4.2.2 Approximate Riemann solvers

The facts that:

1. in Godunov-type methods the structure of the Riemann solution is ``lost" in the cell averaging process
2. the exact solution of a Riemann problem is computationally expensive, particularly in multidimensions
3. Multidimensional problems: coupling of all flow velocity components through the Lorentz factor

   $\rightarrow$ shocks: increase in the number of algebraic jump conditions

   $\rightarrow$ rarefactions: solving a system of ODEs

motivated the proposal of approximate (linearized) Riemann solvers

• Linearized Riemann solvers are based in the exact solution of Riemann problems corresponding to a new system of equations obtained by a suitable linearization of the original one

• The local linearization of the Jacobian matrices of the original system and its subsequent spectral decomposition is on the basis of all these solvers

Standard solvers for classical fluid dynamics:

• Roe solver (JCP 43, 357-372, 1981)
• Osher solver (Math. Comp. 38, 339-374, 1982)
• HLLE solver (Harten, Lax, van Leer & Einfeldt) (SIAM Review 25, 35-61, 1983; SIAM J. Num. Anal. 25, 294-318, 1988)
• Marquina solver (JCP 125, 42-58, 1996)