Annual-Report-2019

NUMERICAL SIMULATION OF SINGULAR
CONSERVATION LAWS AND RELATED APPLICATIONS
Prof. Guoxian Chen
LE STUDIUM / Marie Skłodowska-Curie
Research Fellow
Smart Loire Valley General Programme
From: Wuhan University - CN
In residence at: Institut Denis Poisson (IDP)
- Orléans
Nationality: Chinese
Dates: January 2018 to January 2019
Prof. Chen is an associate professor in the school
of mathematic and statistics in Wuhan University,
and a Le Studium/Marie Skłodowska-Curie
Research Fellow in the University of Orleans.
He received his bachelor degree from Jishou
University in 2001, master degree from Capital
Normal University in 2004, and PhD from Peking
University in 2008 under the supervision of Prof.
Pingwen Zhang and Prof. Huazhong Tang. Then he
worked as a Postdoctoral Researcher at Hongkong
University of Science and Technology, assistant
professor at Wuhan University, Postdoctoral
Researcher at RWTH Aachen University.
He works in the area of numerical analysis and
scientific computing and computational fluid
dynamics. He focus on the numerical simulations
of singular conservation laws and related
applications, such as shallow water equations
with bottom topography, Euler equations with
gravitational potential, and the multicomponent
flows, etc..
Prof. Magali Ribot
Magali Ribot defended her PhD Thesis on
numerical analysis for PDEs in Lyon in 2003.
She became an assistant professor in Nice in 2004
and professor in Orléans in 2015. She is working
mainly in the field of numerical analysis for PDEs,
modeling for biology and fluid dynamics.
More precisely, she is interested in well-balanced
and asymptotic preserving schemes, in mixture
models coupled with fluid dynamics equations and
in the comparison of models of different types.
She is the co-head of the PDE group in Orléans
and she is organizing regularly some workshops
and seminars related to mathematics for biology.
Our project focused on the following goals:
1. Apply the subcell reconstruction to discretize the new shallow water
model;
2. Extend the new model on networks with application to irrigation;
3. Apply the new framework to design some new well-balanced methods
to chemotaxis systems or systems with more general potentials;
4. Insert our new algorithm to softwares FullSWOF and SWASHES
developed within the MAPMO. This project gives the opportunity to
invite colleagues from France and abroad, in particular enhancing the
contacts between Orléans and RWTH Aachen University in a first step.
The work carried out in 2019 at IDP resulted in the following achievements:
1. Insertion of the subcell hydrostatic reconstruction method to softwares
FullSWOF and SWASHES;
2. Application of the subcell hydrostatic reconstruction method to onedimensional
parabolic-hyperbolic chemotaxis systems. The 1D code
was obtained;
3. We designed a scheme for the Euler equations under gravitational
fields based on our subcell hydrostatic reconstruction framework. To
give a proper definition of the nonconservative product terms due to
the gravitational potential, we first separate the singularity to be an
infinitely thin layer, on where the potential is smoothed by defining an
intermediate potential without disturbing its monotonicity ; and then
the physical variables are extended and controlled to be consistent with
the Rayleigh-Taylor stability, which contribute the positivity-preserving
property to keep the nonnegativity of both gas density and pressure
even with vacuum states. By using the hydrostatic equilibrium state
variables the well-balanced property is obtained to maintain the steady
state even with vacuum fronts. In addition, we proved the full discrete
entropy inequality, which preserve the convergence of the solution to
the physical solution, with an error term which tends to zero as the
mesh size approaches to zero if the potential is Lipschitz continuous.
The new scheme is very natural to understand and easy to implement.
The numerical experiments demonstrate the scheme’s robustness to
resolve the nonlinear waves and vacuum fronts;
4. Submission of one paper to SIAM journal on numerical analysis,
another paper is preparing and publication of one paper in Nature
Materials 2020.
Figure 1. The generation of lattice deformation in the 2D crystals. Illustration of the generation
of lattice deformation in the 2D crystals and the sphere diameter effect in the SDE process
Figure 2. The simulated reshaping process of liquid phases on a solid surface with (a) poor
wettability, (c) normal wettability, and (e) good wettability. (b, d, f) The correspond force field
distributions at the liquid-gas interface, respectively.
Computer Science, Mathematics & Mathematical Physics 2019
85