Advection Diffusion Reaction Problems Research Articles

An Efficient Numerical Method for Fractional Advection–Diffusion–Reaction Problem with RLC Fractional Derivative

An Efficient Numerical Method for Fractional Advection–Diffusion–Reaction Problem with RLC Fractional Derivative

A stabilized element-free Galerkin method for the advection–diffusion–reaction problem

A stabilized element-free Galerkin (EFG) method is presented and analyzed for the numerical solution of the advection–diffusion–reaction problem. Through deriving a stabilized Nitsche-type weak form for the problem, the presented EFG method is valid for small diffusivity and large reaction coefficient. Theoretical error of the method is analyzed. Numerical results verify the effectiveness of the method.

A fully adaptive explicit stabilized integrator for advection–diffusion–reaction problems

A novel second order family of explicit stabilized Runge–Kutta–Chebyshev methods for advection–diffusion–reaction equations is introduced. The new methods outperform existing schemes for relatively high Peclet number due to their favorable stability properties and explicitly available coefficients. The construction of the new schemes is based on stabilization using second kind Chebyshev polynomials first used in the construction of the stochastic integrator SK-ROCK. An adaptive algorithm to implement the new scheme is proposed. This algorithm is able to automatically select the suitable step size, number of stages, and damping parameter at each integration step. Numerical experiments that illustrate the efficiency of the new algorithm are presented.

Orthogonal grid physics-informed neural networks: A neural network-based simulation tool for advection–diffusion–reaction problems

Stable and accurate reconstruction of pollutant transport is a crucial and challenging problem, including the inverse problem of identifying pollution sources and physical coefficients and the forward problem of inferring pollutant transport. Governed by advection, diffusion, and reaction processes, this transport phenomenon can be represented by the advection–diffusion–reaction (ADR) equation. In this paper, the physics-informed neural networks (PINNs) are applied to solve the forward and inverse ADR problems. To further enhance the stability and accuracy of the original PINN, two improvements are developed. The first adjusts the orthogonal grid (OG) point selection method and the other suggests adding an additional regulation function, namely, first derivative constraint (FDC). The new method is referred to as OG-PINN with FDC. To verify the effectiveness of the proposed method, five forward and inverse ADR problems are solved, and the results are compared with the analytical and reference solutions. For forward problems, the improved method can solve various ADR problems accurately and stably. For inverse problems, the ability of the OG-PINN for model parameter learning and initial distribution prediction is demonstrated and analyzed. The former gives the missed physical information in the ADR equation from the data, and the latter is used to trace the source of pollutants. The proposed method is quantitatively reliable for investigating various advection–diffusion–reaction processes.

Adaptive Refinement in Advection–Diffusion Problems by Anomaly Detection: A Numerical Study

We consider advection–diffusion–reaction problems, where the advective or the reactive term is dominating with respect to the diffusive term. The solutions of these problems are characterized by the so-called layers, which represent localized regions where the gradients of the solutions are rather large or are subjected to abrupt changes. In order to improve the accuracy of the computed solution, it is fundamental to locally increase the number of degrees of freedom by limiting the computational costs. Thus, adaptive refinement, by a posteriori error estimators, is employed. The error estimators are then processed by an anomaly detection algorithm in order to identify those regions of the computational domain that should be marked and, hence, refined. The anomaly detection task is performed in an unsupervised fashion and the proposed strategy is tested on typical benchmarks. The present work shows a numerical study that highlights promising results obtained by bridging together standard techniques, i.e., the error estimators, and approaches typical of machine learning and artificial intelligence, such as the anomaly detection task.

Automatically adaptive, stabilized finite element method via residual minimization for heterogeneous, anisotropic advection–diffusion–reaction problems

We describe a stable finite element formulation for advection–diffusion–reaction problems that allows for the easy implementation of robust automatic adaptivity. We consider locally vanishing, heterogeneous, and anisotropic diffusivities, as well as advection-dominated diffusion problems. We apply a general stabilized finite element framework (Calo et al., 2020) that seeks a discrete solution through a residual minimization process on a proper stable discontinuous Galerkin (dG) dual norm. This technique leads to a sequence of saddle-point problems that are discretely stable and deliver a robust error estimate that drives mesh adaptivity. In this work, we demonstrate the method’s efficiency in extreme scenarios, where the solutions’ quality and performance are comparable to classical discontinuous Galerkin formulations in the respective discrete space norm on a particular mesh. We focus on the practical implementation of the adaptive technology to a broad range of engineering applications, from singularly perturbed linear advection-dominated diffusion to highly non-linear problems. This technique starts from coarse meshes and adapts itself to achieve a user-specified solution quality.

Optimal fluid stretching for mixing-limited reactions in rough channel flows

We study the reactive displacement of two miscible fluids in channel flows and establish a quantitative link between fluid stretching and chemical reactivity. At the mixing interface, the two fluids react according to the instantaneous irreversible bimolecular reaction $A + B \rightarrow C$ . We simulate the advection–diffusion–reaction problem using a random walk based reactive particle method that is free of numerical dispersion. The relative contributions of stretching and diffusion to mixing-limited reaction is controlled by changing the Péclet number, and the channel roughness is also systematically varied. We observe optimal ranges of fluid stretching that maximize reactivity, which are captured by a Lagrangian stretching measure based on an effective time period that honours the stretching history. We show that the optimality originates from the competition between the enhanced mixing by fluid stretching and the mass depletion of the reactants. We analytically derive the spatial distribution of reaction products using a lamellar formulation and successfully predict the optimal ranges of fluid stretching, which are consistent across different levels of channel roughness. These findings provide a mechanistic understanding of how the interplay between fluid stretching, diffusion and channel roughness controls mixing-limited reactions in rough channel flows, and show how reaction hot spots can be predicted from the concept of optimal fluid stretching.

Goal-oriented adaptivity for a conforming residual minimization method in a dual discontinuous Galerkin norm

We propose a goal-oriented mesh-adaptive algorithm for a finite element method stabilized via residual minimization on dual discontinuous-Galerkin norms. By solving a saddle-point problem, this residual minimization delivers a stable continuous approximation to the solution on each mesh instance and a residual projection onto a broken polynomial space, which is a robust error estimator to minimize the discrete energy norm via automatic mesh refinement. In this work, we propose and analyze a goal-oriented adaptive algorithm for this stable residual minimization. We solve the primal and adjoint problems considering the same saddle-point formulation and different right-hand sides. By solving a third stable problem, we obtain two efficient error estimates to guide goal-oriented adaptivity. We illustrate the performance of this goal-oriented adaptive strategy on advection–diffusion–reaction problems.

Investigation of the advection–diffusion–reaction problem in an inhomogeneous medium with a thin curvilinear channel

The mathematical model of the advection–diffusion–reaction problem in an inhomoge­neous medium with a thin curvilinear channel is considered. The variational formula­tion of the problem in dimensionless variables is obtained. The conditions of existence and uniqueness of the solution of the formulated problem are investigated and found. Cite as: N. V. Mazuriak, “Investigation of the advection–diffusion–reaction problem in an inhomogeneous medium with a thin curvilinear channel,” Prykl. Probl. Mekh. Mat. , Issue 17, 160–170 (2019) (in Ukrainian), https://doi.org/10.15407/apmm2019.17.160-170

Simulation of linear and nonlinear advection–diffusion–reaction problems by a novel localized scheme

This paper presents a new meshless scheme based on the compactly supported radial basis functions for solving linear and nonlinear advection–diffusion–reaction equations. The simulation of proposed problems is divided into the rough approximation obtained from the boundary conditions and corrected approximation using compactly supported radial basis functions where a sparse system matrix will be obtained. The behaviors of the method are verified by several examples which indicate its accuracy and efficiency.

A hybrid smoothed dissipative particle dynamics (SDPD) spatial stochastic simulation algorithm (sSSA) for advection–diffusion–reaction problems

We have developed a new algorithm which merges discrete stochastic simulation, using the spatial stochastic simulation algorithm (sSSA), with the particle based fluid dynamics simulation framework of smoothed dissipative particle dynamics (SDPD). This hybrid algorithm enables discrete stochastic simulation of spatially resolved chemically reacting systems on a mesh-free dynamic domain with a Lagrangian frame of reference. SDPD combines two popular mesoscopic techniques: smoothed particle hydrodynamics and dissipative particle dynamics (DPD), linking the macroscopic and mesoscopic hydrodynamics effects of these two methods. We have implemented discrete stochastic simulation using the reaction–diffusion master equations (RDME) formalism, and deterministic reaction–diffusion equations based on the SDPD method. We validate the new method by comparing our results to four canonical models, and demonstrate the versatility of our method by simulating a flow containing a chemical gradient past a yeast cell in a microfluidics chamber.

An accurate meshless formulation for the simulation of linear and fully nonlinear advection diffusion reaction problems

In this paper, a new meshless semi-analytical collocation technique is presented for the simulation of time-dependent linear and fully nonlinear advection diffusion reaction equations (ADREs). The time-dependent nonlinear problem is reduced to the system of linear problems in each time layer by using the quasilinearization technique in combination with the Crank–Nicolson scheme for the temporal approximation. Then it may be solved by the meshless backward substitution method. In the backward substitution method the approximation is given to satisfy the boundary conditions as a prior. Then the approximation is substituted back to the governing equations where the unknown weighted parameters are obtained using the collocation approach. The developed method has been tested on seven concrete problems for a range of free parameters. The comparison of the obtained numerical results with those available in literatures reveals that the proposed method can accurately approximate linear and nonlinear advection diffusion reaction problems. This method can also be easily extended onto other linear and nonlinear transient problems in 1D, 2D, and 3D due to its general idea and easy implementation.

New Multi-implicit Space–Time Spectral Element Methods for Advection–Diffusion–Reaction Problems

Novel multi-implicit space–time spectral element methods are described for approximating solutions to advection–diffusion–reaction problems characterized by multiple time scales. The new methods are spectrally accurate in space and time and they are designed to be easy to implement and robust. In other words, given an existing stable low order operator split method for approximating solutions to PDEs exhibiting multiple scales, the algorithms described in this article enable one to easily extend a low order method to be a robust space–time spectrally accurate method. In space, two spectrally accurate advective flux reconstructions are proposed: extended element-wise flux reconstruction and non-extended element-wise flux reconstruction. In time, for the hyperbolic term(s), a low-order explicit I-stable building block time integration scheme is introduced in order to obtain a stable and efficient building block for the spectrally accurate space–time scheme. In this article, multiple spectrally accurate space discretization strategies, and multiple spectrally accurate time discretization strategies are compared to one another. It is found that all methods described are spectrally accurate with each method having distinguishing properties.

HiMod Reduction of Advection–Diffusion–Reaction Problems with General Boundary Conditions

We extend the hierarchical model reduction procedure previously introduced in Ern et al. (in: Kunisch, Of, Steinbach (eds) Numerical mathematics and advanced applications, Springer, Berlin, pp 703–710, 2008) and Perotto et al. (Multiscale Model Simul 8(4):1102–1127, 2010) to deal with general boundary conditions, enforcing their prescription in the basis function set. This is achieved by solving a Sturm–Liouville Eigenvalue problem. We analyze this approach and provide a convergence analysis for the associated error in the case of a linear advection–diffusion–reaction problem in rectangles (2D) and slabs (3D). Numerical results confirm the theoretical investigation and the reliability of the proposed approach.

A parameter-free dynamic diffusion method for advection–diffusion–reaction problems

In this paper, we present a two-scale finite element formulation, named Dynamic Diffusion (DD), for advection–diffusion–reaction problems. By decomposing the velocity field in coarse and subgrid scales, the latter is used to determine the smallest amount of artificial diffusion to minimize the coarse-scale kinetic energy. This is done locally and dynamically, by imposing some constraints on the resolved scale solution, yielding a parameter-free consistent method. The subgrid scale space is defined by using bubble functions, whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. Convergence tests on a two-dimensional example are reported, yielding optimal rates. In addition, numerical experiments show that DD method is robust for a wide scope of application problems.

Stabilization of Forced Heat Convection: Applications to Enhanced Geothermal Systems (EGS)

The natural permeability of geothermal reservoirs is low and needs to be enhanced to ensure an efficient use and economic viability. Next to chemical enhancement, the main technique used for that purpose is hydraulic fracturing. Here, hydraulic fracturing is introduced in a thermo-poroelastic framework. The main addition to this framework is a fracturing model, phrased in terms of Terzaghi’s effective stress that governs the evolution of size and aperture of the fractures in all directions of space. At any geometrical point, a fracture-induced anisotropic permeability tensor is calculated: Next to the injection pressure and thermal shrinking, the directional properties of this tensor are strongly influenced by geological stresses. The fully integrated framework is henceforth used in simulating thermal recovery from enhanced geothermal reservoirs. Evidently, the credibility of the numerical simulations cannot be sufficiently trusted with large spurious wiggles in the temperature field and consequently in those of the effective stresses. This paper provides several approaches to stabilize convection of heat due to extreme injection conditions at early stages, sudden increase in permeability due to hydraulic fracturing, and near the production wells at late injection stages. Emphasis is paid to the subgrid scale/gradient subgrid scale method where the transient problem is placed into a stabilized advection–diffusion–reaction problem.

A Lagrangian–Eulerian finite element algorithm for advection–diffusion–reaction problems with phase change

This paper presents a particle-based Lagrangian–Eulerian algorithm for the solution of the unsteady advection–diffusion–reaction heat transfer equation with phase change. The algorithm combines a Lagrangian formulation for the advection + reaction problem with the Eulerian-based heat source method for the diffusion + phase change problem. The coupling between the Lagrangian and Eulerian subproblems is achieved with a phase change detector scheme based on a local latent heat balance and a consistent/conservative interpolation technique between Lagrangian particles and the Eulerian grid. This technique makes use of an auxiliary (finer) Eulerian grid that provides a simple and efficient method of tracking internal heterogeneities (e.g. phase boundaries), allows the use of higher order integration quadratures, and facilitates the implementation of multiscale techniques. The performance of the proposed algorithm is compared against one- and two-dimensional benchmark problems, i.e. pure rigid-body advection, isothermal and non-isothermal phase change, two-phase advective heat transfer and chemical reactions coupled with diffusion and advection. The numerical results confirm that the proposed solution method is accurate, oscillation-free and useful for and applicable to a wide range of fully coupled problems in science and engineering.

A quasi-analytical approach to the advection–diffusion–reaction problem, using operator splitting

Solutions of the combined advection–diffusion–reaction (ADR) transport equation are generally restricted to numerical methods. However, each subprocess, advection, diffusion and reaction can all be solved individually using analytical techniques for a variety of situations. We present a simple numerical technique that uses the split operator method, and the Semi-Lagrangian scheme, to combine these analytical solutions. The result is a coherent, quasi-analytical solution to the combined ADR transport equation.

A hybrid discontinuous Galerkin method for advection–diffusion–reaction problems

A hybrid discontinuous Galerkin (HDG) method for the Poisson problem introduced by Jeon and Park can be viewed as a hybridizable discontinuous Galerkin method using a Baumann–Oden type local solver. In this work, an upwind HDG method with super-penalty is proposed to solve advection–diffusion–reaction problems. A super-penalty formulation facilitates an optimal order convergence in the L2 norm as well as the energy norm. Several numerical examples are presented to show the performance of the method.

A posteriori error analysis of IMEX multi-step time integration methods for advection–diffusion–reaction equations

Implicit–Explicit (IMEX) schemes are an important and widely used class of time integration methods for both parabolic and hyperbolic partial differential equations. We develop accurate a posteriori error estimates for a user-defined quantity of interest for two classes of multi-step IMEX schemes for advection–diffusion–reaction problems. The analysis proceeds by recasting the IMEX schemes into a variational form suitable for a posteriori error analysis employing adjoint problems and computable residuals. The a posteriori estimates quantify distinct contributions from various aspects of the spatial and temporal discretizations, and can be used to evaluate discretization choices. Numerical results are presented that demonstrate the accuracy of the estimates for a representative set of problems.