Numerical analysis of a chemotaxis–swimming bacteria model on a general triangular mesh

A chemorepulsion model with superlinear production: analysis of the continuous problem and two approximately positive and energy-stable schemes

“…In fact, it follows by testing (36) by v − and using that v 0 ≥ 0. Finally, we will prove that (u, v) satisfies the energy inequality (8). Indeed, integrating (24) in time from t 0 to t 1 , with t 1 > t 0 ≥ 0, and taking into account that…”

“…It was proved that the schemes UVε and USε are unconditionally energy-stables with respect to modified energies defined in terms of the variables of each scheme, and some energy inequalities are satisfied (see Theorems 4.7 and 4.15). However, it is not clear how to prove the energy-stability of these schemes with respect to the "exact" energy E e (u, v) given in (81), which comes from the continuous problem (3) (see (8) and ( 9)). Therefore, it is interesting to compare numerically the schemes with respect to this energy E e (u, v), and to study the behavior of the following "residual" of the discrete energy law…”

“…Likewise, the positivity or approximated positivity properties have been studied on numerical schemes for chemotaxis models. In [8], Chamoun and collaborators proved a discrete maximum principle for a combined FV-FE scheme approaching a chemotaxis-fluid model. The positivity of only time-discrete schemes and approximated positivity of a fully discrete FE scheme associated with a chemorepulsion model with quadratic signal production were proved in [17] and [19], respectively; while, for the case of linear production, we refer to [20].…”

Numerical analysis for a chemotaxis-Navier–Stokes system

“…Some unconditionally energy stable fully discrete schemes for a parabolic repulsive-productive chemotaxis model (with linear production term) were recently analyzed in [13]. On the other hand, when the interaction with a fluid is assumed, as far as we know, the literature related to the numerical analysis of chemotaxis-Navier-Stokes system is scarce, see [3,22]. In [22], some numerical evidence that solutions to the elliptic-parabolic 2D-Keller-Segel-Stokes system exist for initial mass larger than 8 , was considered.…”

“…In [22], some numerical evidence that solutions to the elliptic-parabolic 2D-Keller-Segel-Stokes system exist for initial mass larger than 8 , was considered. Even for the 2D-Keller-Segel-Stokes, the convergence of a numerical scheme given by the combination of the finite volume method and the nonconforming FE method has been studied in [3]. For the chemotaxis-Navier-Stokes system (1.1) (with other boundary conditions), we only know that some numerical simulations have been made in order to investigate the patterns formation and predict numerically the nonlinear dynamic of the chemotaxis-fluid system (see for instance, [4,6,19,21,30]).…”