A global method for coupling transport with chemistry in heterogeneous porous media

Modeling reactive transport in porous media, using a local chemical equilibrium assumption, leads to a system of advection-diffusion PDE's coupled with algebraic equations. When solving this coupled system, the algebraic equations have to be solved at each grid point for each chemical species and at each time step. This leads to a coupled non-linear system. In this paper a global solution approach that enables to keep the software codes for transport and chemistry distinct is proposed. The method applies the Newton-Krylov framework to the formulation for reactive transport used in operator splitting. The method is formulated in terms of total mobile and total fixed concentrations and uses the chemical solver as a black box, as it only requires that on be able to solve chemical equilibrium problems (and compute derivatives), without having to know the solution method. An additional advantage of the Newton-Krylov method is that the Jacobian is only needed as an operator in a Jacobian matrix times vector product. The proposed method is tested on the MoMaS reactive transport benchmark.Comment: Computational Geosciences (2009) http://www.springerlink.com/content/933p55085742m203/?p=db14bb8c399b49979ba8389a3cae1b0f&pi=1

A monolithic conservative level set method with built-in redistancing

We introduce a new level set method for representing evolving interfaces. In the case of divergence-free velocity fields, the new method satisfies a conservation principle. Conservation is important for many applications such as modeling two-phase incompressible flow. In the present implementation, the conserved quantity is defined as the integral of a smoothed characteristic function. The new approach embeds level sets into a volume of fluid formulation. The evolution of an approximate signed distance function is governed by a conservation law for its (smoothed) sign. The non-linear level set transport equation is regularized by adding a flux correction term that assures a non-singular Jacobian and penalizes deviations from a distance function. The result is a locally conservative level set method with built-in elliptic redistancing. The continuous model is monolithic in the sense that the level set transport model, the volume of fluid law of mass conservation, and the minimization problem that preserves the approximate distance function property are incorporated into a single equation. There is no need for any extra stabilization, artificial compression, flux limiting, redistancing, mass correction, and other numerical fixes which are commonly used in level set or volume of fluid methods. In addition, there is just one free parameter that controls the strength of regularization and penalization in the model. The accuracy and conservation properties of the monolithic finite element / level set method are illustrated by the results of numerical studies for passive advection of free interfaces

An enhanced non-oscillatory BFECC algorithm for finite element solution of advective transport problems

In this paper, the so-called “back and forth error compensation correction (BFECC)” methodology is utilized to improve the solvers developed for the advection equation. Strict obedience to the so-called “discrete maximum principle” is enforced by incorporating a gradient–based limiter into the BFECC algorithm. The accuracy of the BFECC algorithm in capturing the steep–fronts in hyperbolic scalar–transport problems is improved by introducing a controlled anti–di¿usivity. This is achieved at the cost of performing an additional backward sub–solution–step and modifying the formulation of the error compensation accordingly. The performance of the proposed methodology is assessed by solving a series of benchmarks utilizing di¿erent combinations of the BFECC algorithms and the underlying numerical schemes. Results are presented for both the structured and unstructured meshes.This work was performed within the framework of AMADEUS project (”Advanced Multi-scAle moDEling of coupled mass transport for improving water management in fUel cellS”, reference number PGC2018-101655-B-I00) supported by the Ministerio de Ciencia, Innovacion e Universidades of Spain. The authors also acknowledge financial support of the mentioned Ministry via the “Severo Ochoa Programme” for Centres of Excellence in R&D (referece: CEX2018-000797-S) given to the International Centre for Numerical Methods in Engineering (CIMNE).Peer ReviewedPostprint (published version

Genetic Exponentially Fitted Method for Solving Multi-dimensional Drift-diffusion Equations

A general approach was proposed in this article to develop high-order exponentially fitted basis functions for finite element approximations of multi-dimensional drift-diffusion equations for modeling biomolecular electrodiffusion processes. Such methods are highly desirable for achieving numerical stability and efficiency. We found that by utilizing the one-one correspondence between continuous piecewise polynomial space of degree $k+1$ and the divergence-free vector space of degree $k$, one can construct high-order 2-D exponentially fitted basis functions that are strictly interpolative at a selected node set but are discontinuous on edges in general, spanning nonconforming finite element spaces. First order convergence was proved for the methods constructed from divergence-free Raviart-Thomas space $RT_0^0$ at two different node set

Adaptive multiresolution computations applied to detonations

A space-time adaptive method is presented for the reactive Euler equations describing chemically reacting gas flow where a two species model is used for the chemistry. The governing equations are discretized with a finite volume method and dynamic space adaptivity is introduced using multiresolution analysis. A time splitting method of Strang is applied to be able to consider stiff problems while keeping the method explicit. For time adaptivity an improved Runge--Kutta--Fehlberg scheme is used. Applications deal with detonation problems in one and two space dimensions. A comparison of the adaptive scheme with reference computations on a regular grid allow to assess the accuracy and the computational efficiency, in terms of CPU time and memory requirements.Comment: Zeitschrift f\"ur Physicalische Chemie, accepte

Reactive Flows in Deformable, Complex Media

Many processes of highest actuality in the real life are described through systems of equations posed in complex domains. Of particular interest is the situation when the domain is variable, undergoing deformations that depend on the unknown quantities of the model. Such kind of problems are encountered as mathematical models in the subsurface, or biological systems. Such models include various processes at different scales, and the key issue is to integrate the domain deformation in the multi-scale context. Having this as the background theme, this workshop focused on novel techniques and ideas in the analysis, the numerical discretization and the upscaling of such problems, as well as on applications of major societal relevance today