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Diffusion-controlled bimolecular reactions arise in various branches of natural sciences [1], chemically reacting systems [2], and contaminant transport [3]. Based on the kinetics of the chemical reaction, they are broadly classified in to three categories: fast, moderate, and slow [4]. Herein, we are interested in the dynamics of a (fast/moderate/slow) bimolecular reaction given by the following stoichiometric relationship [5]: nA A + nB B −→ nC C In this discussion, we shall present a computational framework to simulate the dynamics of an irreversible bimolecular advection–diffusion–reaction (ADR) system within the context of low-order mixed finite element method. Most importantly, we shall study the effect of advection process on the formation of product C and its role in understanding fast/moderate/slow chemical reaction kinetics. In continuous setting, it is well-known in the literature on partial differential equations [6–8] that the solution to the governing equations ADR system possesses various important physical and mathematical properties such as non-negativity, maximum principles, comparison principles, monotonicity, and monotone property. Herein, we shall discuss whether such properties are inherited or lost during the procedure of finite element discretization. Furthermore, through various representative examples, we shall quantify the errors incurred in satisfying the local and global species balance property. Finally, we shall provide a general methodology to satisfy all these discrete properties (including local and global species balance property) simultaneously within the context of low-order finite elements. REFERENCES [1] Farkas, M. Dynamical Models in Biology. Academic Press: San Diego, USA, 2001. [2] Epstein, I.R., Pojman, J.A. An Introduction to Nonlinear Chemical Dynamics. Oxford University Press: New York, USA, 1998. [3] Zheng, C., Bennett, G.D. Applied Contaminant Transport Modeling, 2nd edition. John Wiley & Sons, Inc.: New York, USA; 2002. [4] Neufeld, Z., García, E.H. Chemical and Biological Processes in Fluid Flows: A Dynamical Systems Approach. Imperial College Press: London, UK; 2010. [5] Nakshatrala, K.B., Mudunuru, M.K.,Valocchi, A.J. A numerical framework for diffusion-controlled bimolecular-reactive systems to enforce maximum principles and non-negative constraint. J. Comp. Phys. 2013, 253, 278–307. [6] Pao, C.V. Nonlinear Parabolic and Elliptic Equations. Springer-Verlag: New York, USA; 1993. [7] Evans, L.C. Partial Differential Equations. American Mathematical Society: Providence, Rhode Island, USA; 1998. [8] Leung, A.W. Nonlinear Systems of Partial Differential Equations: Applications to Life and Physical Sciences. World -Scientific Publishing Ltd.: Massachusetts, USA; 2009.

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Impact of advection process in simulating the dynamics of diffusion-controlled irreversible fast/moderate/slow bimolecular-reactive systems

Diffusion-controlled bimolecular reactions arise in various branches of natural sciences [1], chemically reacting systems [2], and contaminant transport [3]. Based on the kinetics of the chemical reaction, they are broadly classified in to three categories: fast, moderate, and slow [4]. Herein, we are interested in the dynamics of a (fast/moderate/slow) bimolecular reaction given by the following stoichiometric relationship [5]: nA A + nB B −→ nC C In this discussion, we shall present a computational framework to simulate the dynamics of an irreversible bimolecular advection–diffusion–reaction (ADR) system within the context of low-order mixed finite element method. Most importantly, we shall study the effect of advection process on the formation of product C and its role in understanding fast/moderate/slow chemical reaction kinetics. In continuous setting, it is well-known in the literature on partial differential equations [6–8] that the solution to the governing equations ADR system possesses various important physical and mathematical properties such as non-negativity, maximum principles, comparison principles, monotonicity, and monotone property. Herein, we shall discuss whether such properties are inherited or lost during the procedure of finite element discretization. Furthermore, through various representative examples, we shall quantify the errors incurred in satisfying the local and global species balance property. Finally, we shall provide a general methodology to satisfy all these discrete properties (including local and global species balance property) simultaneously within the context of low-order finite elements. REFERENCES [1] Farkas, M. Dynamical Models in Biology. Academic Press: San Diego, USA, 2001. [2] Epstein, I.R., Pojman, J.A. An Introduction to Nonlinear Chemical Dynamics. Oxford University Press: New York, USA, 1998. [3] Zheng, C., Bennett, G.D. Applied Contaminant Transport Modeling, 2nd edition. John Wiley & Sons, Inc.: New York, USA; 2002. [4] Neufeld, Z., García, E.H. Chemical and Biological Processes in Fluid Flows: A Dynamical Systems Approach. Imperial College Press: London, UK; 2010. [5] Nakshatrala, K.B., Mudunuru, M.K.,Valocchi, A.J. A numerical framework for diffusion-controlled bimolecular-reactive systems to enforce maximum principles and non-negative constraint. J. Comp. Phys. 2013, 253, 278–307. [6] Pao, C.V. Nonlinear Parabolic and Elliptic Equations. Springer-Verlag: New York, USA; 1993. [7] Evans, L.C. Partial Differential Equations. American Mathematical Society: Providence, Rhode Island, USA; 1998. [8] Leung, A.W. Nonlinear Systems of Partial Differential Equations: Applications to Life and Physical Sciences. World -Scientific Publishing Ltd.: Massachusetts, USA; 2009.