[TurbJet]

Greetings,

(Approximate) Riemann solvers are originally designed for piecewise-constant states on both sides, which can be considered as "piecewise-constant" reconstruction when it comes to variable reconstruction/interpolation. However in Toro's book [1], Chap. 13.4, when he introduces the high-order reconstruction, and specifically, the MUSCL type methods, he says, and I quote:

... the piece-wise linear reconstruction process ... As a consequence of having modified the data, at each interface i+1/2 one now may consider the so-called Generalized Riemann Problem (or GRP) to compute an intercell Godunov-type flux...

From here, the solution procedure seems different from conventional piecewise-constant-Riemann solver, e.g., the MUSCL-Hancock method (MHM) mentioned in the book advances the solution for half of time step and then apply the conventional Riemann solver to finish the rest half.

However, from what I can tell, many codes use high-order reconstruction to calculate the left & right states at cell faces, and then directly apply those conventional Riemann solvers, such as HLLC/Roe, etc, no GRP show up.

This seems contradict to what Toro says, and so I am confused: can those conventional Riemann solvers be applied to, e.g., linear-piecewise-reconstructed states, i.e., not constant states on both sides? Or there some compromises/assumptions/approximations are made that I am not aware of?

Any suggestions would be helpful. Thanks!

Ref:
[1] E.F.Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, 2009.

[Eifoehn4]

Dear TurbJet,

it is a great pity that most CFD literature introduce the Riemann problem theory as a constant initial value problem only.

Originally Posted by TurbJet

This seems contradict to what Toro says, and so I am confused: can those conventional Riemann solvers be applied to, e.g., linear-piecewise-reconstructed states, i.e., not constant states on both sides? Or there some compromises/assumptions/approximations are made that I am not aware of?

Sure, they can be applied without any problems. The use of such Riemann solvers along with high order methods is only a question of your approximation accuracy or to be more precise a different way of modelling.

The standard mathematical models in CFD are the incompressible or compressible Navier-Stokes equations, which of course are only approximations of the reality for a certain range of interest. In the latter, heat conduction or viscosity effects are approximated independently with regard to the hyperbolic part by means of Fourier or Stokes law. This results in infinitely fast waves for these effects.

However, most physical processes are in general not of parabolic/elliptic nature. There are pretty much situations in which a separate modelling of the hyperbolic and the parabolic/elliptic part is no longer justified. In such cases, both effects have a very large mutual influence on each other. Here, GRP solvers are often superior, since they consider the fully coupled system and explicitly consider jumps in the derivatives.

In summary, you can consider this to be just a bad modelling when using the normal Riemann solver, however sufficient in most CFD applications.

Regards

[Eifoehn4]

Sorry for the double post.