Many microorganisms live within surface-associated consortia, termed biofilms, that can form intricate porous structures interspersed with a network of fluid channels. In such systems, transport phenomena, including flow and advection, regulate various aspects of cell behavior by controlling nutrient supply, evacuation of waste products, and permeation of antimicrobial agents. This study presen...

In previous chapters we considered the processes of advection and molecular diffusion and have seen some example problems with so called “turbulent diffusion” coefficients, where we use the same governing equations, but with larger diffusion (mixing) coefficients. In natural rivers, a host of processes lead to a non-uniform velocity field, which allows mixing to occur much faster than by molecu...

Reaction-diffusion-advection equations provide precise interpretations for many important phenomena in complex interactions between natural and artificial systems. This paper studies second-order semi-discretizations the numerical solution of reaction-diffusion-advection modeling quenching types singularities occurring numerous applications. Our investigations particularly focus at cases where ...

We have derived reaction-dispersal-aggregation equations from Markovian reaction-random walks with density-dependent jump rate or density-dependent dispersal kernels. From the corresponding diffusion limit we recover well-known reaction-diffusion-aggregation and reaction-diffusion-advection-aggregation equations. It is found that the ratio between the reaction and jump rates controls the onset ...

We prove the existence and uniqueness of a positive solution to a logistic system of differential difference equations that arises as a population model for a single species which is composed of several habitats connected by linear migration rates. Our proof is based on the proof of a similar result for a reaction-advection-diffusion equation.

We have implemented a numerical code (ReLPM, Real Leja Points Method) for polynomial interpolation of the matrix exponential propagators exp (∆tA)v and φ(∆tA)v, φ(z) = (exp (z) − 1)/z. The ReLPM code is tested and compared with Krylov-based routines, on large scale sparse matrices arising from the spatial discretization of 2D and 3D advection-diffusion equations.

We present a fourth-order accurate algorithm for solving Poisson’s equation, the heat equation, and the advection-diffusion equation on a hierarchy of block-structured, adaptively refined grids. For spatial discretization, finite-volume stencils are derived for the divergence operator and Laplacian operator in the context of structured adaptive mesh refinement and a variety of boundary conditio...

Advection has been the standard transport mechanism in flow visualization. Diffusion, in contrast, has not been considered important in visual flow field analysis so far, although it is inherent to many physical processes. We present a novel technique that allows for interactive 3D visualization of both advection and diffusion in unsteady fluid flow. We extend texture-based flow visualization, ...

• Rigorous homogenization of parabolic problems in evolving micro-domains. Reaction-diffusion-advection equations with nonlinear boundary conditions. Strong two-scale compactness result Kolmogorov-Simon-type. We consider a reaction-diffusion-advection problem perforated medium, reactions the bulk and at microscopic boundary, slow diffusion scaling. The microstructure changes time; microstructur...

Predictive simulations are crucial for the success of many subsurface applications, and it is highly desirable to obtain accurate non-negative solutions for transport equations in these numerical simulations. To this end, optimization-based methodologies based on quadratic programming (QP) have been shown to be a viable approach to ensuring discrete maximum principles and the non-negative const...