Positivity-preserving numerical schemes for lubrication-type equations

Lubrication equations are fourth order degenerate diffusion equations of th e form h(t) + del . (f(h)del Delta 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 e quations exhibit complex dynamics which can be difficult to simulate accura tely. In particular, one must ensure that the numerical approximation of th e interface does not show a false premature rupture. Generic finite differe nce schemes have the potential to manifest such instabilities especially wh en underresolved. We present new numerical methods, in one and two space di mensions, that preserve positivity of the solution, regardless of the spati al 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 b e nonnegativity preserving. We prove that positivity-preserving finite diff erence schemes have unique positive solutions at all times. We prove stabil ity and convergence of both positivity-preserving and generic methods, in o ne 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 le ads to the design of other positivity-preserving schemes.