Numerical methods for Balance laws with the singularity in fluid mechanics, geophysics, biology
Several phenomena in geophysics, biology or fluid mechanics are classically modeled by hyperbolic systems of conservation laws. In realistic applications, such as hydrology and chemotaxis, stiff source terms and nonconservative products arise in the equations. We call such system balance laws. This leads to serious numerical challenges concerning efficiency and stability of the numerical simulations. This talk aims to design a scheme for the balance laws with singularity based on our recently designed subcell hydrostatic reconstruction framework [SIAM Journal on Numerical Analysis, 55(2):758-784, 2017.]. We give a proper definition of the nonconservative product terms due to the gravitational potential by smoothing the singularity and physical variables extension. We also proved the stability of the scheme including 1) the well-balanced property to maintain the steady-state numerically; 2) the positivity-preserving property such that both the density and pressure be nonnegative; 3) a relaxed fully discrete entropy inequality which is introduced to preserve the convergence to the physical solutions.
LE STUDIUM / Marie Skłodowska-Curie Research Fellowship
Pr Guoxian Chen
FROM: Wuhan University (Computational Science Hubei Key Laboratory) - CN
IN RESIDENCE AT: Institut Denis Poisson, UMR 7013, University of Orléans / University of Tours / CNRS - FR