Some new error estimates of a semidiscrete finite volume element method for a parabolic integro-differential equation with nonsmooth initial data

Sinha, Rajen Kumar; Ewing, Richard E.; Lazarov, Raytcho

doi:10.1137/040612099

Cited by 33 publications

(33 citation statements)

References 33 publications

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“…This procedure is efficient for Equation (35), because all the matrices involve are symmetric positive definite and well-conditioned. However, the matrix involved in the flow system (26) and (27) is strongly indefinite [40], and it is well known that iterative methods for indefinite systems are not so efficient as those for problems with positive definite matrices. Many different approaches have been proposed to address this issue, and without being exhaustive, we refer to [46][47][48][49][50].…”

Section: The Numerical Schemementioning

confidence: 99%

“…The mathematical and numerical analyses of initial boundary value problems (IBVPs) based on integro-differential equations of this type were studied, for example, in [20][21][22][23][24][25] and [26]. The ability of the integro-differential model (IDM) (9) to capture the dynamics of tracer transport has already been tested by the authors in [27] by fitting the model to experimental breakthrough curves (BTCs).…”

Section: Introductionmentioning

confidence: 99%

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An integro‐differential model for non‐Fickian tracer transport in porous media: validation and numerical simulation

Ferreira

Pinto

2015

Math Methods in App Sciences

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Communicated by J. Vigo-AguiarDiffusion processes have traditionally been modeled using the classical parabolic advection-diffusion equation. However, as in the case of tracer transport in porous media, significant discrepancies between experimental results and numerical simulations have been reported in the literature. Therefore, in order to describe such anomalous behavior, known as non-Fickian diffusion, some authors have replaced the parabolic model with the continuous time random walk model, which has been very effective. Integro-differential models (IDMs) have been also proposed to describe non-Fickian diffusion in porous media. In this paper, we introduce and test a particular type of IDM by fitting breakthrough curves resulting from laboratory tracer transport. Comparisons with the traditional advection-diffusion equation and the continuous time random walk are also presented. Moreover, we propose and numerically analyze a stable and accurate numerical procedure for the two-dimensional IDM composed by a integro-differential equation for the concentration and Darcy's law for flow. In space, it is based on the combination of mixed finite element and finite volume methods over an unstructured triangular mesh.where the total mass flux J is given by where J adv is the flux due to convective transport J adv D vu, and by assuming that the diffusion-dispersion mass flux J dis satisfies the so-called Fick's law,Equation (1), being of the parabolic type, induces a pathologic behavior in the concentration u, namely, it presents infinite speed of propagation, which has no physical meaning. Other limitations associated with the definition of D.v/ have also been reported [6][7][8].Regarding the use of such an equation to simulate tracer transport, several gaps have been observed when simulation results were compared with laboratory experiments. These observations have been extensively reviewed [9][10][11][12][13][14][15][16][17][18].Several approaches have been developed in order to overcome the limitations associated with Equation (1); we refer, for example, to [7] and [8]. A common approach consists of the introduction of a hyperbolic or non-Fickian correction term. A particular model of this type can be obtained by assuming that the diffusion-dispersion mass flux J dis satisfies the following differential equation:where is a delay parameter [19]. Note that the left-hand side of Equation (5) is a first-order approximation of the left-hand side of J dis .x, t C / D D.v/ru.x, t/, which means that the dispersion mass flux at the point x and time t C depends on the gradient of the concentration at the same point but at a delayed time. Equation (5) leads towith the kernel term K er .t/ D 1 e t , provided that J dis .0/ D 0. Combining the partition (4) with Equation (3), where J dis is given by Equation (6), we obtain the integro-differential equationMoreover, if we assume that the diffusion-dispersion mass flux has both a Fickian component J F , and a non-Fickian component J NF , defined byandThe intent of this section is...

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Section: The Numerical Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An integro‐differential model for non‐Fickian tracer transport in porous media: validation and numerical simulation

Ferreira

Pinto

2015

Math Methods in App Sciences

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“…Generally speaking, FVEM is a numerical technique that lies somewhere between finite difference and finite element methods, it has a flexibility similar to that of finite element method for handing complicated solution domain geometries and boundary conditions [5,7,15] and has a simplicity for implementation comparable to finite difference method with triangulations of a simple structure [17,30]. However, the theoretical analysis of FVEM lags far behind that of finite element and finite difference methods, readers can refer to [11][12][13]16,24,28,31,34] for some recent developments.…”

Section: Introductionmentioning

confidence: 97%

The semidiscrete finite volume element method for nonlinear convection–diffusion problem

Zhang

2011

Applied Mathematics and Computation

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“…Initial boundary value problems based on (9) were largely studied from a mathematical point of view. Without being exhaustive, we mention [6][7][8][9][10][11][12][13][14][15]. An equation of the type of (9) was established in [16] using different arguments and with…”

Section: Introductionmentioning

confidence: 99%

“…To model the evolution of a concentration in a one-dimensional porous medium, a hyperbolic equation can be obtained from (12) and the initial mass conservation law,…”

Section: Introductionmentioning

confidence: 99%

Nonfickian effect in time and space for diffusion processes

Ferreira

Pena

2015

Numer. Methods Partial Differential Eq.

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Diffusion processes are usually simulated using the classical diffusion equation. In certain scenarios, such equation induces anomalous behavior and consequently several improvements were introduced in the literature to overcome them. One of the most popular was the replacement of the diffusion equation by an integro-differential equation. Such equation can be established considering a modification of Fick's mass flux where a delay in time is introduced. In this article, we consider mathematical models for diffusion processes that take into account a memory effect in time and space.

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Some new error estimates of a semidiscrete finite volume element method for a parabolic integro-differential equation with nonsmooth initial data

Cited by 33 publications

References 33 publications

An integro‐differential model for non‐Fickian tracer transport in porous media: validation and numerical simulation

An integro‐differential model for non‐Fickian tracer transport in porous media: validation and numerical simulation

The semidiscrete finite volume element method for nonlinear convection–diffusion problem

Nonfickian effect in time and space for diffusion processes

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