Implicit-explicit-compact methods for advection diffusion reaction equations

Singh, Sanjeev; Sircar, Sarthok

doi:10.1016/j.compfluid.2020.104709

Cited by 11 publications

(12 citation statements)

References 41 publications

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“…where the weight, r α j = ( j + 1) 1−α − j 1−α . Following an earlier work by the authors which utilized a standard integer-order advection-reaction-diffusion (ADR) equations [28], we retain 'IMEX method philosophy' by explicitly discretizing the advection and the reaction term (i. e., 'K 1 ∇ • u(x,t)' term and ' f (x,t)' term, respectively in equation ( 2)), while the diffusive term (i. e., 'K 2 (x,t)∇ 2 u(x,t)' term in equation ( 2)) is treated semi-implicitly, as follows,…”

Section: Time Integrationmentioning

confidence: 99%

Implicit-explicit time integration method for fractional advection-reaction-diffusion equations

Ghosh¹,

Chauhan²,

Sircar³

2023

Preprint

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We propose a novel family of asymptotically stable, implicit-explicit, adaptive, time integration method (denoted with the θ -method) for the solution of the fractional advection-diffusion-reaction (FADR) equations. This family of time integration method generalized the computationally explicit L 1 -method adopted by Brunner (J. Comput. Phys. 229 6613-6622 (2010)) as well as the fully implicit method proposed by Jannelli (Comm. Nonlin. Sci. Num. Sim., 105, 106073 ( 2022)). The spectral analysis of the method (involving the group velocity and the phase speed) indicates a region of favorable dispersion for a limited range of Peclet number. The numerical inversion of the coefficient matrix is avoided by exploiting the sparse structure of the matrix in the iterative solver for the Poisson equation. The accuracy and the efficacy of the method is benchmarked using (a) the twodimensional (2D) fractional diffusion equation, originally proposed by Brunner, and (b) the incompressible, subdiffusive dynamics of a planar viscoelastic channel flow of the Rouse chain melts (FADR equation with fractional time-derivative of order α = 1 /2) and the Zimm chain solution (α = 2 /3). Numerical simulations of the viscoelastic channel flow effectively capture the non-homogeneous regions of high viscosity at low fluid inertia (or the so-called 'spatiotemporal macrostructures'), experimentally observed in the flow-instability transition of subdiffusive flows.

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Section: Time Integrationmentioning

confidence: 99%

Implicit-explicit time integration method for fractional advection-reaction-diffusion equations

Ghosh¹,

Chauhan²,

Sircar³

2023

Preprint

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“…This allows the spatial mesh to be refined adaptively without the need for reducing the time-step size. In the literature, IMEX methods have continued to be used in various context, for example, they have recently been used for convection-diffusion-reaction 33,[38][39][40] and Navier-Stokes 34,41,42 equations. In Meng et al, 41 a fractional-step algorithm for an incompressible Navier-Stokes equation on moving overlapping meshes has been proposed.…”

Section: Temporal Discretizationmentioning

confidence: 99%

A p‐adaptive, implicit‐explicit mixed finite element method for diffusion‐reaction problems

Wakeni

Aggarwal

Kaczmarczyk

et al. 2022

Numerical Meth Engineering

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A new class of implicit‐explicit (IMEX) methods combined with a p$$ p $$‐adaptive mixed finite element formulation is proposed to simulate the diffusion of reacting species. Hierarchical polynomial functions are used to construct a conforming base for the flux vectors, and a non‐conforming base for the mass concentration of the species. The mixed formulation captures the distinct nonlinearities associated with the flux constitutive equations and the reaction terms. The IMEX method conveniently treats these two sources of nonlinearity implicitly and explicitly, respectively, within a single time‐stepping framework. A reliable a posteriori error estimate is proposed and analyzed. A p$$ p $$‐adaptive algorithm based on the proposed a posteriori error estimate is also constructed. The combination of the proposed residual‐based a posteriori error estimate and hierarchical finite element spaces allows for the formulation of an efficient p$$ p $$‐adaptive algorithm. A series of numerical examples demonstrate the performance of the approach for problems involving travelling waves, and possessing discontinuities and singularities. The flexibility of the formulation is illustrated via selected applications in pattern formation and electrophysiology.

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“…Advection-diffusion-reaction Equation (ADRE) as numerical solution, widely used within mathematical modelling, to describe physical, biochemical and physical-chemical processes in AL [19,[25][26][27].…”

Section: Advection-diffusion Reaction Equationmentioning

confidence: 99%

A Framework Based on Finite Element Method (FEM) for Modelling and Assessing the Affection of the Local Thermal Weather Factors on the Performance of Anaerobic Lagoons for the Natural Treatment of Swine Wastewater

Brito-Espino

Martín

Báez

et al. 2021

Water

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Anaerobic lagoons are natural wastewater treatment systems suitable for swine farms in small communities due to its low operational and building costs, as well as for the environmental sustainability that these technologies enable. The local weather is one of the factors which greatly influences the efficiency of the organic matter degradation within anaerobic lagoons, since microbial growth is closely related to temperature. In this manuscript, we propose a mathematical model which involves the two-dimensional Stokes, advection–diffusion-reaction and heat transfer equations for an unstirred fluid flow. Furthermore, the Anaerobic Digestion Model No1 (ADM1), developed by the International Water Association (IWA), has been implemented in the model. The partial differential equations resulting from the model, which involve a large number of state variables that change according to the position and the time, are solved through the use of the Finite Element Method. The results of the simulations indicated that the methodology is capable of predicting reasonably well the steady-state of the concentrations for all processes that take place in the anaerobic digestion and for each one of the variables considered; cells, organic matter, nutrients, etc. In view of the results, it can be concluded that the model has significant potential for the design and the study of anaerobic cells’ behaviour within free flow systems.

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Implicit-explicit-compact methods for advection diffusion reaction equations

Cited by 11 publications

References 41 publications

Implicit-explicit time integration method for fractional advection-reaction-diffusion equations

Implicit-explicit time integration method for fractional advection-reaction-diffusion equations

A p‐adaptive, implicit‐explicit mixed finite element method for diffusion‐reaction problems

A Framework Based on Finite Element Method (FEM) for Modelling and Assessing the Affection of the Local Thermal Weather Factors on the Performance of Anaerobic Lagoons for the Natural Treatment of Swine Wastewater

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