Changes between Version 10 and Version 11 of u/erica/ApproximateRS

Timestamp:

05/13/13 14:01:19 (12 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/ApproximateRS

@@ -12 +12 @@
 
 == Two-Rarefaction == 
-
+waves generated 
 This method is computationally costly (in terms of cost it goes - TRRS > TSRS > PVRS, where TRRS = two-rarefaction Riemann solver, TSRS = two-shock Riemann solver, PVRS = primitive variable Riemann solver), although extremely robust. If you assume 2 rarefactions, then you can derive closed-form 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. 
 
@@ -24 +24 @@
 
 = Results and Discussion =
-Numerical results are presented at the end of chapter 9. These results show that the 2 methods tested, 1) TSRS everywhere, and 2) a non-iterative 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 non-iterative 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 use are able to use a simple PVRS over 90% of grid, and a more sophisticated scheme over the remaining domain.
+Numerical results are presented at the end of chapter 9. These results show that the 2 methods tested, 1) TSRS everywhere, and 2) a non-iterative 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 non-iterative 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.
 
 = Further directions =