Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations

Westdickenberg, Michael; Wilkening, Jon

doi:10.1051/m2an/2009043

Cited by 49 publications

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References 14 publications

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“…In turn, the PME has many special features: the finite speed of propagation, the free boundary, a possible waiting time phenomenon [5,18]. Various numerical methods have been studied for the PME, such as finite difference approach [8], tracking algorithm method [3], a local discontinuous Galerkin finite element method [24], Variational Particle Scheme (VPS) [23] and an adaptive moving mesh finite element method [13]. Many theoretical analyses have been derived in the existing literature [1,12,14,16,17,18], etc.Relevant detailed descriptions can be found in a recent paper [5], in which the numerical methods for the PME were constructed by an Energetic Variational Approach (EnVarA) to naturally keep the physical laws, such as the conservation of mass, energy dissipation and force balance.…”

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confidence: 99%

Convergence Analysis of a Numerical Scheme for the Porous Medium Equation by an Energetic Variational Approach

Duan

Liu²,

Wang

et al. 2020

NMTMA

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The porous medium equation (PME) is a typical nonlinear degenerate parabolic equation.We have studied numerical methods for PME by an energetic variational approach in [C. Duan et al, J. Comput. Phys., 385 (2019) 13-32], where the trajectory equation can be obtained and two numerical schemes have been developed based on different dissipative energy laws. It is also proved that the nonlinear scheme, based on f log f as the total energy form of the dissipative law, is uniquely solvable on an admissible convex set and preserves the corresponding discrete dissipation law. Moreover, under certain smoothness assumption, we have also obtained the second order convergence in space and the first order convergence in time for the scheme. In this paper, we provide a rigorous proof of the error estimate by a careful higher order asymptotic expansion and two step error estimates. The latter technique contains a rough estimate to control the highly nonlinear term in a discrete W 1,∞ norm, and a refined estimate is applied to derive the optimal error order.One of the typical nonlinear degenerate parabolic equations is the porous medium equation (PME):and the time t ∈ R + , and m is a constant larger than 1. It has been applied in many physical and biological models, such as an isentropic gas flow through a porous medium, the viscous gravity currents, nonlinear heat transfer and image processing [18], etc.It is well known that the PME is degenerate at points where f = 0. In turn, the PME has many special features: the finite speed of propagation, the free boundary, a possible waiting time phenomenon [5,18]. Various numerical methods have been studied for the PME, such as finite difference approach [8], tracking algorithm method [3], a local discontinuous Galerkin finite element method [24], Variational Particle Scheme (VPS) [23] and an adaptive moving mesh finite element method [13]. Many theoretical analyses have been derived in the existing literature [1,12,14,16,17,18], etc.Relevant detailed descriptions can be found in a recent paper [5], in which the numerical methods for the PME were constructed by an Energetic Variational Approach (EnVarA) to naturally keep the physical laws, such as the conservation of mass, energy dissipation and force balance. Meanwhile, based on different dissipative energy laws, two different numerical schemes have been studied. In more details, based on the total energy form f log f and 1 2f , a fully discrete nonlinear scheme and a linear numerical scheme could be appropriately designed for the trajectory equation, respectively. It has also been proved that the former one is uniquely solvable on an admissible convex set, and both schemes preserve the corresponding discrete dissipation law. Numerical experiments have demonstrated that both schemes have yielded a good approximation for the solution without oscillation and the free boundary. The notable advantage is that the waiting time problem could be naturally treated, which has been a well-known difficult issue for all the existing methods. In ad...

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Convergence Analysis of a Numerical Scheme for the Porous Medium Equation by an Energetic Variational Approach

Duan

Liu²,

Wang

et al. 2020

NMTMA

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“…However, many existing numerical solutions may contain oscillations near the free boundary, such as PCSFE method (Predictor-Correction Algorithm and Standard Finite element method) [37]. In recent years, a local discontinuous Galerkin finite element method by Zhang & Wu [37] and Variational Particle Scheme (VPS) by Westdickenberg & Wilkening [35] have been used to solve the PME. These two methods can effectively eliminate non-physical oscillation in the computed solution near the free boundary, and lead to a high-order convergence rate within the smooth part of the solution support.…”

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confidence: 99%

“…Note that, when f vanishes, the energy in first case is regular while the energy in next two cases is singular. Taking the advantage of the singularity, we can prove that the numerical schemes based on last two energy forms have some good properties which are not possessed by the VPS scheme [35] from the first one, such as conservation of positivity, unique solvability on an admissible convex set, convergence of the corresponding Newton's iteration.…”

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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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“…The following result is largely contained in [13] and [29], and included for completeness. (i) Suppose that ρ 0 ∈ P 2 (T) and ρ 0 is absolutely continuous with ρ 0 dρ 0 /dx ∈ L 2 (ρ 0 ), and that q ∈ D 1 (ρ 0 ).…”

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Statistical mechanics of the periodic Benjamin–Ono equation

Blower

Brett

Doust

2019

Journal of Mathematical Physics

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The periodic Benjamin-Ono equation is an autonomous Hamiltonian system with a Gibbs measure on L 2 (T). The paper shows that the Gibbs measures on bounded balls of L 2 satisfy some logarithmic Sobolev inequalities. The space of n-soliton solutions of the periodic Benjamin-Ono equation, as discovered by Case, is a Hamiltonian system with an invariant Gibbs measure. As n → ∞, these Gibbs measures exhibit a concentration of measure phenomenon. Case introduced soliton solutions that are parameterised by atomic measures in the complex plane. The limiting distributions of these measures gives the density of a compressible gas that satisfies the isentropic Euler equations.Date: 1st February 2019.

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Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations

Cited by 49 publications

References 14 publications

Convergence Analysis of a Numerical Scheme for the Porous Medium Equation by an Energetic Variational Approach

Convergence Analysis of a Numerical Scheme for the Porous Medium Equation by an Energetic Variational Approach

Numerical methods for porous medium equation by an energetic variational approach

Statistical mechanics of the periodic Benjamin–Ono equation

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