A monotone finite volume method for advection–diffusion equations on unstructured polygonal meshes

Lipnikov, Konstantin; Svyatsky, Daniil; Vassilevski, Yuri

doi:10.1016/j.jcp.2010.01.035

Cited by 87 publications

(91 citation statements)

References 30 publications

(27 reference statements)

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“…This is not necessarily the case on a general mesh. Much more complicated coefficients are then needed in 2D [26, 27] depending on the solution ϕ which are difficult to generalize to 3D. The FVM in [17] is cell centered while it is vertex centered in (9).…”

Section: Mesoscopic Model For Diffusionmentioning

confidence: 99%

“…[24, 25, 26, 27, 28, 29]. It is shown in [51] that it is impossible on a quadrilateral mesh in 2D to fulfill all conditions by a scheme.…”

Section: Mesoscopic Model For Diffusionmentioning

confidence: 99%

“…The construction of such maximum preserving and consistent FEM and FVM for unstructured meshes in 2D and 3D is the subject of a number of papers e.g. [24, 25, 26, 27, 28, 29]. In order for the solution to satisfy the discrete maximum principle for Lagrangian FEM there are either geometrical restrictions on the mesh, such as non-obtuse angles, or the coefficients depend on the solution.…”

Section: Introductionmentioning

confidence: 99%

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Simulation of stochastic diffusion via first exit times

Lötstedt

Meinecke

2015

Journal of Computational Physics

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In molecular biology it is of interest to simulate diffusion stochastically. In the mesoscopic model we partition a biological cell into unstructured subvolumes. In each subvolume the number of molecules is recorded at each time step and molecules can jump between neighboring subvolumes to model diffusion. The jump rates can be computed by discretizing the diffusion equation on that unstructured mesh. If the mesh is of poor quality, due to a complicated cell geometry, standard discretization methods can generate negative jump coefficients, which no longer allows the interpretation as the probability to jump between the subvolumes. We propose a method based on the mean first exit time of a molecule from a subvolume, which guarantees positive jump coefficients. Two approaches to exit times, a global and a local one, are presented and tested in simulations on meshes of different quality in two and three dimensions.

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Section: Mesoscopic Model For Diffusionmentioning

confidence: 99%

“…[24, 25, 26, 27, 28, 29]. It is shown in [51] that it is impossible on a quadrilateral mesh in 2D to fulfill all conditions by a scheme.…”

Section: Mesoscopic Model For Diffusionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Simulation of stochastic diffusion via first exit times

Lötstedt

Meinecke

2015

Journal of Computational Physics

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“…They proved that the scheme is monotone on triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients. An interpolation-free nonlinear monotone scheme is presented in [19], and it has been extended to the advection diffusion equations on unstructured polygonal meshes in [20]. A nonlinear finite volume scheme satisfying extremum principle for diffusion operators on triangular cells is presented in [27].…”

Section: Introductionmentioning

confidence: 99%

The finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes

Sheng

Yuan

2011

Journal of Computational Physics

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“…To address this problem, a new class of finite volume methods recently appeared in [5] and was further developed in [4,[6][7][8][9][10], prompted by underground nuclear waste storage research. In order to generate monotone matrices, these methods use two-point flux approximations, determined in a non-linear fashion.…”

Section: Introductionmentioning

confidence: 99%