  Numerical methods for the optimal transport problem is an active area of
research. Recent work of Kitagawa and Abedin shows that the solution of a
time-dependent equation converges exponentially fast as time goes to infinity
to the solution of the optimal transport problem. This suggests a fast
numerical algorithm for computing optimal maps; we investigate such an
algorithm here in the 1-dimensional case. Specifically, we use a finite
difference scheme to solve the time-dependent optimal transport problem and
carry out an error analysis of the scheme. A collection of numerical examples
is also presented and discussed.
