With a correction term Forward Euler becomes Crank-Nicolson

Use the other alternative \( u''=-au' \): $$ u'=-au - {\half}a\Delta t u'\quad\Rightarrow\quad \left( 1 + {\half}a\Delta t\right) u' = -au\tp$$ Apply Forward Euler: $$ \left( 1 + {\half}a\Delta t\right)\frac{u^{n+1}-u^n}{\Delta t} = -au^n,$$ which after some algebra can be written as $$ u^{n+1} = \frac{1 - {\half}a\Delta t}{1+{\half}a\Delta t}u^n\tp$$ This is a Crank-Nicolson scheme (of second order)!