I want to numerically solve the initial value problem of ordinary differential equation for function $u=u(t)$: $$ u'(t)=L(u). $$ I find an second-order implicit method: $$ u_{n+1}=u_n+\Delta t L(u_{n+1}-\frac{\Delta t}{2}L(u_{n+1})). $$ Here $\Delta t$ is the time step size and $u_n$ denotes the numerical solution at $n$-th time step.
It seems that this implicit method cannot be written as a form of Runge-Kutta method. Do someone know what's the name of this method? Could it be extended to higher order?