Positivity-Preserving Numerical Schemes for Lubrication-Type Equations

Lubrication equations are fourth order degenerate diffusion equations of the form ht + ∇ · (f(h)∇∆h) = 0, describing thin films or liquid layers driven by surface tension. Recent studies of singularities in which h → 0 at a point, describing rupture of the fluid layer, show that such equations exhibit complex dynamics which can be difficult to simulate accurately. In particular, one must ensure that the numerical approximation of the interface does not show a false premature rupture. Generic finite difference schemes have the potential to manifest such instabilities especially when underresolved. We present new numerical methods, in one and two space dimensions, that preserve positivity of the solution, regardless of the spatial resolution, whenever the PDE has such a property. We also show that the schemes can preserve positivity even when the PDE itself is only known to be nonnegativity preserving. We prove that positivity-preserving finite difference schemes have unique positive solutions at all times. We prove stability and convergence of both positivitypreserving and generic methods, in one and two space dimensions, to positive solutions of the PDE, showing that the generic methods also preserve positivity and have global solutions for sufficiently fine meshes. We generalize the positivity-preserving property to a finite element framework and show, via concrete examples, how this leads to the design of other positivity-preserving schemes.