  This work continues a line of works on developing partially explicit methods
for multiscale problems. In our previous works, we have considered linear
multiscale problems, where the spatial heterogeneities are at subgrid level and
are not resolved. In these works, we have introduced contrast-independent
partially explicit time discretizations for linear equations. The
contrast-independent partially explicit time discretization divides the spatial
space into two components: contrast dependent (fast) and contrast independent
(slow) spaces defined via multiscale space decomposition. Following this
decomposition, temporal splitting is proposed that treats fast components
implicitly and slow components explicitly. The space decomposition and temporal
splitting are chosen such that it guarantees a stability and formulate a
condition for the time stepping. This condition is formulated as a condition on
slow spaces. In this paper, we extend this approach to nonlinear problems. We
propose a splitting approach and derive a condition that guarantees stability.
This condition requires some type of contrast-independent spaces for slow
components of the solution. We present numerical results and show that the
proposed methods provide results similar to implicit methods with the time step
that is independent of the contrast.
