  We present a generalisation of the all-Mach solver of Fuster & Popinet (2018)
to account for heat diffusion between two different compressible phases. By
solving a two-way coupled system of equations for pressure and temperature, the
current code is shown to increase the robustness and accuracy of the solver
with respect to classical explicit discretization schemes. Different test cases
are proposed to validate the implementation of the thermal effects: an
Epstein-Plesset like problem for temperature is shown to compare well with a
spectral method solution. The code also reproduces free small amplitude
oscillations of a spherical bubble where analytical solutions capturing the
transition between isothermal and adiabatic regimes are available. We show
results of a single sonoluminescent bubble (SBSL) in standing waves, where the
result of the DNS is compared with that of other methods in the literature.
Moreover, the Rayleigh collapse problem is studied in order to evaluate the
importance of thermal effects on the peak pressures reached during the collapse
of spherical bubbles. Finally, the collapse of a bubble near a rigid boundary
is studied reporting the change of heat flux as a function of the stand-off
distance.
