Positivity preservation for density and temperature

Establish positivity preservation for the number density and temperature in the fully discrete linear scheme for the dynamic van der Waals model, despite the essentially explicit treatment of the logarithmic term.

Background

The proposed fully discrete scheme treats the logarithmic contribution essentially explicitly in the decoupled density update. This explicit treatment is important for maintaining a linear solution procedure, but it prevents the authors from proving that the numerical density and temperature remain positive. Positivity is assumed in the entropy-production theorem, which requires the numerical solutions for density and internal energy to be pointwise positive.

The unresolved issue is therefore to develop a fully discrete method that retains the scheme’s linearity, decoupling, and entropy-production property while also guaranteeing positivity of the relevant thermodynamic variables. The conclusion identifies positivity preservation of density and temperature as an additional priority alongside simultaneous preservation of entropy production and energy conservation.