| 7 | 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. |