Roe-type relativistic solvers

3.4 Roe-type relativistic solvers

Linearized Riemann solvers are based on the exact solution of Riemann problems of a modified system of conservation equations obtained by a suitable linearization of the original system. This idea was put forward by Roe [248

], who developed a linearized Riemann solver for the equations of ideal (classical) gas dynamics. Eulderink et al. [83

, 84

] have extended Roe’s Riemann solver to the general relativistic system of equations in arbitrary spacetimes. Eulderink uses a local linearization of the Jacobian matrices of the system fulfilling the properties demanded by Roe in his original paper.

Let $ℬ = ∂F ∕∂u$ be the Jacobian matrix associated with one of the fluxes $F$ of the original system, and $u$ the vector of unknowns. Then, the locally constant matrix $^ℬ$ , depending on $u L$ and $u R$ (the left and right state defining the local Riemann problem), must have the following four properties:

It constitutes a linear mapping from the vector space $u$ to the vector space $F$ .
As $u → u → u L R$ , $^ℬ(u ,u ) → ℬ(u ) L R$ .
For any $uL$ , $uR$ , $^ ℬ(uL, uR)(uR − uL ) = F (uR ) − F(uL )$ .
The eigenvectors of $^ℬ$ are linearly independent.

Conditions 1 and 2 are necessary if one is to recover smoothly the linearized algorithm from the nonlinear version. Condition 3 (supposing Condition 4 is fulfilled) ensures that if a single discontinuity is located at the interface, then the solution of the linearized problem is the exact solution of the nonlinear Riemann problem.

Once a matrix $^ ℬ$ satisfying Roe’s conditions has been obtained for every numerical interface, the numerical fluxes are computed by solving the locally linear system. Roe’s numerical flux is then given by

with

where $^λ(p)$ , $^r(p)$ , and $^l(p)$ are the eigenvalues and the right and left eigenvectors of $ℬ^$ , respectively (p runs from 1 to the number of equations of the system).

Roe’s linearization for the relativistic system of equations in a general spacetime can be expressed in terms of the average state [83 , 84 ]

with

and

where $g$ is the determinant of the metric tensor $gμν$ . The role played by the density $ρ$ in case of the Cartesian non-relativistic Roe solver as a weight for averaging, is taken over in the relativistic variant by $k$ , which apart from geometrical factors tends to $ρ$ in the non-relativistic limit. A Riemann solver for special relativistic flows and the generalization of Roe’s solver to the Euler equations in arbitrary coordinate systems are easily deduced from Eulderink’s work. The results obtained in 1D test problems for ultra-relativistic flows (up to Lorentz factors of 625) in the presence of strong discontinuities and large gravitational background fields demonstrate the excellent performance of the Eulderink–Roe solver [84

Relaxing Condition 3 above, Roe’s solver is no longer exact for shocks but still produces accurate solutions. Moreover, the remaining conditions are fulfilled by a large number of averages. The 1D general relativistic hydrodynamic code developed by Romero et al. [250 ] uses flux formula (36 ) with an arithmetic average of the primitive variables at both sides of the interface. It has successfully passed a long series of tests including the spherical version of the relativistic shock reflection (see Section 6.1).

Roe’s original idea has been exploited in the so-called local characteristic approach (see, e.g., [307 ]). This approach relies on a local linearization of the system of equations by defining at each point a set of characteristic variables, which obey a system of uncoupled scalar equations. This approach has proven to be very successful, because it allows for the extension to systems of scalar nonlinear methods. Based on the local characteristic approach are the methods developed by Marquina et al. [176 ] and Dolezal and Wong [74 ], which both use high-order reconstructions of the numerical characteristic fluxes, namely PHM [176 ] and ENO [74 ] (see Section 9.5).