Bound-Preserving High Order Schemes for Hyperbolic Equations: Survey and Recent Developments

2021 ◽  
Author(s):  
Chi-Wang Shu
Keyword(s):  
Hyperbolic Equations ◽  
High Order ◽  
High Order Schemes ◽  
Recent Developments

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.

Fast high order ADER schemes for linear hyperbolic equations

2004 ◽  
Vol 197 (2) ◽  
pp. 532-539 ◽  
Author(s):  
Thomas Schwartzkopff ◽  
Michael Dumbser ◽  
Claus-Dieter Munz
Keyword(s):  
Hyperbolic Equations ◽  
High Order ◽  
Linear Hyperbolic Equations

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.

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