A Hyperbolic Theory for Advection-Diffusion Problems: Mathematical Foundations and Numerical Modeling

Gómez, Héctor; Colominas, Ignasi; Navarrina, F.; París, José; Casteleiro, Manuel

doi:10.1007/s11831-010-9042-5

Cited by 29 publications

(24 citation statements)

References 72 publications

(88 reference statements)

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“…Consider the hyperbolic advection-diffusion model of [24] for mass transport of a solute species in the presence of a known, divergence-free (incompressible) background velocity field, a.x; t/, in which x 2 E d denotes spatial position and t 2 R denotes time. Let u denote the solute species concentration and q its diffusive flux.…”

Section: Continuum Modelmentioning

confidence: 99%

“…Let u denote the solute species concentration and q its diffusive flux. Then the system of governing equations from [24] can be written as…”

Section: Continuum Modelmentioning

confidence: 99%

“…This paper is concerned with a finite element method for solving hyperbolic advection-diffusion problems, such as (5). A survey of the literature on numerical methods that address hyperbolic diffusion, for both pure diffusion [14,[19][20][21], and combined advection-diffusion [1,22,23], can be found in [24]. For the pure diffusion case, numerical models based on discontinuous Galerkin (DG) finite element methods, [22,25], are most relevant to the method proposed here.…”

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confidence: 99%

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Adaptive spacetime discontinuous Galerkin method for hyperbolic advection-diffusion with a non-negativity constraint

Pal

Abedi

Madhukar

et al. 2015

Int. J. Numer. Meth. Engng

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Applications where the diffusive and advective time scales are of similar order give rise to advectiondiffusion phenomena that are inconsistent with the predictions of parabolic Fickian diffusion models. Non-Fickian diffusion relations can capture these phenomena and remedy the paradox of infinite propagation speeds in Fickian models. In this work, we implement a modified, frame-invariant form of Cattaneo's hyperbolic diffusion relation within a spacetime discontinuous Galerkin advection-diffusion model. An hadaptive spacetime meshing procedure supports an asynchronous, patch-by-patch solution procedure with linear computational complexity in the number of spacetime elements. This localized solver enables the selective application of optimization algorithms in only those patches that require inequality constraints to ensure a non-negative concentration solution. In contrast to some previous methods, we do not modify the numerical fluxes to enforce non-negative concentrations. Thus, the element-wise conservation properties that are intrinsic to discontinuous Galerkin models are defined with respect to physically meaningful Riemann fluxes on the element boundaries. We present numerical examples that demonstrate the effectiveness of the proposed model, and we explore the distinct features of hyperbolic advection-diffusion response in subcritical and supercritical flows. Ã q D Kr ;and show that the resulting hyperbolic system exhibits finite propagation velocities and is Galileaninvariant. We refer to (5) as the modified Maxwell-Cattaneo (mMC) model and use it as our reference hyperbolic model for advection-diffusion from here on. This paper is concerned with a finite element method for solving hyperbolic advection-diffusion problems, such as (5). A survey of the literature on numerical methods that address hyperbolic diffusion, for both pure diffusion [14,[19][20][21], and combined advection-diffusion [1,22,23], can be found in [24]. For the pure diffusion case, numerical models based on discontinuous Galerkin (DG) finite element methods, [22,25], are most relevant to the method proposed here. In particular, the adaptive spacetime DG method described in [26] is a direct antecedent to the method advanced in this work. For the mMC advection-diffusion problem, the DG method in [22] is the closest HYPERBOLIC ADVECTION-DIFFUSION WITH NON-NEGATIVITY CONSTRAINT Unconditional stability for linear systems, such as (5). Falk and Richter prove stability of SDG solutions for linear, symmetric hyperbolic systems in [44], and Lowrie et al. report similar findings in [43] for linear systems of conservation laws, such as the one in this work. Compact computational stencils that do not expand with higher-order approximations; cf. discussion in first paragraph in Section 4. Element-wise conservation with respect to computed Riemann fluxes.

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Section: Continuum Modelmentioning

confidence: 99%

“…Let u denote the solute species concentration and q its diffusive flux. Then the system of governing equations from [24] can be written as…”

Section: Continuum Modelmentioning

confidence: 99%

mentioning

confidence: 99%

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Adaptive spacetime discontinuous Galerkin method for hyperbolic advection-diffusion with a non-negativity constraint

Pal

Abedi

Madhukar

et al. 2015

Int. J. Numer. Meth. Engng

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“…There are experimental evidences that diffusive process takes place with finite velocity [5][6][7]. This issue can be ignored in some applications, but otherwise it is necessary to consider the wave nature of diffusive process [8].…”

Section: Introductionmentioning

confidence: 99%

Numerical simulation on hyperbolic diffusion equations using modified cubic B-spline differential quadrature methods

Mittal

Dahiya

2015

Computers & Mathematics with Applications

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Available online xxxx Keywords:Hyperbolic diffusion problem Cubic B-spline functions Modified cubic B-spline differential quadrature method System of ordinary differential equations Runge-Kutta 4th order method a b s t r a c tIn this article, a modified cubic B-spline differential quadrature method (MCB-DQM) is proposed to solve a hyperbolic diffusion problem in which flow motion is affected by both convection and diffusion. One dimensional hyperbolic non-homogeneous heat, wave and telegraph equations are also considered along with two dimensional hyperbolic diffusion problem. The method reduces the hyperbolic problem into a system of nonlinear ordinary differential equations. The system is then solved by the optimal four stage three order strong stability-preserving time stepping Runge-Kutta (SSP-RK43) scheme. The reliability and efficiency of the method has been tested on seven examples. The stability of the method is also discussed and found to be unconditionally stable.

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“…However, as has been stated many times and supported by new experimental evidence, this model can be insufficient in many cases [1,2]. Numerous approaches for deriving a more general formulation have been attempted such as hyperbolic diffusion theories [3][4][5][6] and Continuous Time Random Walks (CTRW) [7].…”

Section: Introductionmentioning

confidence: 99%

Least-Squares Spectral Method for the solution of a fractional advection–dispersion equation

Carella

Dorao

2013

Journal of Computational Physics

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A Hyperbolic Theory for Advection-Diffusion Problems: Mathematical Foundations and Numerical Modeling

Cited by 29 publications

References 72 publications

Adaptive spacetime discontinuous Galerkin method for hyperbolic advection-diffusion with a non-negativity constraint

Adaptive spacetime discontinuous Galerkin method for hyperbolic advection-diffusion with a non-negativity constraint

Numerical simulation on hyperbolic diffusion equations using modified cubic B-spline differential quadrature methods

Least-Squares Spectral Method for the solution of a fractional advection–dispersion equation

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