Falle and Komissarov upwind scheme

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,

where $v$ 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 and Komissarov [89 ] have considered two different algorithms to solve the local Riemann problems in SRHD by extending the methods devised in [87 ]. In a first algorithm, the intermediate states of the Riemann problem at both sides of the contact discontinuity, $vL∗$ and $vR ∗$ , are obtained by solving the system

where $r−L$ is the right eigenvector of $𝒜(vL )$ associated with sound waves moving upstream, and $r+R$ is the right eigenvector of $𝒜(vR )$ 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 $bL$ and $bR$ from the above expressions, and hence the linear approximation to the intermediate state $v (v ,v ) ∗ L R$ .

In the second algorithm proposed by Falle and Komissarov [89 ], a linearization of system (41 ) is obtained by constructing a constant matrix $𝒜^(vL, vR ) = 𝒜 (12(vL + vR ))$ . The solution of the corresponding Riemann problem is that of a linear system with matrix $𝒜^$ , i.e.,

or, equivalently,

with

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

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