  In this work, we present an efficient approach to solve nonlinear
high-contrast multiscale diffusion problems. We incorporate the
explicit-implicit-null (EIN) method to separate the nonlinear term into a
linear term and a damping term, and then utilise the implicit and explicit time
marching scheme for the two parts respectively. Due to the multiscale property
of the linear part, we further introduce a temporal partially explicit
splitting scheme and construct suitable multiscale subspaces to speed up the
computation. The approximated solution is splitted into these subspaces
associated with different physics. The temporal splitting scheme employs
implicit discretization in the subspace with small dimension that representing
the high-contrast property and uses explicit discretization for the other
subspace. We exploit the stability of the proposed scheme and give the
condition for the choice of the linear diffusion coefficient. The convergence
of the proposed method is provided. Several numerical tests are performed to
show the efficiency and accuracy of the proposed approach.
