The Solution of the Riemann Problem for a Hyperbolic System of Conservation Laws Modeling Polymer Flooding

Johansen, T. H.; Winther, Ragnar

doi:10.1137/0519039

Cited by 88 publications

(58 citation statements)

References 7 publications

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“…Reference [9]). Figure 3 shows both components of the computed solution (dotted lines) at time t = 1 of the viscous case with = 0.0025, obtained using the FEOS method with two splitting steps only.…”

Section: Examplementioning

confidence: 96%

Fast explicit operator splitting method for convection–diffusion equations

Chertock

Kurganov

Petrova

2006

Numerical Methods in Fluids

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SUMMARYSystems of convection-diffusion equations model a variety of physical phenomena which often occur in real life. Computing the solutions of these systems, especially in the convection dominated case, is an important and challenging problem that requires development of fast, reliable and accurate numerical methods. In this paper, we propose a second-order fast explicit operator splitting (FEOS) method based on the Strang splitting. The main idea of the method is to solve the parabolic problem via a discretization of the formula for the exact solution of the heat equation, which is realized using a conservative and accurate quadrature formula. The hyperbolic problem is solved by a second-order finite-volume Godunov-type scheme.We provide a theoretical estimate for the convergence rate in the case of one-dimensional systems of linear convection-diffusion equations with smooth initial data. Numerical convergence studies are performed for one-dimensional nonlinear problems as well as for linear convection-diffusion equations with both smooth and nonsmooth initial data. We finally apply the FEOS method to the one-and two-dimensional systems of convection-diffusion equations which model the polymer flooding process in enhanced oil recovery. Our results show that the FEOS method is capable to achieve a remarkable resolution and accuracy in a very efficient manner, that is, when only few splitting steps are performed.

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Section: Examplementioning

confidence: 96%

Fast explicit operator splitting method for convection–diffusion equations

Chertock

Kurganov

Petrova

2006

Numerical Methods in Fluids

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“…Setting β to zero, System (1) reduces to the model studied by [3], [5] and [8]. Settingβ < 0, System (1) reduces to the model studied by [4].…”

Section: Transport Model With Adsorptionmentioning

confidence: 99%

“…However, the solution for a very similar Riemann problem in the context of polymer flooding was found in [4]. In [11] and here, structural stability is given in the sense of [1], i.e., the wave groups that comprise the Riemann solutions persist if both L and R states are allowed to vary in open sets.…”

Section: Introductionmentioning

confidence: 96%

Loss of hyperbolicity changes the number of wave groups in Riemann problems

Matos

Silva

Marchesin

2016

Bull Braz Math Soc, New Series

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Abstract.The main goal of our work is to show that there exists a class of 2×2 Riemann problems for which the solution comprises a single wave group for an open set of initial conditions. This wave group comprises a 1-rarefaction joined to a 2-rarefaction, not by an intermediate state, but by a doubly characteristic shock, 1-left and 2-right characteristic. In order to ensure that perturbations of initial conditions do not destroy the adjacency of the waves, local transversality between a composite curve foliation and a rarefaction curve foliation is necessary.

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“…Since the case when f is monotone was already studied in [16,15], we concentrate on the nonmonotone case which is more complicated and corresponds to taking into account gravity. Here we assume that ϕ = 0 for the nonlinearities of the system (4).…”

Section: Inriamentioning

confidence: 99%

“…This system, or similar systems of equations, is nonstrictly hyperbolic and is studied in several papers [31,16,15,13]. For example in [16] the authors solve Riemann problems associated to this system when gravity is neglected and therefore the fractional flow function is an increasing function of the unknown. In this case, the eigenvalues of the corresponding Jacobian matrix are positive and hence it is less difficult to construct Godunov type schemes which turn out to be upwind schemes.…”

Section: Introductionmentioning

confidence: 99%

The DFLU flux for systems of conservation laws

Adimurthi

Gowda

Jaffré

2013

Journal of Computational and Applied Mathematics

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Abstract:The DFLU numerical flux was introduced in order to solve hyperbolic scalar conservation laws with a flux function discontinuous in space. We show how this flux can be used to solve certain class of systems of conservation laws such as systems modeling polymer flooding in oil reservoir engineering. Furthermore, these results are extended to the case where the flux function is discontinuous in the space variable. Such a situation arises for example while dealing with oil reservoirs which are heterogeneous. Numerical experiments are presented to illustrate the efficiency of this new scheme compared to other standard schemes like upstream mobility, Lax-Friedrichs and Force schemes. Key-words:Finite volumes, finite differences, Riemann solvers, system of conservation laws, flow in porous media, polymer flooding. Le flux DFLU pour les systèmes de lois de conservationRésumé : Le flux numérique DFLU a été introduit pour résoudre des lois de conservation scalaires hyperboliques dont la fonction de flux est discontinues en espace. Nous montrons comment ce flux peut être utilisé pour résoudre une certaine classe de systèmes de lois de conservation tels que les systèmes modélisant l'injection de polymères en ingéniérie de réservoirs pétroliers. En outre, ces résultats s'étendent au cas de fonctions de flux discontinus par rapport à la variable d'espace. Une telle situation apparaît par exemple quand on considère des réservoirs pétroliers qui sont hétérogènes. Des expériences numériques sont présentées pour illustrer l'efficacité de ce nouveau schéma comparé à d'autres shémas standard tels que les schémas Mobilités Amont, Lax-Friedrichs et Force.

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The Solution of the Riemann Problem for a Hyperbolic System of Conservation Laws Modeling Polymer Flooding

Cited by 88 publications

References 7 publications

Fast explicit operator splitting method for convection–diffusion equations

Fast explicit operator splitting method for convection–diffusion equations

Loss of hyperbolicity changes the number of wave groups in Riemann problems

The DFLU flux for systems of conservation laws

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