3.5 Falle and Komissarov upwind 3 High-Resolution Shock-Capturing Methods3.3 Two-shock approximation for relativistic

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 [155Jump To The Next Citation Point In The Article], who developed a linearized Riemann solver for the equations of ideal (classical) gas dynamics. Eulderink at al. [49Jump To The Next Citation Point In The Article, 50Jump To The Next Citation Point In The Article] 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 tex2html_wrap_inline5961 be the Jacobian matrix associated with one of the fluxes tex2html_wrap_inline5963 of the original system, and tex2html_wrap_inline5699 the vector of unknowns. Then, the locally constant matrix tex2html_wrap_inline5967, depending on tex2html_wrap_inline5881 and tex2html_wrap_inline5883 (the left and right state defining the local Riemann problem) must have the following four properties:

  1. It constitutes a linear mapping from the vector space tex2html_wrap_inline5699 to the vector space tex2html_wrap_inline5963 .
  2. As tex2html_wrap_inline5977 .
  3. For any tex2html_wrap_inline5881, tex2html_wrap_inline5981 .
  4. The eigenvectors of tex2html_wrap_inline5967 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 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 tex2html_wrap_inline5967 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

  equation455

with

  equation475

where tex2html_wrap_inline5987, tex2html_wrap_inline5989, and tex2html_wrap_inline5991 are the eigenvalues and the right and left eigenvectors of tex2html_wrap_inline5967, 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 [49Jump To The Next Citation Point In The Article, 50Jump To The Next Citation Point In The Article]

equation494

with

equation503

and

equation506

where g is the determinant of the metric tensor tex2html_wrap_inline5999 . The role played by the density tex2html_wrap_inline5637 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 tex2html_wrap_inline5637 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 625) in the presence of strong discontinuities and large gravitational background fields demonstrate the excellent performance of the Eulderink-Roe solver [50Jump To The Next Citation Point In The Article].

Relaxing condition 3 above, Roe's solver is no longer exact for shocks but still produces accurate solutions, and moreover, the remaining conditions are fulfilled by a large number of averages. The 1D general relativistic hydrodynamic code developed by Romero et al. [157Jump To The Next Citation Point In The Article] uses flux formula (26Popup Equation) 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., [198]). 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. [106Jump To The Next Citation Point In The Article] and Dolezal & Wong [42Jump To The Next Citation Point In The Article], which both use high-order reconstructions of the numerical characteristic fluxes, namely PHM [106Jump To The Next Citation Point In The Article] and ENO [42Jump To The Next Citation Point In The Article] (see Section  9.4).



3.5 Falle and Komissarov upwind 3 High-Resolution Shock-Capturing Methods3.3 Two-shock approximation for relativistic

image Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de