  We consider a biphasic continuum model for avascular tumour growth in two
spatial dimensions, in which a cell phase and a fluid phase follow conservation
of mass and momentum. A limiting nutrient that follows a diffusion process
controls the birth and death rate of the tumour cells. The cell volume
fraction, cell velocity--fluid pressure system, and nutrient concentration are
the model variables. A coupled system of a hyperbolic conservation law, \cred{a
viscous fluid model}, and a parabolic diffusion equation governs the dynamics
of the model variables. The tumour boundary moves with the normal velocity of
the outermost layer of cells, and this time--dependence is a challenge in
designing and implementing a stable and fast numerical scheme. We recast the
model into a form where the hyperbolic equation is defined on a fixed extended
domain and retrieve the tumour boundary as the interface at which the cell
volume fraction decreases below a threshold value. This procedure eliminates
the need to track the tumour boundary explicitly and the computationally
expensive re--meshing of the time--dependent domains. A numerical scheme based
on finite volume methods for the hyperbolic conservation law, Lagrange
$\mathbb{P}_2 - \mathbb{P}_1$ Taylor--Hood finite element method for the
\cred{viscous} system, and mass--lumped finite element method for the parabolic
equations is implemented in two spatial dimensions, and several cases are
studied. We demonstrate the versatility of the numerical scheme in catering for
irregular and asymmetric initial tumour geometries. We present numerical
simulations for both cases and the results are consistent with theoretical and
heuristic expectations such as early linear growth rate and preservation of
radial symmetry when the boundary conditions are symmetric.
