An unconditionally energy stable and positive upwind DG scheme for the Keller-Segel model

The well-suited discretization of the Keller-Segel equations for chemotaxis has become a very challenging problem due to the convective nature inherent to them. This paper aims to introduce a new upwind, mass-conservative, positive and energy-dissipative discontinuous Galerkin scheme for the Keller-Segel model. This approach is based on the gradient-flow structure of the equations. In addition, we show some numerical experiments in accordance with the aforementioned properties of the discretization. The numerical results obtained emphasize the really good behaviour of the approximation in the case of chemotactic collapse, where very steep gradients appear.Comment: 24 pages, 17 figures, 4 table

A posteriori error analysis of a positivity preserving scheme for the power-law diffusion Keller-Segel model

We study a finite volume scheme approximating a parabolic-elliptic Keller-Segel system with power law diffusion with exponent $\gamma \in [1,3]$ and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the $L^\infty(0,T;H^1(\Omega))$ norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability estimates and suitable conforming reconstructions of the numerical solution. We perform numerical experiments showing that our error bounds are linear in mesh width and elucidating the behaviour of the error estimator under changes of $\gamma$.Comment: 26 pages, 2 figures, 3 table

Well-balanced finite volume schemes for hydrodynamic equations with general free energy

Well balanced and free energy dissipative first- and second-order accurate finite volume schemes are proposed for a general class of hydrodynamic systems with linear and nonlinear damping. The natural Liapunov functional of the system, given by its free energy, allows for a characterization of the stationary states by its variation. An analog property at the discrete level enables us to preserve stationary states at machine precision while keeping the dissipation of the discrete free energy. These schemes allow for analysing accurately the stability properties of stationary states in challeging problems such as: phase transitions in collective behavior, generalized Euler-Poisson systems in chemotaxis and astrophysics, and models in dynamic density functional theories; having done a careful validation in a battery of relevant test cases.Comment: Videos from the simulations of this work are available at https://sergioperezresearch.wordpress.com/well-balance

A posteriori error control for a Discontinuous Galerkin approximation of a Keller-Segel model

We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional, in the sense that an a posteriori computable quantity needs to be small enough - which can be ensured by mesh refinement - and optimal in the sense that the error estimator decays with the same order as the error under mesh refinement. A specific feature of our error estimator is that it can be used to prove existence of a weak solution up to a certain time based on numerical results.Comment: 31 pages, 1 figure, 5 table

Bound-preserving finite element approximations of the Keller-Segel equations

This paper aims to develop numerical approximations of the Keller--Segel equations that mimic at the discrete level the lower bounds and the energy law of the continuous problem. We solve these equations for two unknowns: the organism (or cell) density, which is a positive variable, and the chemoattractant density, which is a nonnegative variable. We propose two algorithms, which combine a stabilized finite element method and a semi-implicit time integration. The stabilization consists of a nonlinear artificial diffusion that employs a graph-Laplacian operator and a shock detector that localizes local extrema. As a result, both algorithms turn out to be nonlinear.Both algorithms can generate cell and chemoattractant numerical densities fulfilling lower bounds. However, the first algorithm requires a suitable constraint between the space and time discrete parameters, whereas the second one does not. We design the latter to attain a discrete energy law on acute meshes. We report some numerical experiments to validate the theoretical results on blowup and non-blowup phenomena. In the blowup setting, we identify a \textit{locking} phenomenon that relates the $L^\infty(\Omega)$-norm to the $L^1(\Omega)$-norm limiting the growth of the singularity when supported on a macroelement.Comment: 27 pages, 22 figure

Finite Difference Approximation with ADI Scheme for Two-dimensional Keller-Segel Equations

Keller-Segel systems are a set of nonlinear partial differential equations used to model chemotaxis in biology. In this paper, we propose two alternating direction implicit (ADI) schemes to solve the 2D Keller-Segel systems directly with minimal computational cost, while preserving positivity, energy dissipation law and mass conservation. One scheme unconditionally preserves positivity, while the other does so conditionally. Both schemes achieve second-order accuracy in space, with the former being first-order accuracy in time and the latter second-order accuracy in time. Besides, the former scheme preserves the energy dissipation law asymptotically. We validate these results through numerical experiments, and also compare the efficiency of our schemes with the standard five-point scheme, demonstrating that our approaches effectively reduce computational costs.Comment: 29 page

Analysis and numerical simulation of tumor growth models

In this dissertation we focus on the numerical analysis of tumor growth models. Due to the difficulty of developing physically meaningful approximations of such models, we divide the main problem into more simple pieces of work that are addressed in the different chapters. First, in Chapter 2 we present a new upwind discontinuous Galerkin (DG) scheme for the convective Cahn–Hilliard model with degenerate mobility which preserves the pointwise bounds and prevents non-physical spurious oscillations. These ideas are based on a well-suited piecewise constant approximation of convection equations. The proposed numerical scheme is contrasted with other approaches in several numerical experiments. Afterwards, in Chapter 3, we extend the previous ideas to a mass-conservative, positive and energy-dissipative approximation of the Keller–Segel model for chemotaxis. Then we carry out several numerical tests in regimes of chemotactic collapse. These ideas are used later in Chapter 4 to develop a well-suited approximation of two different models related to chemotaxis: a generalization of the classical Keller–Segel model and a model of the neuroblast migration process to the olfactory bulb in rodents’ brains. Now we propose and study a phase-field tumor growth model in Chapter 5. Then, we develop an upwind DG scheme preserving the mass conservation, pointwise bounds and energy stability of the continuous model and we show both the good properties of the approximation and the qualitative behavior of the model in several numerical tests. Next, in Chapter 6, we present two new coupled and decoupled approximations of a Cahn–Hilliard–Navier–Stokes model with variable densities and degenerate mobility that preserve the physical properties of the model. Both approaches are compared in different computational tests including benchmark problems. Consequently, we propose, in Chapter 7, an extension of the previous tumor model including the effects of the surrounding fluid by means of a Cahn–Hilliard–Darcy model for which obtaining a physically meaningful approximation seems rather plausible using the previous ideas. Finally, this and other future lines of research are described, along with the conclusions and the scientific production of the dissertation, in Chapter 8

Well-balanced finite volume schemes for hydrodynamic equations with general free energy

Well balanced and free energy dissipative first- and second-order accurate finite volume schemes are proposed for a general class of hydrodynamic systems with linear and nonlinear damping. The natural Liapunov functional of the system, given by its free energy, allows for a characterization of the stationary states by its variation. An analog property at the discrete level enables us to preserve stationary states at machine precision while keeping the dissipation of the discrete free energy. These schemes allow for analysing accurately the stability properties of stationary states in challeging problems such as: phase transitions in collective behavior, generalized Euler-Poisson systems in chemotaxis and astrophysics, and models in dynamic density functional theories; having done a careful validation in a battery of relevant test cases.Comment: Videos from the simulations of this work are available at https://sergioperezresearch.wordpress.com/well-balance