Riemann solver
numerical method used to solve a Riemann problem
A Riemann solver is a numerical method used to solve a hyperbolic partial differential equation based on the solution of the corresponding Riemann problem. They are heavily used in computational fluid dynamics and computational magnetohydrodynamics.
Definition
Generally speaking, Riemann solvers are specific methods for computing the numerical flux across a discontinuity in the Riemann problem. They form an important part of high-resolution schemes; typically the right and left states for the Riemann problem are calculated using some form of nonlinear reconstruction, such as a flux limiter or a WENO method, and then used as the input for the Riemann solver.
Exact solvers
Sergei K. Godunov is credited with introducing the first exact Riemann solver for the Euler equations, by extending the previous CIR (Courant-Isaacson-Rees) method to non-linear systems of hyperbolic conservation laws. Modern solvers are able to simulate relativistic effects and magnetic fields.
More recent research shows that an exact series solution to the Riemann problem exists, which may converge fast enough in some cases to avoid the iterative methods required in Godunov's scheme.
Approximate solvers
As iterative solutions are too costly, especially in magnetohydrodynamics, some approximations have to be made. Some popular solvers are:
Roe solver
Philip L. Roe used the linearisation of the Jacobian, which he then solves exactly.
HLLE solver
The HLLE solver (developed by Ami Harten, Peter Lax, Bram van Leer and B. Einfeldt) is an approximate solution to the Riemann problem, which is only based on the integral form of the conservation laws and the largest and smallest signal velocities at the interface.
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