Unconditionally stable compact theta schemes for solving the linear and semi-linear fourth-order diffusion equations

This paper is concerned with numerical methods for solving a class of fourth-order diffusion equations. Combining the compact difference operator in space discretization and the linear θ method in time, the compact theta scheme for the linear problem is first proposed. By virtue of the Fourier method, the suggested scheme is shown to be unconditionally stable and convergent in the discrete L2-norm for any θ ∈ [1/2, 1]. And then this idea is generalized to the semi-linear case, the corresponding compact theta scheme is constructed and analyzed in detail. Numerical experiments corresponding to the linear and semi-linear situations are carried out to support our theoretical statements.