Riemann solver
Encyclopedia
A Riemann solver is a numerical method used to solve a Riemann problem
. They are heavily used in computational fluid dynamics
and computational magnetohydrodynamics
.
is credited with introducing the first exact Riemann solver for the Euler equations, by extending the previous CIR (Courant-Isaacson-Reeves) method to non-linear systems of hyperbolic conservation laws. Modern solvers are able to simulate relativistic effects and magnetic fields.
For the hydrodynamic case latest research results showed the possibility to avoid the iterations to calculate the exact solution for the
Euler equations.
used the linearisation of the Jacobian, which he then solves exactly.
, Lax
, van Leer
and Einfeldt) solver 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. The stability and robustness of the HLLE solver is closely related to the signal velocities and a single central average state, as proposed by Einfeldt in the original paper. The description of the HLLE scheme in the book mentioned below is incomplete and partially wrong. The reader is referred to the original paper. Actually, the HLLE scheme is based on a new stability theory for discontinuities in fluids, which was never published.
of the Roe solver and the excessive diffusion of the HLLE solver at the same time. They developed robust and accurate Riemann solvers by combining the Roe solver and the HLLE/Rusanov solvers: they show that being applied in two orthogonal directions the two Riemann solvers can be combined into a single Roe-type solver (the Roe solver with modified wave speeds). In particular, the one derived from the Roe and HLLE solvers, called Rotated-RHLL solver, is extremely robust (carbuncle-free for all possible test cases on both structured and unstructured grids) and accurate (as accurate as the Roe solver for the boundary layer calculation).
Riemann problem
A Riemann problem, named after Bernhard Riemann, consists of a conservation law together with piecewise constant data having a single discontinuity. The Riemann problem...
. They are heavily used in computational fluid dynamics
Computational fluid dynamics
Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...
and computational magnetohydrodynamics
Computational Magnetohydrodynamics
Computational magnetohydrodynamics is a rapidly developing branch of magnetohydrodynamics that uses numerical methods and algorithms to solve and analyze problems that involve electrically conducting fluids. Most of the methods used in CMHD are borrowed from the well established techniques...
.
Exact solvers
GodunovSergei K. Godunov
Sergei Konstantinovich Godunov is professor at the Sobolev Institute of Mathematics of the Russian Academy of Sciences in Novosibirsk, Russia....
is credited with introducing the first exact Riemann solver for the Euler equations, by extending the previous CIR (Courant-Isaacson-Reeves) method to non-linear systems of hyperbolic conservation laws. Modern solvers are able to simulate relativistic effects and magnetic fields.
For the hydrodynamic case latest research results showed the possibility to avoid the iterations to calculate the exact solution for the
Euler equations.
Approximate solvers
As iterative solutions are too costly, especially in Magnetohydrodynamics, some approximations have to be made. The most popular solvers are:Roe solver
RoePhilip L. Roe
Philip L. Roe is a Professor of Aerospace Engineering at the University of Michigan in Ann Arbor. He is known for his work in the field of Computational Fluid Dynamics and Magnetohydrodynamics. Roe made fundamental contributions to the development of high-resolution schemes for hyperbolic...
used the linearisation of the Jacobian, which he then solves exactly.
HLLE solver
The HLLE (HartenAmi Harten
Amiram Harten was an American/Israeli applied mathematician. Harten made fundamental contribution to the development of high-resolution schemes for the solution of hyperbolic partial differential equations...
, Lax
Peter Lax
Peter David Lax is a mathematician working in the areas of pure and applied mathematics. He has made important contributions to integrable systems, fluid dynamics and shock waves, solitonic physics, hyperbolic conservation laws, and mathematical and scientific computing, among other fields...
, van Leer
Bram van Leer
Bram van Leer is the Arthur B. Modine Professor of aerospace engineering at the University of Michigan, in Ann Arbor. He specialises in Computational fluid dynamics , fluid dynamics, and numerical analysis where he has made substantial contributions.-Research work:Professor van Leer developed...
and Einfeldt) solver 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. The stability and robustness of the HLLE solver is closely related to the signal velocities and a single central average state, as proposed by Einfeldt in the original paper. The description of the HLLE scheme in the book mentioned below is incomplete and partially wrong. The reader is referred to the original paper. Actually, the HLLE scheme is based on a new stability theory for discontinuities in fluids, which was never published.
HLLC solver
The HLLC (Harten-Lax-van Leer-Contact) solver was introduced by Toro. It restores the missing Rarefaction wave by some estimates, like linearisations, these can be simple but also more advanced exists like using the Roe average velocity for the middle wave speed. They are quite robust and efficient but somewhat more diffusive.Rotated-hybrid Riemann solvers
These solvers were introduced by Nishikawa and Kitamura, in order to overcome the carbuncle problemsof the Roe solver and the excessive diffusion of the HLLE solver at the same time. They developed robust and accurate Riemann solvers by combining the Roe solver and the HLLE/Rusanov solvers: they show that being applied in two orthogonal directions the two Riemann solvers can be combined into a single Roe-type solver (the Roe solver with modified wave speeds). In particular, the one derived from the Roe and HLLE solvers, called Rotated-RHLL solver, is extremely robust (carbuncle-free for all possible test cases on both structured and unstructured grids) and accurate (as accurate as the Roe solver for the boundary layer calculation).
See also
- Godunov's schemeGodunov's schemeIn numerical analysis and computational fluid dynamics, Godunov's scheme is a conservative numerical scheme, suggested by S. K. Godunov in 1959, for solving partial differential equations...
- Computational fluid dynamicsComputational fluid dynamicsComputational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...
- Computational MagnetohydrodynamicsComputational MagnetohydrodynamicsComputational magnetohydrodynamics is a rapidly developing branch of magnetohydrodynamics that uses numerical methods and algorithms to solve and analyze problems that involve electrically conducting fluids. Most of the methods used in CMHD are borrowed from the well established techniques...