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Adaptive Smoothed Particle Hydrodynamics (ASPH) is a generalized form of Smoothed Particle Hydrodynamics (SPH) which allows improved treatment of strongly anisotropic hydrodynamic phenomena. ASPH utilizes locally ellipsoidal kernels rather than the traditional spherical kernels of SPH. One objection to ASPH which has been raised is the issue of angular momentum conservation, which is not rigorously guaranteed under ASPH. A particular example of this is that of an initially rigidly rotating water bar, posed by Benz and Davies. In this paper we present evidence that ASPH does indeed conserve angular momentum (of order a few percent) when presented with a well-posed problem. However, problems where surfaces are present do not conserve angular momentum. The error is traced directly to the neglect of the surface terms when deriving the (A)SPH form of the hydrodynamic equations. The surface terms themselves do not guarantee conservation of angular momentum, but only reduce the magnitude of the error. Finally, we propose a new scheme for using a mass averaged kernel function instead of the traditional volume averaging of SPH. The combination of surface terms and mass weighting may offer a viable solution to allow (A)SPH based techniques to correctly deal with discontinuous structures such as surfaces.
