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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