A linear finite volume method for nonlinear cross-diffusion systems

Murakawa, Hideki

doi:10.1007/s00211-016-0832-z

Cited by 21 publications

(19 citation statements)

References 21 publications

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“…A class of doubly nonlinear degenerate parabolic equations generalizing on bounded domains was studied by using maximal monotone operators in Damlamian, Kenmochi, Kubo and Lu, and so on; see also Droniou, Eymard, and Talbot. Another approach to nonlinear diffusion equations via cross‐diffusion systems was recently built by Murakawa,() whose approach is versatile and easy to implement. In comparison with the Cahn‐Hilliard approximation as in Colli and Fukao and Fukao, the methods by previous studies() require the growth condition that

\int_{0}^{r} β false(s false) d s \geq c false| r |^{2} - d

for all

r \in double-struckR

with some constants c , d >0.…”

Section: Introduction and Resultsmentioning

confidence: 99%

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Nonlinear diffusion equations as asymptotic limits of Cahn‐Hilliard systems on unbounded domains via Cauchy's criterion

Fukao

Kurima

Yokota

2018

Math Methods in App Sciences

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This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial‐boundary problem for the nonlinear diffusion equation in an unbounded domain normalΩ⊂RN( N∈double-struckN), written as ∂u∂t+(−Δ+1)β(u)=ginΩ×(0,T), which represents the porous media, the fast diffusion equations, etc, where β is a single‐valued maximal monotone function on double-struckR, and T>0. In Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) existence and uniqueness of solutions for were directly proved under a growth condition for β even though the Stefan problem was excluded from examples of . This paper completely removes the growth condition for β by confirming Cauchy's criterion for solutions of the following approximate problem ε with approximate parameter ε>0: ∂uε∂t+(−Δ+1)(ε(−Δ+1)uε+β(uε)+πε(uε))=ginΩ×(0,T), which is called the Cahn‐Hilliard system, even if normalΩ⊂RN( N∈double-struckN) is an unbounded domain. Moreover, it can be seen that the Stefan problem excluded from Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) is covered in the framework of this paper.

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\int_{0}^{r} β false(s false) d s \geq c false| r |^{2} - d

for all

r \in double-struckR

with some constants c , d >0.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Furthermore, we infer from (6) that1 (t) = (u 1 (t)) − (t) = (u 2 (t)) − (t) = 2 (t) for all t ∈ [0, T].Proof of Theorem 1. (Error estimate) The estimate(12) can be proved by the same argument as in the proof of Kurima and Yokota 14 , theorem 1.3.…”

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confidence: 84%

Nonlinear diffusion equations as asymptotic limits of Cahn‐Hilliard systems on unbounded domains via Cauchy's criterion

Fukao

Kurima

Yokota

2018

Math Methods in App Sciences

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“…Lemma 9 (Discrete Aubin-Lions). Let ‖⋅‖ 1, m be the norm on   m defined in (11) with the dual norm ‖⋅‖ −1, m given by (12), and let (u m ) ⊂   m ,△ t m be a sequence of piecewise constants in time functions with values in   m satisfying…”

Section: How To Cite This Articlementioning

confidence: 99%

“…A two-point flux approximation with a nonlinear positivity-preserving approximation of the cross-diffusion coefficients, modeling the segregation of a two-species population, was suggested in Andreianov and coworkers [10], assuming positive definiteness of the diffusion matrix. The Laplacian structure of the population model (still for positive definite matrices) was exploited in Murakawa [11] to design a convergent linear finite-volume scheme, which avoids fully implicit approximations. A semi-implicit finite-volume discretization for a biofilm model with a nonlocal time integrator was proposed in Rahman and Eberl [12].…”

Section: Introductionmentioning

confidence: 99%

Finite‐volume scheme for a degenerate cross‐diffusion model motivated from ion transport

Cancès

Chainais-Hillairet

Gerstenmayer

et al. 2018

Numerical Methods Partial

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An implicit Euler finite‐volume scheme for a degenerate cross‐diffusion system describing the ion transport through biological membranes is proposed. The strongly coupled equations for the ion concentrations include drift terms involving the electric potential, which is coupled to the concentrations through the Poisson equation. The cross‐diffusion system possesses a formal gradient‐flow structure revealing nonstandard degeneracies, which lead to considerable mathematical difficulties. The finite‐volume scheme is based on two‐point flux approximations with “double” upwind mobilities. The existence of solutions to the fully discrete scheme is proved. When the particles are not distinguishable and the dynamics is driven by cross diffusion only, it is shown that the scheme preserves the structure of the equations like nonnegativity, upper bounds, and entropy dissipation. The degeneracy is overcome by proving a new discrete Aubin–Lions lemma of “degenerate” type. Numerical simulations of a calcium‐selective ion channel in two space dimensions show that the scheme is efficient even in the general case of ion transport.

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“…After discretizing this scheme in space, we obtain an easy-to-implement scheme for the cross-diffusion system. A finite volume method based on the linear scheme (1.8) has been analyzed by the author [16]. Implementation is almost the same as that for the implicit method for the linear heat equation.…”

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confidence: 99%

Error Estimates for Discrete-Time Approximations of Nonlinear Cross-Diffusion Systems

Murakawa

2014

SIAM J. Numer. Anal.

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This paper is concerned with nonlinear and linear discrete-time algorithms for crossdiffusion systems. The nonlinear scheme corresponds to backward differences in time, and the linear algorithm is a very-easy-to-implement scheme proposed by the author [ESAIM Math. Model. Numer. Anal., 45 (2011), pp. 1141-1161. The main purpose of this paper is to derive convergence rates of the discrete-time schemes. We obtain the same orders for both the nonlinear and the linear scheme. Moreover, these orders are optimal. We also establish uniqueness and regularity results of weak solutions of the cross-diffusion systems.

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A linear finite volume method for nonlinear cross-diffusion systems

Cited by 21 publications

References 21 publications

Nonlinear diffusion equations as asymptotic limits of Cahn‐Hilliard systems on unbounded domains via Cauchy's criterion

Nonlinear diffusion equations as asymptotic limits of Cahn‐Hilliard systems on unbounded domains via Cauchy's criterion

Finite‐volume scheme for a degenerate cross‐diffusion model motivated from ion transport

Error Estimates for Discrete-Time Approximations of Nonlinear Cross-Diffusion Systems

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