A numerical procedure for the porous media equation

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

doi:10.1016/0898-1221(85)90156-7

Cited by 17 publications

(18 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…the classical references [23,37,41]. In one space dimension, a key result in the context of deterministic approximations is the one by Russo [48], which applies to the linear diffusion equation, in which the diffusion operator is replaced by a nearest neighbour interaction term (see also later generalizations to nonlinear diffusion in [35]). A recent result which uses the same approach to nonlinear drift diffusion equations is presented in [36].…”

mentioning

confidence: 99%

Rigorous Derivation of Nonlinear Scalar Conservation Laws from Follow-the-Leader Type Models via Many Particle Limit

Francesco

Rosini

2015

Arch Rational Mech Anal

100

View full text Add to dashboard Cite

Abstract. We prove that the unique entropy solution to a scalar nonlinear conservation law with strictly monotone velocity and nonnegative initial condition can be rigorously obtained as the large particle limit of a microscopic follow-the-leader type model, which is interpreted as the discrete Lagrangian approximation of the nonlinear scalar conservation law. More precisely, we prove that the empirical measure (respectively the discretised density) obtained from the follow-the-leader system converges in the 1-Wasserstein topology (respectively in L 1 loc ) to the unique Kruzkov entropy solution of the conservation law. The initial data are taken in L ∞ , nonnegative, and with compact support, hence we are able to handle densities with vacuum. Our result holds for a reasonably general class of velocity maps (including all the relevant examples in the applications, e.g. in the Lighthill-Whitham-Richards model for traffic flow) with possible degenerate slope near the vacuum state. The proof of the result is based on discrete BV estimates and on a discrete version of the one-sided Oleinik-type condition. In particular, we prove that the regularizing effect L ∞ → BV for nonlinear scalar conservation laws is intrinsic of the discrete model. Keywords: Micro-macro limit and Scalar conservation laws and Follow-the-leader models and Oleinik condition and Entropy solutions and Particle method. 35L65 and 35L45 and 90B20 and 65N75 and 82C22 . AMS Subject classification:

show abstract

mentioning

confidence: 99%

Rigorous Derivation of Nonlinear Scalar Conservation Laws from Follow-the-Leader Type Models via Many Particle Limit

Francesco

Rosini

2015

Arch Rational Mech Anal

100

View full text Add to dashboard Cite

show abstract

“…In principle, the proof of Lemma 9 below provides a criterion on the smallness of τ ą 0 that would guarantee the unique solvability of (26). We shall not make this criterion explicit, since in practice, we observe that the Newton method applied to (26) and initialized with x n´1 ∆ always converges to "the right" solution x n ∆ , even for comparatively large steps τ and in rather degenerate situations; we refer the reader to our numerical results in Section 5.…”

Section: Discretization In Timementioning

confidence: 99%

14. Convergence of a fully discrete variational scheme for a thin-film equation

Osberger¹,

Matthes²

2017

Topological Optimization and Optimal Transport

View full text Add to dashboard Cite

Abstract. This paper is concerned with a rigorous convergence analysis of a fully discrete Lagrangian scheme for the Hele-Shaw flow, which is the fourth order thin-film equation with linear mobility in one space dimension. The discretization is based on the equation's gradient flow structure in the L 2 -Wasserstein metric. Apart from its Lagrangian character -which guarantees positivity and mass conservation -the main feature of our discretization is that it dissipates both the Dirichlet energy and the logarithmic entropy. The interplay between these two dissipations paves the way to proving convergence of the discrete approximations to a weak solution in the discrete-to-continuous limit. Thanks to the time-implicit character of the scheme, no CFL-type condition is needed. Numerical experiments illustrate the practicability of the scheme.

show abstract

“…Proof. The monotonicity (38) follows (by induction on n) from the definition of x n ∆ as minimizer of F ∆ (·; x n−1 ∆ ):…”

Section: A Priori Estimates and Compactnessmentioning

confidence: 99%

A Convergent Lagrangian Discretization for a Nonlinear Fourth-Order Equation

Matthes

Osberger

2015

Found Comput Math

View full text Add to dashboard Cite

Abstract. A fully discrete Lagrangian scheme for numerical solution of the nonlinear fourth order DLSS equation in one space dimension is analyzed. The discretization is based on the equation's gradient flow structure in the L 2 -Wasserstein metric. We prove that the discrete solutions are strictly positive and mass conserving. Further, they dissipate both the Fisher information and the logarithmic entropy. Numerical experiments illustrate the practicability of the scheme.Our main result is a proof of convergence of fully discrete to weak solutions in the limit of vanishing mesh size. Convergence is obtained for arbitrary non-negative initial data with finite entropy, without any CFL type condition. The key ingredient in the proof is a discretized version of the classical entropy dissipation estimate.

show abstract

A numerical procedure for the porous media equation

Cited by 17 publications

References 7 publications

Rigorous Derivation of Nonlinear Scalar Conservation Laws from Follow-the-Leader Type Models via Many Particle Limit

Rigorous Derivation of Nonlinear Scalar Conservation Laws from Follow-the-Leader Type Models via Many Particle Limit

14. Convergence of a fully discrete variational scheme for a thin-film equation

A Convergent Lagrangian Discretization for a Nonlinear Fourth-Order Equation

Contact Info

Product

Resources

About