  This paper investigates numerical methods for simulations of the short-term
behavior of a geothermal energy storage. Such simulations are needed for the
optimal control and management of residential heating systems equipped with an
underground thermal storage. There a given volume under or aside of a building
is filled with soil and insulated to the surrounding ground. The thermal energy
is stored by raising the temperature of the soil inside the storage. It is
charged and discharged via heat exchanger pipes filled with a moving fluid.
Simulations of geothermal energy storages aim to determine how much energy can
be stored in or taken from the storage within a given short period of time. The
latter depends on the dynamics of the spatial temperature distribution in the
storage which is governed by a linear heat equation with convection and
appropriate boundary and interface conditions. We consider semi- and full
discretization of that PDE using upwind finite difference schemes and study
associated stability problems. Numerical results based on the derived methods
are presented in the companion paper [12].
