  In this work, we present a modification of the phase-field tumor growth model
given in [26] that leads to bounded, more physically meaningful, volume
fraction variables. In addition, we develop an upwind discontinuous Galerkin
(DG) scheme preserving the mass conservation, pointwise bounds and energy
stability of the continuous model. Finally, some computational tests in
accordance with the theoretical results are introduced. In the first test, we
compare our DG scheme with the finite element (FE) scheme related to the same
time approximation. The DG scheme shows a well-behavior even for strong
cross-diffusion effects in contrast with FE where numerical spurious
oscillations appear. Moreover, the second test exhibits the behavior of the
tumor-growth model under different choices of parameters and also of mobility
and proliferation functions.
