  We are interested in the numerical solution of coupled nonlinear partial
differential equations (PDEs) in two and three dimensions. Under certain
assumptions on the domain, we take advantage of the Kronecker structure arising
in standard space discretizations of the differential operators and illustrate
how the resulting system of ordinary differential equations (ODEs) can be
treated directly in matrix or tensor form. Moreover, in the framework of the
proper orthogonal decomposition (POD) and the discrete empirical interpolation
method (DEIM) we derive a two- and three-sided model order reduction strategy
that is applied directly to the ODE system in matrix and tensor form
respectively. We discuss how to integrate the reduced order model and, in
particular, how to solve the tensor-valued linear system arising at each
timestep of a semi-implicit time discretization scheme. We illustrate the
efficiency of the proposed method through a comparison to existing techniques
on classical benchmark problems such as the two- and three-dimensional Burgers
equation.
