  This work focuses on the development of a two-step field-split nonlinear
preconditioner to accelerate the convergence of two-phase flow and transport in
heterogeneous porous media. We propose a field-split algorithm named
Field-Split Multiplicative Schwarz Newton (FSMSN), consisting in two steps:
first, we apply a preconditioning step to update pressure and saturations
nonlinearly by solving approximately two subproblems in a sequential fashion;
then, we apply a global step relying on a Newton update obtained by linearizing
the system at the preconditioned state. Using challenging test cases, FSMSN is
compared to an existing field-split preconditioner, Multiplicative Schwarz
Preconditioned for Inexact Newton (MSPIN), and to standard solution strategies
such as the Sequential Fully Implicit (SFI) method or the Fully Implicit Method
(FIM). The comparison highlights the impact of the upwinding scheme in the
algorithmic performance of the preconditioners and the importance of the
dynamic adaptation of the subproblem tolerance in the preconditioning step. Our
results demonstrate that the two-step nonlinear preconditioning approach-and in
particular, FSMSN-results in a faster outer-loop convergence than with the SFI
and FIM methods. The impact of the preconditioners on computational
performance-i.e., measured by wall-clock time-will be studied in a subsequent
publication.
