An adaptive stabilized method for advection–diffusion–reaction equation

Araya, Rodolfo; Aguayo, Jorge; Muñoz, Santiago Palmero

doi:10.1016/j.cam.2020.112858

Cited by 7 publications

(5 citation statements)

References 15 publications

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“… 2019 ; Araya et al. 2020 ). The two-dimensional advection–diffusion reaction equation with Dirichlet-type boundary conditions can be written as follows: On the other hand, the TFADRE is a variant of the classical advection–diffusion reaction equation in which the integer-order derivative is replaced by the Caputo fractional derivative.…”

Section: The Mathematical Modelmentioning

confidence: 99%

Efficient numerical simulations based on an explicit group approach for the time fractional advection–diffusion reaction equation

et al. 2023

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The time-fractional advection–diffusion reaction equation (TFADRE) is a fundamental mathematical model because of its key role in describing various processes such as oil reservoir simulations, COVID-19 transmission, mass and energy transport, and global weather production. One of the prominent issues with time fractional differential equations is the design of efficient and stable computational schemes for fast and accurate numerical simulations. We construct in this paper, a simple and yet efficient modified fractional explicit group method (MFEGM) for solving the two-dimensional TFADRE with suitable initial and boundary conditions. The proposed method is established using a difference scheme based on L1 discretization in temporal direction and central difference approximations with double spacing in spatial direction. For comparison purposes, the Crank–Nicolson finite difference method (CNFDM) is proposed. The stability and convergence of the presented methods are theoretically proved and numerically affirmed. We illustrate the computational efficiency of the MFEGM by comparing it to the CNFDM for four numerical examples including fractional diffusion and fractional advection–diffusion models. The numerical results show that the MFEGM is capable of reducing iteration count and CPU timing effectively compared to the CNFDM, making it well-suited to time fractional diffusion equations.

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Section: The Mathematical Modelmentioning

confidence: 99%

Efficient numerical simulations based on an explicit group approach for the time fractional advection–diffusion reaction equation

et al. 2023

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“…The class of adaptive numerical methods is a practical technique often in conjunction with classical numerical schemes, such as finite elements and finite volumes, to reduce the overall computational cost while obtaining a highly accurate numerical solution. Generally speaking, three classes of adaptive approaches are available in the literature, including local mesh refinement (

h

‐adaptive), 1‐5 moving mesh approach (

r

‐adaptive), 6,7 and a numerical method by locally increasing the degree polynomial of basis functions on selected elements (

p

‐adaptive) 8 . For example, an

h

‐adaptive finite element (FEM) method consists of a loop of three key phases: solution, error estimation, and local mesh refinement.…”

Section: Introductionmentioning

confidence: 99%

Locally adaptive bubble function enrichment for multiscale finite element methods: Application to convection‐diffusion problems

Hwang

2023

Numerical Methods in Fluids

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SummaryWe develop a new class of the multiscale finite element method (MsFEM) to solve the convection‐diffusion problems. In the proposed framework, we decompose the solution function space into two parts in MsFEM with locally adaptive bubble function enrichment (LABFE). The first part is the one that the multiscale basis functions can resolve, and the second part is an unresolved part that is taken care of by a set of bubble functions. These bubble functions are defined similarly to construct multiscale basis functions. We exchange the local‐global information through updated local boundary conditions for these bubble functions. The new multiscale solution recovered from the solution of global numerical formulation provides feedback for updating the local boundary conditions on each coarse element. As the approach iterates, the quality of MsFEM‐LABFE solutions improves since these multiscale basis functions with bubble function enrichment are expected to capture the multiscale feature of the approximate solution more accurately. However, the most expansive part of the algorithm is reconstructing the bubble functions on each coarse element. To reduce the overhead of the bubble function reconstruction, we update the local bubble function only when the intermediate multiscale solution is not well resolved within the region corresponding to the sharp local gradient or the discontinuity of the solution. We illustrate the effectiveness of the proposed method through some numerical examples for convection‐diffusion benchmark problems.

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“…This second approach is the most popular strategy of evaluating error behavior and it is adopted in the present work. In contrast to the gradient-based h-adaptive finite element methods as those investigated in [23,24,25], linear systems in the proposed enriched Galerkin-characteristics finite element method keep the same structure and size at each adaptation step. Indeed, for the gradient-based h-adaptive methods, an initial coarse mesh is needed to compute a primary solution for evaluating the gradient.…”

Section: Introductionmentioning

confidence: 99%

An enriched Galerkin-characteristics finite element method for convection-dominated and transport problems

Ouardghi

El‐Amrani

Seaı̈d

2021

Applied Numerical Mathematics

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An adaptive stabilized method for advection–diffusion–reaction equation

Cited by 7 publications

References 15 publications

Efficient numerical simulations based on an explicit group approach for the time fractional advection–diffusion reaction equation

Efficient numerical simulations based on an explicit group approach for the time fractional advection–diffusion reaction equation

Locally adaptive bubble function enrichment for multiscale finite element methods: Application to convection‐diffusion problems

An enriched Galerkin-characteristics finite element method for convection-dominated and transport problems

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