Numerical approach to the waiting time for the one-dimensional porous medium equation

Nakaki, Tatsuyuki; Tomoeda, Kenji

doi:10.1090/qam/2019614

Cited by 4 publications

(3 citation statements)

References 15 publications

(10 reference statements)

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“…For the waiting phenomenon, Mimura et al [21], Bertsch & Dal Passo [4] and Tomoeda & Mimura [32] estimated the waiting time by the interface, but the numerical interface actually has a velocity. Nakaki & Tomoeda [22] transformed the PME into another problem whose solution will blow up at a finite time, which is just the waiting time of PME. But the solution cannot be obtained after the waiting time.…”

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confidence: 99%

Numerical methods for porous medium equation by an energetic variational approach

Duan

Wang

et al. 2019

Journal of Computational Physics

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We study numerical methods for porous media equation (PME). There are two important characteristics: the finite speed propagation of the free boundary and the potential waiting time, which make the problem not easy to handle. Based on different dissipative energy laws, we develop two numerical schemes by an energetic variational approach. Firstly, based on f log f as the total energy form of the dissipative law, we obtain the trajectory equation, and then construct a fully discrete scheme. It is proved that the scheme is uniquely solvable on an admissible convex set by taking the advantage of the singularity of the total energy. Next, based on 1 2f as the total energy form of the dissipation law, we construct a linear numerical scheme for the corresponding trajectory equation. Both schemes preserve the corresponding discrete dissipation law. Meanwhile, under some smoothness assumption, it is proved, by a higher order expansion technique, that both schemes are second-order convergent in space and first-order convergent in time. Each scheme yields a good approximation for the solution and the free boundary. No oscillation is observed for the numerical solution around the free boundary. Furthermore, the waiting time problem could be naturally treated, which has been a well-known difficult issue for all the existence methods. Due to its linear nature, the second scheme is more efficient.The porous medium equation (PME) can be found in many physical and biological phenomena, such as the flow of an isentropic gas through a porous medium [18], the viscous gravity currents [12], nonlinear heat transfer and image processing; e.g., see [33]. The aim of this paper is to provide numerical methods for the PMEwhere f := f (x, t) is a non-negative scalar function of space x ∈ R d and the time t ∈ R, the space dimension is given by d ≥ 1, and m is a constant larger than 1.The PME is a nonlinear degenerate parabolic equation since the diffusivity D(f ) = mf m−1 = 0 at points where f = 0. In turn, the PME has a special feature: the finite speed of propagation, called finite propagation [33]. If the initial data has a compact support, the solution of Cauchy problem of the PME will have a compact support at any given time t > 0. In comparison with the heat equation, which can smooth out the initial data, the solution of the PME becomes non-smooth even if the initial data is smooth with compact support. If an initial data is zero in some open domain in Ω, it causes the appearance of the free boundary (in some cases, called interface) that separates the regions where the solution is positive from the regions where the value is zero in the domain. Moreover, for certain initial data, the solution of the PME can exhibit a waiting time phenomenon where the free boundary remains stationary until a finite positive time (called waiting time).After that time instant, the interface begins to move with a finite speed. Many theoretical analyses have been available in the existing literature, including the earlier works by Oleǐnik et al. [25], Ka...

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confidence: 99%

Numerical methods for porous medium equation by an energetic variational approach

Duan

Wang

et al. 2019

Journal of Computational Physics

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“…Most of previous studies estimate the waiting time from the trajectory of numerical interface [55,6,43], but the numerical waiting time may not be clearly estimated in some cases. As an alternative approach, Nakaki and Tomoeda estimate the waiting time for the one-dimensional PME by transforming the original equation into another equation whose solution will blow up at the waiting time of the original PME [41]. Recently, Duan et al proposed an elegant criterion to determine the waiting time in one-dimensional case, and they manually set the velocity of interface to be zero before the numerical waiting time.…”

Section: Waiting Timementioning

confidence: 99%

On Lagrangian schemes for porous medium type generalized diffusion equations: A discrete energetic variational approach

Chun

Wang

2020

Journal of Computational Physics

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“…On Ω = (0, π), the initial condition given by ρ 0 (x) = sin 2/(m−1) (x) if 0 ≤ x ≤ π, 0 otherwise, produces this behavior. It is shown in [40] that the corresponding solution has a waiting time of t * = m−1 2m(m+1) . As we choose m = 2, here t * = 0.08 3.…”

Section: Applications and Numerical Testsmentioning

confidence: 99%

An entropy structure preserving space-time formulation for cross-diffusion systems: Analysis and Galerkin discretization

Braukhoff,

Perugia,

Stocker

2020

Preprint

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Cross-diffusion systems are systems of nonlinear parabolic partial differential equations that are used to describe dynamical processes in several application, including chemical concentrations and cell biology. We present a space-time approach to the proof of existence of bounded weak solutions of cross-diffusion systems, making use of the system entropy to examine long-term behavior and to show that the solution is nonnegative, even when a maximum principle is not available. This approach naturally gives rise to a novel space-time Galerkin method for the numerical approximation of cross-diffusion systems that conserves their entropy structure. We prove existence and convergence of the discrete solutions, and present numerical results for the porous medium, the Fisher-KPP, and the Maxwell-Stefan problem.

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Numerical approach to the waiting time for the one-dimensional porous medium equation

Cited by 4 publications

References 15 publications

Numerical methods for porous medium equation by an energetic variational approach

Numerical methods for porous medium equation by an energetic variational approach

On Lagrangian schemes for porous medium type generalized diffusion equations: A discrete energetic variational approach

An entropy structure preserving space-time formulation for cross-diffusion systems: Analysis and Galerkin discretization

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