Analysis on a Numerical Scheme with Second-Order Time Accuracy for Nonlinear Diffusion Equations (Journal of Computational Mathematics, 2021)
A nonlinear fully implicit finite difference scheme with second-order time evolution fornonlinear diffusion problem is studied. The scheme is constructed with two-layer coupleddiscretization (TLCD) at each time step. It does not stir numerical oscillation, while permits large time step length, and produces more accurate numerical solutions than the othertwo well-known second-order time evolution nonlinear schemes, the Crank-Nicolson (CN)scheme and the backward difference formula second-order (BDF2) scheme. By developinga new reasoning technique, we overcome the difficulties caused by the coupled nonlineardiscrete diffusion operators at different time layers, and prove rigorously the TLCD schemeis uniquely solvable, unconditionally stable, and has second-order convergence in both space and time. Numerical tests verify the theoretical results, and illustrate its superiorityover the CN and BDF2 schemes.