  A Monge-Amp\`ere (MA) equation arises when seeking an optimally transported
mesh that equidistributes a given monitor function in Cartesian space. This MA
equation is a fully nonlinear PDE, with a source term that is a function of the
gradient of the solution. This nonlinear source term is an additional
computational challenge that has received little attention from MA applications
in other fields. There are two major components needed to find a solution to
the MA equation: a spatial discretisation and an algorithm to find a solution
of the resulting nonlinear algebraic equations. There have been a number of
different approaches proposed in the literature to solve the MA equation but
none of which perform consistent comparisons across both algorithmic and
discretisation differences. In this study we explore different algorithmic
methods for the MA equation all within the context of a finite volume spatial
discretisation. We introduce a new linearisation of the MA equation that
neglects the nonlinearities arising from the source term and show that it leads
to a method that is fast, robust and free of tuning parameters. We present
numerical experiments that show methods based on this linearisation of the MA
equation are more computationally efficient than those that rely on other
techniques such as a parabolic relaxation. Further, the equations resulting
from a full linearisation of the MA equation, equivalent to using Newton's
method, can be seen as analogous to an advection-diffusion equation. This
allows many tools that exist for computational fluid dynamics to be re-factored
easily to solve the MA equation. The robustness and efficiency of the newly
introduced method gives hope that an adaptive solver for geophysical flows
using mesh redistribution can be computationally feasible in the near future.
