3.6 Relativistic HLL method3 High-Resolution Shock-Capturing Methods3.4 Roe-type relativistic solvers

3.5 Falle and Komissarov upwind scheme 

Instead of starting from the conservative form of the hydrodynamic equations, one can use a primitive-variable formulation in quasi-linear form

  equation523

where tex2html_wrap_inline6007 is any set of primitive variables. A local linearization of the above system allows one to obtain the solution of the Riemann problem, and from this the numerical fluxes needed to advance a conserved version of the equations in time.

Falle & Komissarov [55Jump To The Next Citation Point In The Article] have considered two different algorithms to solve the local Riemann problems in SRHD by extending the methods devised in [53]. In a first algorithm, the intermediate states of the Riemann problem at both sides of the contact discontinuity, tex2html_wrap_inline5763 and tex2html_wrap_inline5765, are obtained by solving the system

equation538

where tex2html_wrap_inline6013 is the right eigenvector of tex2html_wrap_inline6015 associated with sound waves moving upstream and tex2html_wrap_inline6017 is the right eigenvector of tex2html_wrap_inline6019 of sound waves moving downstream. The continuity of pressure and of the normal component of the velocity across the contact discontinuity allows one to obtain the wave strengths tex2html_wrap_inline6021 and tex2html_wrap_inline6023 from the above expressions, and hence the linear approximation to the intermediate state tex2html_wrap_inline6025 .

In the second algorithm proposed by Falle & Komissarov [55Jump To The Next Citation Point In The Article], a linearization of system (31Popup Equation) is obtained by constructing a constant matrix tex2html_wrap_inline6027 . The solution of the corresponding Riemann problem is that of a linear system with matrix tex2html_wrap_inline6029, i.e.,

equation586

or, equivalently,

equation597

with

equation608

where tex2html_wrap_inline5987, tex2html_wrap_inline5989, and tex2html_wrap_inline5991 are the eigenvalues and the right and left eigenvectors of tex2html_wrap_inline6029, respectively (p runs from 1 to the total number of equations of the system).

In both algorithms, the final step involves the computation of the numerical fluxes for the conservation equations

equation625



3.6 Relativistic HLL method3 High-Resolution Shock-Capturing Methods3.4 Roe-type relativistic solvers

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
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