Riemann solver

Riemann solver

A Riemann solver is a numerical method used to solve a Riemann problem. They are heavily used in Computational fluid dynamics and Computational Magnetohydrodynamics.

Exact Solvers

Godunov is credited to introduce the first exact Riemann solver for the Euler equations [Citation| last = Godunov| first = S | title = A Finite Differences... | journal = Math Sb.| volume = 47 | pages = 357-393 | year = 1959] , by extending the previous CIR 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 theEuler 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

Roe used the linearisation of the Jacobian, which he then solves exactly. [Citation| last = Roe| first = P.L. | title = Approximate Riemann solvers, parameter vectors and difference schemes| journal = J. Comput. Phys.| volume = 43 | pages = 357-372| year = 1981]

HLLC solver

This solver was introduce by Toro [Citation| last = Toro| first = Eleuterio | title = Restoration of the contact surface in the HLL-Riemann solver| journal = Shock Waves| volume = 4 | pages = 25-34| year = 1994] it restores the missing Rarefaction waveby some estimates, like linearizations, these can be simple but also more advanced exists like using the Roe average velocityfor the middle wave speed. They are quite robust and efficient but somewhat more diffusive. [cite journal| last = Quirk| first = James | title = A Contribution to the Great Riemann Solver Debate| journal = International Journal for Numerical Methods in Fluids| volume = 18 | pages = 555-574| year = 1994 |doi = 10.1002/fld.1650180603]

Notes

References

*

ee also

* Godunov's scheme
* Computational fluid dynamics
* Computational Magnetohydrodynamics

External links


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