AN EFFICIENT RECURSIVE ALGORITHM IN THE COMPUTATIONAL SIMULATION OF THE BOUNDED GROWTH OF BIOLOGICAL FILMS

In this work, we present a simple, finite-difference method to approximate positive and bounded solutions of a parabolic partial differential equation with nonlinear diffusion, which describes the growth dynamics of colonies of bacteria. A theorem on the existence and uniqueness of positive and bounded solutions of the model considered is available in the standard literature; however, analytical solutions for this model are difficult to calculate in exact form. The linear approach used in this manuscript provides a convenient way to represent the method in vector form through the multiplication of the new approximations by a square matrix that, under suitable conditions, turns out to be an M-matrix. The facts that every M-matrix is invertible and that all the entries of their inverses are positive numbers, are employed to elucidate conditions, which guarantee that positive and bounded initial profiles evolve into positive and bounded new approximations. The method is relatively simple, the temporal step-size is variable in general, and its efficient computational implementation makes use of the stabilized bi-conjugate gradient method. We provide numerical simulations in order to evince that the method preserves the practice comprising the positive and the bounded characters of the approximations.