  For time-dependent problems with high-contrast multiscale coefficients, the
time step size for explicit methods is affected by the magnitude of the
coefficient parameter. With a suitable construction of multiscale space, one
can achieve a stable temporal splitting scheme where the time step size is
independent of the contrast. Consider the parabolic equation with heterogeneous
diffusion parameter, the flow rates vary significantly in different regions due
to the high-contrast features of the diffusivity. In this work, we aim to
introduce a multirate partially explicit splitting scheme to achieve efficient
simulation with the desired accuracy. We first design multiscale subspaces to
handle flow with different speed. For the fast flow, we obtain a
low-dimensional subspace with respect to the high-diffusive component and adopt
an implicit time discretization scheme. The other multiscale subspace will take
care of the slow flow, and the corresponding degrees of freedom are treated
explicitly. Then a multirate time stepping is introduced for the two parts. The
stability of the multirate methods is analyzed for the partially explicit
scheme. Moreover, we derive local error estimators corresponding to the two
components of the solutions and provide an upper bound of the errors. An
adaptive local temporal refinement framework is then proposed to achieve higher
computational efficiency. Several numerical tests are presented to demonstrate
the performance of the proposed method.
