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# Approximating pstar and ustar

The Godunov method relies on the solution of pressure and velocity inside of the star region set up within local Riemann problems, which before now, has been done using an iterative approach inside of the Exact Riemann Solver (ERS). The iterative approach is computationally costly, and so if we could devise schemes which approximate pstar and ustar, we can make a more efficient time marching procedure.

# Linearization of the Euler equations / Primitive Variable Riemann Solver

There are 2 methods which make use of approximating the Euler equations themselves in linearized form, to get simple algebraic expressions for pstar and ustar. The first makes a substitution in the coefficient matrix of the Euler equations, such that all fluid variables are replaced with constants, making the system of equations a constant coefficient system, which can be solved exactly algebraically. An obvious choice for a constant may be the mean of the left and right states' variable to be substituted, i.e. rho = mean(rho_L, rho_R). The 2nd method for this type of approximation is matching states using the characteristic equations, to solve for the star regions variables (p, u, and rho). After the star region is specified, the variables are fed into the sampling routine of the Exact Riemann Solver (ERS), slightly modified as it no longer has to solve for rho in the star region, and the Godunov method continues as before. These methods for approximation are the cheapest computationally, but also the least robust. You would not want to use such a scheme over the entire computational domain for most practical problems.

# Non-linear Wave Assumption Approximations

If one assumes the generation of either 2 shocks or 2 rarefaction from the Riemann problem, then there are 2 possibilities for approximating pstar, from which follow ustar and rho.

## Two-Rarefaction

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.

## Two-Shock

Unlike the TRRS, there does not exist an exact closed form expression for pstar and ustar, derivable from the pressure functions for the case of a two-shock scenario. Instead, one could approximate one of the unknowns in the equations equal to a constant, and then the system would be solvable. This leads to robust results that are not too computationally costly.

# Hybrid Schemes

One could make use of various approximation methods in different regions of the flow as needed. In regions of smooth flow, without sharp gradients, one could use the computationally cheap (and also least robust) PVRS methods. In regions with sharp gradients (near reflective boundaries, or contacts, shocks, and shears), one would opt for the more robust methods of TRRS or TSRS. Such adaptive, or hybrid schemes, can either be iterative or not, meaning near sharp gradients, pstar can either be solved for using the iteration method of the ERS, or estimated using one of the TSRS or TRRS, whereas both iterative and non-iterative schemes would use the same simple PVRS in smoother flow regions. It is interesting to note that if one were to choose an iterative hybrid scheme, that most times only 1 iteration step produces sufficient results, provided there is a good enough initial estimate for pEst.

# 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.

# Further directions

These schemes can be further improved by adding approximation methods to the Godunov method, to be discussed in later chapters such as 15.

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