  Convergence failure and slow convergence rate are among the biggest
challenges with solving the system of non-linear equations numerically. While
using strictly small time steps sizes and unconditionally stable fully implicit
scheme mitigate the problem, the computational load becomes enormous. We
introduce a sequential local refinement scheme in space-time domain that
improves convergence rate and prevents convergence failure while not
restricting to small time step, thus boosting computational efficiency. We rely
on the non-linear two-phase flow model. The algorithm starts by solving the
coarsest mesh. Then regions with certain features such as saturation front is
refined to the finest resolution sequentially. Such process prevents
convergence failure. After each refinement, the solution from the previous mesh
is used to estimate initial guess of the current mesh for faster convergence.
Numerical results are presented to confirm accuracy of our algorithm as
compared to the traditional fine time step approach. We also observe 5 times
speedup in the runtime by using our algorithm.
