Non-uniqueness of cosmic ray two-fluid equations at shocks and possible remedies. (arXiv:1906.07200v1 [astro-ph.HE])
<a href="http://arxiv.org/find/astro-ph/1/au:+Gupta_S/0/1/0/all/0/1">Siddhartha Gupta</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Sharma_P/0/1/0/all/0/1">Prateek Sharma</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Mignone_A/0/1/0/all/0/1">Andrea Mignone</a>
Cosmic rays (CRs) are frequently modeled as an additional fluid in
hydrodynamic (HD) and magnetohydrodynamic (MHD) simulations of astrophysical
flows. The standard CR two-fluid model is described in terms of three
conservation laws (expressing conservation of mass, momentum and total energy)
and one additional equation (for the CR pressure) that cannot be cast in a
satisfactory conservative form. The presence of non-conservative terms with
spatial derivatives in the model equations prevents a unique weak solution
behind a shock. We investigate a number of methods for the numerical solution
of the two-fluid equations and find that, in the presence of shock waves, the
results depend on the choice of the numerical methods (spatial reconstruction,
time stepping, and the CFL number) and the adopted discretization.
Nevertheless, all methods converge to a unique result only if the energy
partition between the thermal and non-thermal fluids at the shock is prescribed
a priori. This highlights the closure problem of the two-fluid equations at
shocks. We suggest a robust method where the solutions are insensitive to the
numerical method. Comparison with the currently used methods, critical test
problems, and future directions are discussed.
Cosmic rays (CRs) are frequently modeled as an additional fluid in
hydrodynamic (HD) and magnetohydrodynamic (MHD) simulations of astrophysical
flows. The standard CR two-fluid model is described in terms of three
conservation laws (expressing conservation of mass, momentum and total energy)
and one additional equation (for the CR pressure) that cannot be cast in a
satisfactory conservative form. The presence of non-conservative terms with
spatial derivatives in the model equations prevents a unique weak solution
behind a shock. We investigate a number of methods for the numerical solution
of the two-fluid equations and find that, in the presence of shock waves, the
results depend on the choice of the numerical methods (spatial reconstruction,
time stepping, and the CFL number) and the adopted discretization.
Nevertheless, all methods converge to a unique result only if the energy
partition between the thermal and non-thermal fluids at the shock is prescribed
a priori. This highlights the closure problem of the two-fluid equations at
shocks. We suggest a robust method where the solutions are insensitive to the
numerical method. Comparison with the currently used methods, critical test
problems, and future directions are discussed.
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