  In this work, we present an efficient way to decouple the multicontinuum
problems. To construct decoupled schemes, we propose Implicit-Explicit time
approximation in general form and study them for the fine-scale and
coarse-scale space approximations. We use a finite-volume method for fine-scale
approximation, and the nonlocal multicontinuum (NLMC) method is used to
construct an accurate and physically meaningful coarse-scale approximation. The
NLMC method is an accurate technique to develop a physically meaningful coarse
scale model based on defining the macroscale variables. The multiscale basis
functions are constructed in local domains by solving constraint energy
minimization problems and projecting the system to the coarse grid. The
resulting basis functions have exponential decay properties and lead to the
accurate approximation on a coarse grid. We construct a fully Implicit time
approximation for semi-discrete systems arising after fine-scale and
coarse-scale space approximations. We investigate the stability of the two and
three-level schemes for fully Implicit and Implicit-Explicit time
approximations schemes for multicontinuum problems in fractured porous media.
We show that combining the decoupling technique with multiscale approximation
leads to developing an accurate and efficient solver for multicontinuum
problems.
