Explicit Euler method for the sign-decay equation
= Explicit Euler method for the sign-decay equation
The iteration $y_{n+1}=y_n-h\operatorname{sign}(y_n)$ 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.