A Fourth-order Positivity Preserving Geometric Flow

Local eventual positivity for a biharmonic heat equation in Rn∗

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.

High Order Positivity-Preserving Discontinuous Galerkin Methods for Radiative Transfer Equations

The positivity-preserving property is an important and challenging issue for the numerical solution of radiative transfer equations. In the past few decades, different numerical techniques have been proposed to guarantee positivity of the radiative intensity in several schemes, however it is difficult to maintain both high order accuracy and positivity. The discontinuous Galerkin (DG) finite el...

Positivity-preserving Lagrangian scheme for multi-material compressible flow

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...

A Positivity-Preserving Numerical Scheme for a Nonlinear Fourth Order Parabolic System

Nonlinear Stability of Stationary Solutions for Surface Diffusion with Boundary Conditions

The volume preserving fourth order surface diffusion flow has constant mean curvature hypersurfaces as stationary solutions. We show nonlinear stability of certain stationary curves in the plane which meet an exterior boundary with a prescribed contact angle. Methods include semigroup theory, energy arguments, geometric analysis and variational calculus.