Positivity-preserving discontinuous Galerkin (DG) methods for solving hyperbolic 5 conservation laws have been extensively studied in the last several years. But nearly all the devel6 oped schemes are coupled with explicit time discretizations. Explicit discretizations suffer from the 7 constraint for the Courant-Friedrichs-Levis (CFL) number. This makes explicit methods impractical 8 for probl...

Robustness of numerical methods has attracted an increasing interest in the community of computational fluid dynamics. One mathematical aspect of robustness for numerical methods is the positivity-preserving property. At high Mach numbers or for flows near vacuum, solving the conservative Euler equations may generate negative density or internal energy numerically, which may lead to nonlinear i...

We consider finite element operators defined on “rough” functions in a bounded polyhedron Ω in RN . Insisting on preserving positivity in the approximations, we discover an intriguing and basic difference between approximating functions which vanish on the boundary of Ω and approximating general functions which do not. We give impossibility results for approximation of general functions to more...

We study the positivity preserving property for the Cauchy problem for the linear fourth order heat equation. Although the complete positivity preserving property fails, we show that it holds eventually on compact sets.

A finite-difference hybrid numerical method for the solution of isothermal fluctuating hydrodynamic equations is proposed. The primary focus to ensure positivity-preserving property scheme, which critical its functionality and reliability especially when simulating vapour systems. Both cases single- two-phase flows are considered by exploiting van der Waals' square-gradient approximation model ...

One of the main challenges in computational simulations of gas detonation propagation is that negative density or negative pressure may emerge during the time evolution, which will cause blow-ups. Therefore, schemes with provable positivity-preserving of certain quantities such as density and pressure are desired. First order and second order positivity-preserving schemes were well studied, e.g...

In this paper we develop high order positivity-preserving finite volume weighted essentially non-oscillatory (WENO) schemes for solving a hierarchical size-structured population model with nonlinear growth, mortality and reproduction rates. We carefully treat the technical complications in boundary conditions and global integration terms to ensure high order accuracy and positivity-preserving p...

For a Lagrangian scheme defined in the cylindrical coordinates, two important issues are whether the scheme can maintain spherical symmetry (symmetry-preserving) and whether the scheme can maintain positivity of density and internal energy (positivity-preserving). While there were previous results in the literature either for symmetry-preserving in the cylindrical coordinates or for positivity-...