  We present and analyze a structure-preserving method for the approximation of
solutions to nonlinear cross-diffusion systems, which combines a Local
Discontinuous Galerkin spatial discretization with the backward Euler time
stepping scheme. The proposed method makes use of the underlying entropy
structure of the system, expressing the main unknown in terms of the entropy
variable by means of a nonlinear transformation. Such a transformation allows
for imposing the physical positivity or boundedness constraints on the
approximate solution in a strong sense. Moreover, nonlinearities do not appear
explicitly within differential operators or interface terms in the scheme,
which significantly improves its efficiency and ease its implementation. We
prove the existence of discrete solutions and their asymptotic convergence to a
weak solution to the continuous problem. Numerical results for some one- and
two-dimensional problems illustrate the accuracy and entropy stability of the
proposed method.
