  We propose and analyze a second-order partitioned time-stepping method for a
two-phase flow problem in porous media. The algorithm is based on a
refactorization of Cauchy's one-leg $\theta$-method. The main part consists of
the implicit backward Euler method on $[t^n, t^{n+\theta}]$, while part two
uses a linear extrapolation on $[t^{n+\theta},t^{n+1}]$ to obtain the solution
at $t^{n+1}$, equivalent to the forward Euler method.
  In the backward Euler step, the decoupled equations are solved iteratively.
We prove that the iterations converge linearly to the solution of the coupled
problem, under some conditions on the data. When $\theta = 1/2$, the algorithm
is equivalent to the symplectic midpoint method. In the absence of the chain
rule for time-discrete setting, we approximate the change in the free energy by
the product of a second-order accurate discrete gradient (chemical potential)
and the one-step increment of the state variables. Similar to the continuous
case, we also prove a discrete Gibbs free energy balance equation, without
numerical dissipation. In the numerical tests we compare this implicit midpoint
method with the classic backward Euler method, and two implicit-explicit
time-lagging schemes. The midpoint method outperforms the other schemes in
terms of rates of convergence, long-time behavior and energy approximation, for
small and large values of the time step.

Numerical Analysis;Numerical Analysis