An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems
File(s)
Accepted version
Author(s)
Sun, Zheng
Carrillo de la Plata, Jose Antonio
Shu, Chi-Wang
Type
Journal Article
Abstract
As an extension of our previous work in [41], we develop a discontinuous Galerkin method for solving cross-diffusion systems with a formal gradient flow structure. These systems are associated with non-increasing entropy functionals. For a class of problems, the positivity (non-negativity) of solutions is also expected, which is implied by the physical model and is
crucial to the entropy structure. The semi-discrete numerical scheme we propose is entropy stable. Furthermore, the scheme is also compatible with the positivity-preserving procedure in [43] in many scenarios, hence the resulting fully discrete scheme is able to produce non-negative solutions. The method can be applied to both one-dimensional problems and two dimensional problems on Cartesian meshes. Numerical examples are given to examine the performance of the method.
crucial to the entropy structure. The semi-discrete numerical scheme we propose is entropy stable. Furthermore, the scheme is also compatible with the positivity-preserving procedure in [43] in many scenarios, hence the resulting fully discrete scheme is able to produce non-negative solutions. The method can be applied to both one-dimensional problems and two dimensional problems on Cartesian meshes. Numerical examples are given to examine the performance of the method.
Date Issued
2019-08
Date Acceptance
2019-02-14
Citation
Kinetic and Related Models, 2019, 12 (4), pp.885-908
ISSN
1937-5093
Publisher
American Institute of Mathematical Sciences
Start Page
885
End Page
908
Journal / Book Title
Kinetic and Related Models
Volume
12
Issue
4
Copyright Statement
© American Institute of Mathematical Sciences. This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Kinetic & Related Models following peer review. The definitive publisher-authenticated version is available online at: http://aimsciences.org//article/doi/10.3934/krm.2019033
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Grant Number
EP/P031587/1
Subjects
Science & Technology
Physical Sciences
Mathematics, Applied
Mathematics
Discontinuous Galerkin method
entropy stability
positivity-preserving
cross-diffusion system
gradient flow
FINITE-ELEMENT-METHOD
NONLINEAR CONTINUITY EQUATIONS
CONSERVATION-LAWS
NUMERICAL-SIMULATION
SCHEME
MODEL
Applied Mathematics
Publication Status
Published
Date Publish Online
2019-05