Solving diffusion equation with AMG approach

[ArturSalamatin]

Recently our group has implemented an additive (splitting) scheme (of Samarskii and Vabishchevich)
https://www.researchgate.net/publication/51962589_Additive_schemes_splitting_schemes_for_some_systems_of_evolutionaryequations

to solve the
ground water flow equation
https://en.wikipedia.org/wiki/Groundwater_flow_equation

The problem is essentially a single-phase approximation for a reservoir flow simulation based on Darcy's law (temperature equation).
It is a linear problem with a linear source term (a horizontal well), and a non-homogeneous
piezoconductivity (hydraulic diffusivity) coefficient.

However, we soon realized that due to relatively large piezoconductivity ( \alpha~0.05 m^2/s)
we have to advance in time with steps on the order of 1-10 minutes (!) to study the
reservoir evolution for months-years.

The time step estimation is determined by the condition
\alpha*\tau < 0.1hh

This requires a lot of computational time.

Is their any way to solve the problem faster? So, the point is to advance in time with larger time steps.
As far as I understand the AMG approach is also based in iterative solvers, which might have similar limitations...

Any recommendations are highly appreciated!