where dx is the distance from the initial discontinuity to the sampling point (x,t), and t is the simulation end time. This s for every point on the grid can be compared to the known present waves, as their speeds are known exactly. In this way, the relative position of the sampling point with respect to the waves on the grid is determined, and hence the fluid variables at that point. |
| 7 | Given the above properties of the Riemann problem, an algebraic expression can be derived which gives p in the central "star-region" (denoted by '*'). The overall structure of the Riemann problem then is to solve this algebraic equation for p*. Once p* is known, u* follows immediately. The remaining rho*_L and rho*_R follow from expressions valid for the specific L- or R- wave present. Specifically, this means that the code will determine at every point (x,t), that point's relative position to the different waves present. Once the position is determined, the solution is given by analytic expressions for the following five possibilities: pre- or post- shock, ahead of rarefaction head, behind rarefaction tail, or within the rarefaction fan. The position of each sampling point (x,t) is determined by a characteristic speed in the grid given by s = dx/t, where dx is the distance from the initial discontinuity to the sampling point (x,t), and t is the simulation end time. This s for every point on the grid can be compared to the known present waves, as their speeds are known exactly. In this way, the relative position of the sampling point with respect to the waves on the grid is determined, and hence the fluid variables at that point. |