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

Analysis of a fully discrete approximation for the classical Keller--Segel model: lower and a priori bounds

This paper is devoted to constructing approximate solutions for the classical Keller--Segel model governing \emph{chemotaxis}. It consists of a system of nonlinear parabolic equations, where the unknowns are the average density of cells (or organisms), which is a conserved variable, and the average density of chemoattractant. The numerical proposal is made up of a crude finite element method together with a mass lumping technique and a semi-implicit Euler time integration. The resulting scheme turns out to be linear and decouples the computation of variables. The approximate solutions keep lower bounds -- positivity for the cell density and nonnegativity for the chemoattractant density --, are bounded in the $L^1(\Omega)$-norm, satisfy a discrete energy law, and have \emph{ a priori} energy estimates. The latter is achieved by means of a discrete Moser--Trudinger inequality. As far as we know, our numerical method is the first one that can be encountered in the literature dealing with all of the previously mentioned properties at the same time. Furthermore, some numerical examples are carried out to support and complement the theoretical results.Comment: 24 pages, 6 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

DISCONTINUOUS GALERKIN METHODS FOR COMPRESSIBLE MISCIBLE DISPLACEMENTS AND APPLICATIONS IN RESERVOIR SIMULATION

This dissertation contains research on discontinuous Galerkin (DG) methods applied to the system of compressible miscible displacements, which is widely adopted to model surfactant flooding in enhanced oil recovery (EOR) techniques. In most scenarios, DG methods can effectively simulate problems in miscible displacements.However, if the problem setting is complex, the oscillations in the numerical results can be detrimental, with severe overshoots leading to nonphysical numerical approximations. The first way to address this issue is to apply the bound-preservingtechnique. Therefore, we adopt a bound-preserving Discontinuous Galerkin methodwith a Second-order Implicit Pressure Explicit Concentration (SIPEC) time marchingmethod to compute the system of two-component compressible miscible displacement in our first work. The Implicit Pressure Explicit Concentration (IMPEC) method is one of the most prevalent time marching approaches used in reservoir simulation for solving coupled flow systems in porous media. The main idea of IMPEC is to treat the pressure equation implicitly and the concentration equations explicitly. However, this treatment results in a first-order accurate scheme. To improve the order of accuracy of the scheme, we propose a correction stage to compensate for the second-order accuracy in each time step, thus naming it the SIPEC method. The SIPEC method is a crucial innovation based on the traditional second-order strong-stability-preserving Runge-Kutta (SSP-RK2) method. However, the SIPEC method is limited to second-order accuracy and cannot efficiently simulate viscous fingering phenomena. High-order numerical methods are preferred to reduce numerical artifacts and mesh dependence. In our second work, we adopt the IMPEC method based on the implicit-explicit Runge-Kutta (IMEX-RK) Butcher tableau to achieve higher order temporal accuracy while also ensuring stability. The high-order discontinuous Galerkin method is employed to simulate the viscous fingering fluid instabilities in a coupled nonlinear system of compressible miscible displacements. Although the bound-preserving techniques can effectively yield physically relevant numerical approximations, their success depends heavily on theoretical analysis, which is not straightforward for high-order methods. Therefore, we introduce an oscillation-free damping term to effectively suppress the spurious oscillations near discontinuities in high-order DG methods. As indicated by the numerical experiments, the incorporation of the bound-preserving DG method with SIPEC time marching and high-order OFDG with IMPEC time marching provides satisfactory results for simulating fluid flow in reservoirs

Local Discontinuous Galerkin Method for the Keller-Segel Chemotaxis Model

© 2017, Springer Science+Business Media New York. In this paper, we apply the local discontinuous Galerkin (LDG) method to 2D Keller–Segel (KS) chemotaxis model. We improve the results upon (Epshteyn and Kurganov in SIAM J Numer Anal, 47:368–408, 2008) and give optimal rate of convergence under special finite element spaces before the blow-up occurs (the exact solutions are smooth). Moreover, to construct physically relevant numerical approximations, we consider P1 LDG scheme and develop a positivity-preserving limiter to the scheme, extending the idea in Zhang and Shu (J Comput Phys, 229:8918–8934, 2010). With this limiter, we can prove the L1-stability of the numerical scheme. Numerical experiments are performed to demonstrate the good performance of the positivity-preserving LDG scheme. Moreover, it is known that the chemotaxis model will yield blow-up solutions under certain initial conditions. We numerically demonstrate how to find the approximate blow-up time by using the L2-norm of the L1-stable numerical solution

Discontinuous Galerkin methods for convection-diffusion equations and applications in petroleum engineering

This dissertation contains research in discontinuous Galerkin (DG) methods applying to convection-diffusion equations. It contains both theoretical analysis and applications. Initially, we develop a conservative local discontinuous Galerkin (LDG) method for the coupled system of compressible miscible displacement problem in two space dimensions. The main difficulty is how to deal with the discontinuity of approximations of velocity, u, in the convection term across the cell interfaces. To overcome the problems, we apply the idea of LDG with IMEX time marching using the diffusion term to control the convection term. Optimal error estimates in Linfinity(0, T; L2) norm for the solution and the auxiliary variables will be derived. Then, high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes will be developed. There are three main difficulties to make the concentration of each component between 0 and 1. Firstly, the concentration of each component did not satisfy a maximum-principle. Secondly, the first-order numerical flux was difficult to construct. Thirdly, the classical slope limiter could not be applied to the concentration of each component. To conquer these three obstacles, we first construct special techniques to preserve two bounds without using the maximum-principle-preserving technique. The time derivative of the pressure was treated as a source of the concentration equation. Next, we apply the flux limiter to obtain high-order accuracy using the second-order flux as the lower order one instead of using the first-order flux. Finally, L2-projection of the porosity and constructed special limiters that are suitable for multi-component fluid mixtures were used. Lastly, a new LDG method for convection-diffusion equations on overlapping mesh introduced in [J. Du, Y. Yang and E. Chung, Stability analysis and error estimates of local discontinuous Galerkin method for convection-diffusion equations on overlapping meshes, BIT Numerical Mathematics (2019)] showed that the convergence rates cannot be improved if the dual mesh is constructed by using the midpoint of the primitive mesh. They provided several ways to gain optimal convergence rates but the reason for accuracy degeneration is still unclear. We will use Fourier analysis to analyze the scheme for linear parabolic equations with periodic boundary conditions in one space dimension. To investigate the reason for the accuracy degeneration, we explicitly write out the error between the numerical and exact solutions. Moreover, some superconvergence points that may depend on the perturbation constant in the construction of the dual mesh were also found out