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.
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