| 177 | | It seems like the types of waves generated in the Riemann problem can be 'guessed' at from the initial conditions, but there is not a hard and strict rule to learn, it's more of a type of intuition to be developed. For simple cases, such as those given in chapter 4 of Toro, you can guess that if you have a left and right state that are flowing into each other, you likely will get out 2 shocks. If they are flowing away from each other, you may get 2 rarefactions. If there is no motion in the left and right states, but the left is at a higher density or pressure, you can imagine it forcing a shock into the right state (a right traveling shock). By the same token, you can imagine this would create a left traveling rarefaction. Instead of using the iterative method to find pstar, you could guess at the value of pstar by using exact solutions if the 2 waves consisted of 2 shocks or 2 rare's. Another approximation could be the geometric mean of the left and right pressures. Once you'd have an estimate for pstar, to further your estimate of the solution, you'd have to consider how far those waves traveled, whether they disturbed the left and right states by the time you are interested, and so on. Obviously this would be quite tricky business, and so it seems reasonable to try to hone your skills for estimating what types of waves may be generated at any given discontinuity, and quantitatively how this may effect the data. |
| | 177 | It seems like the types of waves generated in the Riemann problem can be 'guessed' at from the initial conditions, but there is not a hard and strict rule to learn, it's more of a type of intuition to be developed. For simple cases, such as those given in chapter 4 of Toro, you can guess that if you have a left and right state that are flowing into each other, you likely will get out 2 shocks. If they are flowing away from each other, you may get 2 rarefactions. If there is no motion in the left and right states, but the left is at a higher density or pressure, you can imagine it forcing a shock into the right state (a right traveling shock). By the same token, you can imagine this would create a left traveling rarefaction. Instead of using the iterative method to find pstar, you could guess at the value of pstar by using exact solutions if the 2 waves consisted of 2 shocks or 2 rare's. Another approximation could be the geometric mean of the left and right pressures. Once you'd have an estimate for pstar, to further your estimate of the solution, you'd have to consider how far those waves traveled, whether they disturbed the left and right states by the time you are interested, and so on. Obviously this would be quite tricky business, and so it seems reasonable to try to hone your skills for estimating what types of waves may be generated at any given discontinuity, and qualitatively how this may effect the data. |
