A second‐order space–time accurate scheme for nonlinear diffusion equation with general capacity term

Abstract

AbstractA fully implicit finite difference (FIFD) scheme with second‐order space–time accuracy is studied for a nonlinear diffusion equation with general capacity term. A new reasoning procedure is introduced to overcome difficulties caused by the nonlinearity of the capacity term and the diffusion operator in the theoretical analysis. The existence of the FIFD solution is investigated at first which plays an important role in the analysis. It is established by choosing a new test function to bound the solution and its temporal and spatial difference quotients in suitable norms in the fixed point arguments, which is different from the traditional way. Based on these bounds, other fundamental properties of the scheme are rigorously analyzed consequently. It shows that the scheme is uniquely solvable, unconditionally stable, and convergent with second‐order space–time accuracy in L∞(L2) and L∞(H1) norms. The theoretical analysis adapts to both one‐ and multidimensional problems, and can be extended to schemes with first‐order time accuracy. Numerical tests are provided to verify the theoretical results and highlight the high accuracy of the second‐order space–time accurate scheme. The reasoning techniques can be extended to a broad family of discrete schemes for nonlinear problems with capacity terms.