Before revision:  We give an analytical solution to a simple problem and verify optimality and uniqueness of the corresponding HJB equation . To manage HJB equationsin more general cases , we propose a fast sweep method resorting to neither pseudo-time integration nor vanishing discount.
After revision:  We give an analytical solution to a simple problem and verify optimality and uniqueness of the corresponding HJB equation having unique non-smoothness and discontinuity. To numerically compute HJB equations , we propose a fast sweep method resorting to neither pseudo-time integration nor vanishing discount.