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Ambipolar diffusion is a process occurring in partially ionised astrophysicalsystems that imparts a complicated mathematical and physical nature to Ohm'slaw. The numerical codes that solve the magnetohydrodynamic (MHD) equationshave to be able to deal with the singularities that are naturally created inthe system by the ambipolar diffusion term. The global aim is to calculate aset of theoretical self-similar solutions to the nonlinear diffusion equationwith cylindrical symmetry that can be used as tests for MHD codes which includethe ambipolar diffusion term. First, following the general methods developed inthe applied mathematics literature, we obtained the theoretical solutions aseigenfunctions of a nonlinear ordinary differential equation. Phase-planetechniques were used to integrate through the singularities at the locations ofthe nulls, which correspond to infinitely sharp current sheets. In the secondhalf of the paper, we consider the use of these solutions as tests for MHDcodes. To that end, we used the Bifrost code, thereby testing the capabilitiesof these solutions as tests as well as (inversely) the accuracy of Bifrost'srecently developed ambipolar diffusion module. The obtained solutions are shownto constitute a demanding, but nonetheless viable, test for MHD codes thatincorporate ambipolar diffusion. The Bifrost code is able to reproduce thetheoretical solutions with sufficient accuracy up to very advanced diffusivetimes. Using the code, we also explored the asymptotic properties of ourtheoretical solutions in time when initially perturbed with either small orfinite perturbations. The functions obtained in this paper are relevant asphysical solutions and also as tests for general MHD codes. They provide a morestringent and general test than the simpleZeldovich-Kompaneets-Barenblatt-Pattle solution.

Ambipolar diffusion: Self-similar solutions and MHD code testing. Cylindrical symmetry