Existence of a fully implicit solution with second-order time evolution for strong nonlinear diffusion equation

Abstract A fully implicit finite difference scheme with second-order time evolution for strong nonlinear diffusion equation of divergence type is studied. Theoretical analysis is established under a coercive condition reflecting the diffusion characteristics of the strong nonlinear equation. Some new reasoning techniques are developed to overcome the difficulties caused by the strong nonlinearity of the divergence diffusion operator. The existence of the fully implicit finite difference solution is proved by constructing an appropriate mapping in the fixed point argument and bounding the solution as well as its first- and second-order spatial difference quotients tactfully.