Sign-decay differential equation

For $y_0>0$, the initial-value problemhas the continuous piecewise differentiable solutionThe vector field is not Lipschitz at zero, but the direction on each side prevents any solution from leaving zero, so this solution is unique in the piecewise differentiable class.

The iteration $y_{n+1}=y_n-h\mathrm{sign}(y_n)$ agrees with the exact linear descent until the first step past zero. It then either remains at zero or alternates between two values of magnitude at most $h$. Consequently its uniform grid-point error is at most $h$ for arbitrarily long finite time intervals.