A numerical scheme and benchmark tests for non-isothermal two-fluid ambipolar diffusion

Stars form in magnetized molecular clouds composed primarily of neutral gas with a trace amount of ions. We present a semi-implicit strategy for incorporating the equations that describe the coupled ion and neutral two-fluid equations, with a full energy equation, into the RIEMANN code that uses a TR-BDF2 algorithm to stably handle the stiffness of the source terms. We demonstrate that the numerical implementation works through the use of a suite of test problems that we catalog here. We show that reproducing the analytic dispersion analysis for the propagation of waves in a two-fluid plasma is an especially strong code test. We also present a two-fluid analogue of the Noh wall-shock problem and demonstrate the performance of the code on the Wardle instability. We also present a novel blast wave test, showing that the results reduce to the single fluid results under strong coupling, yet differing considerably when the coupling is weak. These test problems demonstrate that the numerical implementation can accurately capture the dissipation rate of waves and reproduce the structure of a C-shock.