An error in the gravitational force that the source of gravity induces on itself (a self-force error) violates both the conservation of linear momentum and the conservation of energy. If such errors are present in a self-gravitating system and are not sufficiently random to average out, the obtained numerical solution will become progressively more unphysical with time: the system will acquire or lose momentum and energy due to numerical effects. In this paper, we demonstrate how self-force errors can arise in the case where self-gravity is solved on an adaptively refined mesh when the refinement is nonuniform. We provide the analytical expression for the self-force error and numerical examples that demonstrate such self-force errors in idealized settings. We also show how these errors can be corrected to an arbitrary order by straightforward addition of correction terms at the refinement boundaries.

重力源在自身上引起的重力误差（自力误差）违反了线性动量守恒和能量守恒。如果这样的误差存在于自引力系统中并且没有足够的随机性来平均化，那么所获得的数值解将随着时间的推移逐渐变得更加不实际：由于数值效应，系统将获得或失去动量和能量。在本文中，我们展示了在细化不均匀时，在自适应细化网格上求解自重力的情况下，自力误差是如何产生的。我们提供了自力误差和数值例子的解析表达式，这些例子证明了理想化设置中的这种自力误差。