Changes between Version 10 and Version 11 of u/erica/ApproximateRS
 Timestamp:
 05/13/13 14:01:19 (11 years ago)
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u/erica/ApproximateRS
v10 v11 12 12 13 13 == TwoRarefaction == 14 14 waves generated 15 15 This method is computationally costly (in terms of cost it goes  TRRS > TSRS > PVRS, where TRRS = tworarefaction Riemann solver, TSRS = twoshock Riemann solver, PVRS = primitive variable Riemann solver), although extremely robust. If you assume 2 rarefactions, then you can derive closedform expressions for pstar and ustar from the pressure function formalism for rarefactions, given in chapter 4 of Toro. From there, you would feed these values of pstar and ustar into the sampling routine of the ERS, and continue with Godunov method as before. Assuming there are actually 2 rarefaction waves, this method would produce the exact solution. Even if there are not 2 rarefactions, this method produces sufficient results. 16 16 … … 24 24 25 25 = Results and Discussion = 26 Numerical results are presented at the end of chapter 9. These results show that the 2 methods tested, 1) TSRS everywhere, and 2) a noniterative hybrid scheme using PVRS in smooth flow, and TSRS in sharp gradient flow, both produce results that exactly match up to the Godunov method that uses the ERS. This shows that 1) you can get the same order of accuracy of the Godunov method, using computationally cheaper methods, and 2) using a noniterative hybrid TSRS method can further reduce the computational cost without reducing accuracy of the solution. It is interesting to note that according to Toro, most problems useare able to use a simple PVRS over 90% of grid, and a more sophisticated scheme over the remaining domain.26 Numerical results are presented at the end of chapter 9. These results show that the 2 methods tested, 1) TSRS everywhere, and 2) a noniterative hybrid scheme using PVRS in smooth flow, and TSRS in sharp gradient flow, both produce results that exactly match up to the Godunov method that uses the ERS. This shows that 1) you can get the same order of accuracy of the Godunov method, using computationally cheaper methods, and 2) using a noniterative hybrid TSRS method can further reduce the computational cost without reducing accuracy of the solution. It is interesting to note that according to Toro, most problems are able to use a simple PVRS over 90% of grid, and a more sophisticated scheme over the remaining domain. 27 27 28 28 = Further directions =