The forward difference for \( u'(t) \)

Now consider a forward difference: $$ u'(t_n) \approx [D_t^+ u]^n = \frac{u^{n+1}-u^n}{\Delta t}\tp$$ Define the truncation error: $$ R^n = [D_t^+ u]^n - u'(t_n)\tp$$ Expand \( u^{n+1} \) in a Taylor series around \( t_n \), $$ u(t_{n+1}) = u(t_n) + u'(t_n)\Delta t + {\half}u''(t_n)\Delta t^2 + \Oof{\Delta t^3} \tp $$ We get $$ R = {\half}u''(t_n)\Delta t + \Oof{\Delta t^2}\tp$$