The classification of 2 × 2 systems of non‐strictly hyperbolic conservation laws, with application to oil recovery

Schaeffer, David G.; Shearer, Michael

doi:10.1002/cpa.3160400202

Cited by 152 publications

(116 citation statements)

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“…Theories of hyperbolic conservation laws treat the boundary condition on one side as a fixed parameter, the boundary condition on the other side as unknown, and the wave velocity c is derived from the Rankine-Hugoniot condition (Duijn et al 2007(Duijn et al , 2013LeVeque 2004;Lax 2006;LeFloch 2002;Cuesta et al 2000;Schaeffer and Shearer 1987). In contrast to this, the present paper treats both boundary resp.…”

Section: Objectives and Definition Of The Modelmentioning

confidence: 99%

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Non-monotonic Travelling Wave Fronts in a System of Fractional Flow Equations from Porous Media

et al. 2016

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Motivated by observations of saturation overshoot, this article investigates generic classes of smooth travelling wave solutions of a system of two coupled nonlinear parabolic partial differential equations resulting from a flux function of high symmetry. All boundary resp. limit value problems of the travelling wave ansatz, which lead to smooth travelling wave solutions, are systematically explored. A complete, visually and computationally useful representation of the five-dimensional manifold connecting wave velocities and boundary resp. limit data is found by using methods from dynamical systems theory. The travelling waves exhibit monotonic, non-monotonic or plateau-shaped behaviour. Special attention is given to the non-monotonic profiles. The stability of the travelling waves is studied by numerically solving the full system of the partial differential equations with an efficient and accurate adaptive moving grid solver.

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Section: Objectives and Definition Of The Modelmentioning

confidence: 99%

“…3 in a clear and concise way. Furthermore this new approach is compared to the traditional method used, for example, in Duijn et al (2007Duijn et al ( , 2013, LeVeque (2004), LeFloch (2002), Cuesta et al (2000) and Schaeffer and Shearer (1987).…”

Section: Reduced Representation Of the Solution Spacementioning

confidence: 99%

Non-monotonic Travelling Wave Fronts in a System of Fractional Flow Equations from Porous Media

et al. 2016

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“…Furthermore, the discussion in Section 2 motivates our assumption that Q be symmetric and strictly positive definite when the quadratic model is put in the normal form of Schaeffer-Shearer [18]. To characterize the set of transitional waves for such a model, we first construct its fundamental wave manifold [8], which parameterizes all shock wave solutions, and we determine various important subsets (see Section 3).…”

Section: Jane Hurley Wenstrom and Bradley J Plohr 253mentioning

confidence: 99%

“…Schaeffer and Shearer [18] showed that if the origin is an isolated umbilic point, then such a system can be transformed, by means of invertible linear transformations of state space and the (x, t)-plane, into a system of the form…”

Section: Schaeffer-shearer Normal Formmentioning

confidence: 99%

“…The present and previous studies [18,19,22,10,11,12,21,7] have been motivated by the observation that a general system of two conservation laws can be approximated by such a quadratic system in the neighborhood of an isolated umbilic point. We seek scale-invariant weak solutions of Riemann problems comprising (continuous) centered rarefaction waves and (discontinuous) centered shock waves (see, e.g., Ref.…”

Section: Introductionmentioning

confidence: 99%

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Classification of homogeneous quadratic conservation laws with viscous terms

Wenstrom

Plohr

2007

Mat. apl. comput.

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Abstract.In this paper, we study systems of two conservation laws with homogeneous quadratic flux functions. We use the viscous profile criterion for shock admissibility. This criterion leads to the occurrence of non-classical transitional shock waves, which are sensitively dependent on the form of the viscosity matrix. The goal of this paper is to lay a foundation for investigating how the structure of solutions of the Riemann problem is affected by the choice of viscosity matrix.Working in the framework of the fundamental wave manifold, we derive a necessary and sufficient condition on the model parameters for the presence of transitional shock waves. Using this condition, we are able to identify the regions in the wave manifold that correspond to transitional shock waves. Also, we determine the boundaries in the space of model parameters that separate models with differing numbers of transitional regions.Mathematical subject classification: 35L65, 35L67.

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The gap lemma and geometric criteria for instability of viscous shock profiles

Gardner

Zumbrun

1998

Comm. Pure Appl. Math.

281

343

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An obstacle in the use of Evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis, since the Evans function has in general been constructed only away from the essential spectrum. A notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles. Here we prove a general theorem, the “gap lemma,” concerning the analytic continuation of the Evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave. This allows geometric stability theory to be applied in many cases where it could not be applied previously. We demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves. A necessary geometric condition for stability is determined in terms of the sign of a certain Melnikov integral of the associated viscous profile. This sign can easily be evaluated numerically. We also compute it analytically for solutions of several important classes of systems. In particular, we show for a wide class of systems that homoclinic (solitary) waves are linearly unstable, confirming these as the first known examples of unstable viscous shock waves. We also show that (strong) heteroclinic undercompressive waves are sometimes unstable. Similar stability conditions are also derived for Lax and overcompressive shocks and for n × n conservation laws, n ≥ 2. © 1998 John Wiley & Sons, Inc.

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The classification of 2 × 2 systems of non‐strictly hyperbolic conservation laws, with application to oil recovery

Cited by 152 publications

References 9 publications

Non-monotonic Travelling Wave Fronts in a System of Fractional Flow Equations from Porous Media

Non-monotonic Travelling Wave Fronts in a System of Fractional Flow Equations from Porous Media

Classification of homogeneous quadratic conservation laws with viscous terms

The gap lemma and geometric criteria for instability of viscous shock profiles

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