  We consider the problem of finding stationary points of the Dirichlet energy
for orthogonal matrix-valued fields. Following the Ginzburg-Landau approach,
this energy is relaxed by penalizing the matrix-valued field when it does not
take orthogonal matrix values. A generalization of the MBO diffusion generated
motion is introduced that effectively finds local minimizers of this energy by
iterating two steps until convergence. In the first step, as in the original
method, the current matrix-valued field is evolved by the diffusion equation.
In the second step, the field is pointwise reassigned to the closest orthogonal
matrix, which can be computed via the singular value decomposition. We extend
the Lyapunov function of Esedoglu and Otto to show that the method is
non-increasing on iterates and hence, unconditionally stable. We also prove
that spatially discretized iterates converge to a stationary solution in a
finite number of iterations. The algorithm is implemented using the closest
point method and non-uniform fast Fourier transform. We conclude with several
numerical experiments on flat tori and closed surfaces, which, unsurprisingly,
exhibit classical behavior from the Allen-Cahn and complex Ginzburg Landau
equations, but also new phenomena.
