Riemann problem [SEP] named after
Riemann problem
A Riemann problem, named after Bernhard Riemann, is a specific initial value problem composed of a conservation equation together with piecewise constant initial data which has a single discontinuity in the domain of interest. The Riemann problem is very useful for the understanding of equations like Euler conservation equations because all properties, such as shocks and rarefaction waves, appear as characteristics in the solution. It also gives an exact solution to some complex nonlinear equations, such as the Euler equations.
In numerical analysis, Riemann problems appear in
Sod shock tube
]
The Sod shock tube problem, named after Gary A. Sod, is a common test for the accuracy of computational fluid codes, like Riemann solvers, and was heavily investigated by Sod in 1978.
The test consists of a one-dimensional Riemann problem with the following parameters, for left and right states of an ideal gas.
formula_1
formula_2
where
The time evolution of this problem can be described by solving the Euler equations,
which leads to three characteristics, describing the