  In this work, fourth-order compact block-centered finite difference (CBCFD)
schemes combined with the Crank-Nicolson discretization are constructed and
analyzed for solving parabolic integro-differential type non-Fickian flows in
one-dimensional and two-dimensional cases. Stability analyses of the
constructed schemes are derived rigorously. We also obtain the optimal
second-order convergence in temporal increment and the fourth-order convergence
in spatial direction for both velocity and pressure. To verify the validity of
the CBCFD schemes, we present some experiments to show that the numerical
results are in agreement with our theoretical analysis.
