Bound/positivity preserving and unconditionally stable schemes for a class of fourth order nonlinear equations

Abstract

We construct two classes of time discretization schemes for fourth order nonlinear equations by combining a function transform approach with the scalar auxiliary variable (SAV)approach. A suitable function transform ensures that the schemes are bound/positivity preserving, while the SAV approach enables us to construct unconditionally stable schemes. The first class of schemes requires solving a coupled second-order linear systems with variable coefficients at each time step, while the second class of schemes only requires solving a fourth-order linear equation with constant coeffcients. We apply this approach to the Cahn-Hilliard equations with logarithmic potential and the more challenging Lubrication-type equations, and present ample numerical examples to validate the accuracy and efficiency of the proposed schemes.