Let be the Jacobian matrix
associated with one of the fluxes
of the original
system, and
the vector of unknowns. Then, the
locally constant matrix
, depending on
and
(the left and right state defining the
local Riemann problem), must have the following four
properties:
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 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
Roe’s linearization for the relativistic system
of equations in a general spacetime can be expressed in terms of
the average state [83, 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. [249] 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).