A coordinate transformation for the porous media equation that renders the free boundary stationary

Gurtin, Morton E.; MacCamy, R. C.; Socolovsky, E. A.

doi:10.1090/qam/757173

Cited by 33 publications

(16 citation statements)

References 13 publications

(11 reference statements)

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“…Such an approach has been proposed by Gurtin et al [29] and Berryman [30] who employed a Lagrangian frame of reference to integrate the equations. The approach here differs in that we utilise a stretched Eulerian grid.…”

Section: Numerical Solutionsmentioning

confidence: 99%

Self-similar gravity currents in porous media: Linear stability of the Barenblatt–Pattle solution revisited

Mathunjwa

Hogg

2006

European Journal of Mechanics - B/Fluids

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We study the linear stability properties of the Barenblatt-Pattle (B-P) self-similar solutions of the porous medium equation which models flow including viscous and porous media gravity currents. Grundy and McLaughlin [R.E. Grundy, R. McLaughlin, Eigenvalues of the Barenblatt-Pattle similarity solution in nonlinear diffusion, Proc. Roy. Soc. London Ser. A 383 (1982) 89-100] have shown that, in both planar and axisymmetric geometries, the B-P solutions are linearly stable to symmetric perturbations. Using a new technique that eliminates singularities in the linear stability analysis, we extend their result and establish that the axisymmetric B-P solution is linearly stable to asymmetric perturbations. This suggests that the axisymmetric B-P solution provides the intermediate asymptotics of gravity currents that evolve from a wide range of initial distributions including those that are not axisymmetric. We use the connection between the perturbation eigenfunctions and the symmetry transformations of the B-P solution to demonstrate that the leading order rate of decay of the perturbations can be maximised by redefining the volume, time and space variables. We show that, in general, radially symmetric perturbations decay faster than asymmetric perturbations of equal amplitude. These theoretical predictions are confirmed by numerical results.  2005 Elsevier SAS. All rights reserved.

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Section: Numerical Solutionsmentioning

confidence: 99%

Self-similar gravity currents in porous media: Linear stability of the Barenblatt–Pattle solution revisited

Mathunjwa

Hogg

2006

European Journal of Mechanics - B/Fluids

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“…The idea of using Lagrangian coordinates in the study of the free-boundary problems for evolution equations was independently proposed in [3,15,20]. The review of results obtained with this method in the study of evolution equation in divergence form can be found in the monograph [21] or in paper [8].…”

Section: The Change Of Independent Variables: a Mechanical Approachmentioning

confidence: 99%

Lagrangian Approach to the Study of Level Sets: Application to a Free Boundary Problem in Climatology

Díaz

Shmarev

2008

Arch Rational Mech Anal

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We study the dynamics and regularity of level sets in solutions of the semilinear parabolic equationwhere Ω ⊂ R n is a ring-shaped domain, a and µ are given positive constants, H(·) is the Heaviside maximal monotone graph:Such equations arise in climatology (the so-called Budyko energy balance model), as well as in other contexts such as combustion. We show that under certain conditions on the initial data the level sets Γ µ = {(x, t) : u(x, t) = µ} are n-dimensional hypersurfaces in the (x, t)-space and show that the dynamics of Γ µ is governed by a differential equation which generalizes the classical Darcy law in filtration theory. This differential equation expresses the velocity of advancement of the level surface Γ µ through spatial derivatives of the solution u. Our approach is based on the introduction of a local set of Lagrangian coordinates: the equation is formally considered as the mass balance law in the motion of a fluid and the passage to Lagrangian coordinates allows us to watch the trajectory of each of the fluid particles.

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“…However, they do not explicitly determine the waiting time for 9 £ (4,1], In numerical points of view, there are some numerical methods to estimate the waiting time t*. Tomoeda and Mimura [18], Mimura, Nakaki, and Tomoeda [15], Gurtin, MacCamy, and Socolovsky [9], and Bertsch and Dal Passo [6] introduce interface tracking algorithms to (1.1)-(1.2), and show the numerical simulations in the case of (1.6) with 9 = 0, 9 = i, and 9=1.…”

Section: Introductionmentioning

confidence: 99%

Untitled

2003

Quart. Appl. Math.

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A coordinate transformation for the porous media equation that renders the free boundary stationary

Cited by 33 publications

References 13 publications

Self-similar gravity currents in porous media: Linear stability of the Barenblatt–Pattle solution revisited

Self-similar gravity currents in porous media: Linear stability of the Barenblatt–Pattle solution revisited

Lagrangian Approach to the Study of Level Sets: Application to a Free Boundary Problem in Climatology

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