scholarly journals Erratum to: The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

2012 ◽  
Vol 53 (2) ◽  
pp. 482-482 ◽  
Author(s):  
M. Medvinsky ◽  
S. Tsynkov ◽  
E. Turkel
Keyword(s):  
Helmholtz Equation ◽  
High Order ◽  
Difference Potentials ◽  
High Order Schemes

scholarly journals Positivity preserving high order schemes for angiogenesis models

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Carpio ◽  
E. Cebrian
Keyword(s):  
Kinetic Equation ◽  
Time Discretization ◽  
Strong Stability ◽  
High Order ◽  
Angiogenic Factor ◽  
Strong Stability Preserving ◽  
Positivity Preserving ◽  
Blood Vessel Formation ◽  
High Order Schemes ◽  
Asymptotic Reduction

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.

scholarly journals Compact high-order schemes for the Euler equations

1988 ◽  
Vol 3 (3) ◽  
pp. 275-288 ◽  
Author(s):  
Saul Abarbanel ◽  
Ajay Kumar
Keyword(s):  
Euler Equations ◽  
High Order ◽  
High Order Schemes

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

2012 ◽  
Vol 12 (1) ◽  
pp. 1-41 ◽  
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.

scholarly journals Whispering Bloch modes

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160103 ◽  
Author(s):  
B. Maling ◽  
R. V. Craster
Keyword(s):  
Helmholtz Equation ◽  
Eigenvalue Problems ◽  
Open Systems ◽  
Rotational Symmetry ◽  
Field Enhancement ◽  
High Order ◽  
Whispering Gallery Modes ◽  
Whispering Gallery ◽  
Field Patterns ◽  
Bloch Modes

We investigate eigenvalue problems for the planar Helmholtz equation in open systems with a high order of rotational symmetry. The resulting solutions have similarities with the whispering gallery modes exploited in photonic micro-resonators and elsewhere, but unlike these do not necessarily require a surrounding material boundary, with confinement instead resulting from the geometry of a series of inclusions arranged in a ring. The corresponding fields exhibit angular quasi-periodicity reminiscent of Bloch waves, and hence we refer to them as whispering Bloch modes (WBMs). We show that if the geometry of the system is slightly perturbed such that the rotational symmetry is broken, modes with asymmetric field patterns can be observed, resulting in field enhancement and other potentially desirable effects. We investigate the WBMs of two specific geometries first using expansion methods and then by applying a two-scale asymptotic scheme.

Difference potentials for the Helmholtz equation in exterior domains

2000 ◽  
Vol 33 (1-4) ◽  
pp. 533-540 ◽  
Author(s):  
Victor S. Ryaben'kii ◽  
Ivan L. Sofronov
Keyword(s):  
Helmholtz Equation ◽  
Exterior Domains ◽  
Difference Potentials
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