On an efficient implementation and mass boundedness conditions for a discrete Dirichlet problem associated with a nonlinear system of singular partial differential equations

In this work, we propose an efficient implementation of a finite-difference method employed to approximate the solutions of a system of partial differential equations that appears in the investigation of the growth of biological films. The associated homogeneous Dirichlet problem is discretized using a linear approach. This discretization yields a positivity- and boundedness-preserving implicit technique which is represented in vector form through the multiplication by a sparse matrix. A straightforward implementation of this methodology would require a substantial amount of computer memory and time, but the problem is conveniently coded using a continual reduction of the zero sub-matrices of the representing matrix. In addition to the conditions that guarantee the positivity and the boundedness of the numerical approximations, we establish some parametric constraints that assure that the same properties for the discrete total mass at each point of the mesh-grid and each discrete time are actually satisfied. Some simulations are provided in order to illustrate both the performance of the implementation, and the preservation of the positivity and the boundedness of the numerical approximations.