  At the continuous level, we consider two types of tumor growth models: the
cell density model, which is based on the fluid mechanical construction, is
more favorable for scientific interpretation and numerical simulations; and the
free boundary model, as the incompressible limit of the former, is more
tractable when investigating the boundary propagation. In this work, we aim to
investigate the boundary propagation speeds in those models based on asymptotic
analysis of the free boundary model and efficient numerical simulations of the
cell density model. We derive, for the first time, some analytical solutions
for the free boundary model with pressures jumps across the tumor boundary in
multi-dimensions with finite tumor sizes. We further show that in the large
radius limit, the analytical solutions to the free boundary model in one and
multiple spatial dimensions converge to traveling wave solutions. The
convergence rate in the propagation speeds are algebraic in multi-dimensions as
opposed to the exponential convergence in 1D. We also propose an accurate front
capturing numerical scheme for the cell density model, and extensive numerical
tests are provided to verify the analytical findings.
