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. 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

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. A obvious choice for a constant, may be the mean of the left and right state's 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, 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

Two-Rarefaction

Two-Shock

Hybrid Schemes

Iterative

Non-Iterative

Results and Discussion

• Same as God
• Most problems use a simple over 90% of grid, and a more sophisticated over the disc
• Can further improve the results using approximation methods to the God method (later chapters)