A finite volume method on general meshes for a degenerate parabolic convection–reaction–diffusion equation

Angelini, O.; Brenner, Konstantin; Hilhorst, Danielle

doi:10.1007/s00211-012-0485-5

Cited by 30 publications

(20 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, the left-hand side of (30) is negative, by induction hypothesis, which gives a contradiction. ▪ Lemmas 4 and 5 imply that we may remove the truncation in (28). Moreover, by definition, we Proof.…”

Section: Proofmentioning

confidence: 94%

See 2 more Smart Citations

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

Cancès

Chainais-Hillairet

Gerstenmayer

et al. 2018

Numerical Methods Partial

View full text Add to dashboard Cite

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.

show abstract

Section: Proofmentioning

confidence: 94%

“…Lemma 5 (Nonnegativity of u k 0 ). Let Assumption (A2) hold and let (u, Φ) be a solution to (15), (16), (20), and (28). Then u k 0,…”

Section: Proofmentioning

confidence: 99%

See 1 more Smart Citation

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

Cancès

Chainais-Hillairet

Gerstenmayer

et al. 2018

Numerical Methods Partial

View full text Add to dashboard Cite

show abstract

“…However, it may be very difficult to obtain an exact solution of the general CDE, especially for those with variable coefficients. On the contrary, the numerical methods, including finite-element method [3,4], finite-difference method [5,6], finite-volume method [7] can be served as an alternative to solve the CDE with the development of computer technology.…”

Section: Introductionmentioning

confidence: 99%

Lattice Boltzmann model for a class of convection–diffusion equations with variable coefficients

Li¹,

Chai²,

Shi³

2015

Computers & Mathematics with Applications

View full text Add to dashboard Cite

a b s t r a c tIn this work, a lattice Boltzmann model for a class of n-dimensional convection-diffusion equations with variable coefficients is proposed through introducing an auxiliary distribution function. The model can exactly recover the convection-diffusion equation without any assumptions. A detailed numerical study on several types of convection-diffusion equations is performed to validate the present model, and the results show that the accuracy of the present model is better than previous models.

show abstract

“…As an adaptive scheme is designed to yield highly nonuniform mesh, discretization of the governing equation on general meshes should be settled first [23]. As for how to locate mesh points adaptively, such methods often appeal to equidistributing a monitor function or solving mesh equations [19].…”

Section: Introductionmentioning

confidence: 99%

A Self‐Adaptive Numerical Method to Solve Convection‐Dominated Diffusion Problems

Cao

Liu

et al. 2017

Mathematical Problems in Engineering

View full text Add to dashboard Cite

Convection-dominated diffusion problems usually develop multiscaled solutions and adaptive mesh is popular to approach high resolution numerical solutions. Most adaptive mesh methods involve complex adaptive operations that not only increase algorithmic complexity but also may introduce numerical dissipation. Hence, it is motivated in this paper to develop an adaptive mesh method which is free from complex adaptive operations. The method is developed based on a range-discrete mesh, which is uniformly distributed in the value domain and has a desirable property of self-adaptivity in the spatial domain. To solve the time-dependent problem, movement of mesh points is tracked according to the governing equation, while their values are fixed. Adaptivity of the mesh points is automatically achieved during the course of solving the discretized equation. Moreover, a singular point resulting from a nonlinear diffusive term can be maintained by treating it as a special boundary condition. Serval numerical tests are performed. Residual errors are found to be independent of the magnitude of diffusive term. The proposed method can serve as a fast and accuracy tool for assessment of propagation of steep fronts in various flow problems.

show abstract

A finite volume method on general meshes for a degenerate parabolic convection–reaction–diffusion equation

Cited by 30 publications

References 30 publications

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

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

Lattice Boltzmann model for a class of convection–diffusion equations with variable coefficients

A Self‐Adaptive Numerical Method to Solve Convection‐Dominated Diffusion Problems

Contact Info

Product

Resources

About