Implicit second order method

The implicit second order method involves the derivatives of the next time level. Due to this reason they are iterative in nature. The second order time integration scheme is given by:

Algorithm

for r:= 1 step 1 until M do
$\phi ^{n + 1} = {4 \over 3}\phi ^n - {1 \over 3}\phi ^{n - 1} + {2 \over 3}\dot \phi \left( {t_{n + 1} ,\phi ^{n + 1} } \right) \bullet \Delta t$
end (r-loop)

Here, M is maximum number of internal iterations.