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A least-squares method for solving the hyperbolic Monge-Amp\`ere equationwith transport boundary condition is introduced. The method relies on aniterative procedure for the gradient of the solution, the so-called mapping. Byformulating error functionals for the interior domain, the boundary, bothseparately and as linear combination, three minimization problems are solvediteratively to compute the mapping. After convergence, a fourth minimizationproblem, to compute the solution of the Monge-Amp\`ere equation, is solved. Theapproach is based on a least-squares method for the elliptic Monge-Amp\`ereequation by Prins et al., and is improved upon by the addition of analyticalsolutions for the minimization on the interior domain and by the introductionof two new boundary methods. Lastly, the iterative method is tested on avariety of examples. It is shown that, when the iterative method converges,second-order global convergence as function of the spatial discretization isobtained.

An Iterative Least-Squares Method for the Hyperbolic Monge-Amp\`ere Equation with Transport Boundary Condition