Divergence-free-preserving high-order schemes for magnetohydrodynamics: An artificial magnetic resistivity method

Abstract Hypoxy induced angiogenesis processes can be described by coupling an integrodifferential kinetic equation of Fokker–Planck type with a diffusion equation for the angiogenic factor. We propose high order positivity preserving schemes to approximate the marginal tip density by combining an asymptotic reduction with weighted essentially non oscillatory and strong stability preserving time discretization. We capture soliton-like solutions representing blood vessel formation and spread towards hypoxic regions.

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Compact high-order schemes for the Euler equations

Journal of Scientific Computing ◽

10.1007/bf01061287 ◽

1988 ◽

Vol 3 (3) ◽

pp. 275-288 ◽

Cited By ~ 45

Author(s):

Saul Abarbanel ◽

Ajay Kumar

Keyword(s):

Euler Equations ◽

High Order ◽

High Order Schemes

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High Order Schemes on Three-Dimensional General Polyhedral Meshes — Application to the Level Set Method

Communications in Computational Physics ◽

10.4208/cicp.260511.050811a ◽

2012 ◽

Vol 12 (1) ◽

pp. 1-41 ◽

Cited By ~ 6

Author(s):

Thibault Pringuey ◽

R. Stewart Cant

Keyword(s):

Level Set ◽

Three Dimensional ◽

Unstructured Meshes ◽

High Order ◽

Convex Polyhedra ◽

Two Phase ◽

High Order Schemes ◽

Flow Problems ◽

Mixed Element ◽

Areas Of Interest

AbstractIn this article, we detail the methodology developed to construct arbitrarily high order schemes — linear and WENO — on 3D mixed-element unstructured meshes made up of general convex polyhedral elements. The approach is tailored specifically for the solution of scalar level set equations for application to incompressible two-phase flow problems. The construction of WENO schemes on 3D unstructured meshes is notoriously difficult, as it involves a much higher level of complexity than 2D approaches. This due to the multiplicity of geometrical considerations introduced by the extra dimension, especially on mixed-element meshes. Therefore, we have specifically developed a number of algorithms to handle mixed-element meshes composed of convex polyhedra with convex polygonal faces. The contribution of this work concerns several areas of interest: the formulation of an improved methodology in 3D, the minimisation of computational runtime in the implementation through the maximum use of pre-processing operations, the generation of novel methods to handle complex 3D mixed-element meshes and finally the application of the method to the transport of a scalar level set.

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