  We describe and test a family of new numerical methods to solve the
Schrodinger equation in self-gravitating systems, e.g. Bose-Einstein
condensates or 'fuzzy'/ultra-light scalar field dark matter. The methods are
finite-volume Godunov schemes with stable, higher-order accurate gradient
estimation, based on a generalization of recent mesh-free finite-mass Godunov
methods. They couple easily to particle-based N-body gravity solvers (with or
without other fluids, e.g. baryons), are numerically stable, and
computationally efficient. Different sub-methods allow for manifest
conservation of mass, momentum, and energy. We consider a variety of test
problems and demonstrate that these can accurately recover solutions and remain
stable even in noisy, poorly-resolved systems, with dramatically reduced noise
compared to some other proposed implementations (though certain types of
discontinuities remain challenging). This is non-trivial because the "quantum
pressure" is neither isotropic nor positive-definite and depends on
higher-order gradients of the density field. We implement and test the method
in the code GIZMO.
