On the maximum principle preserving schemes for the generalized Allen-Cahn equation
nonlinear convection-diffusion-reaction equation
The authors investigate the 1D Allen-Cahn equation (a nonlinear convection-diffusion-reaction equation) along with homogeneous Neumann or Dirichlet conditions, or with periodic boundary conditions. Continuing their former work (see [\textit{T. Tang} and \textit{J. Yang}, J. Comput. Math. 34, No. 5, 451--461 (2016; \url{doi:10.4208/jcm.1603-m2014-0017})]) and considering a polynomial and a logarithmic potential, they give a semi-discrete scheme and a fully discretized semi-implicit difference scheme for the numerical solution of the problem. Assuming suitable (sufficient) restrictions on the time step, they prove (based on properties of diagonally dominant matrices) maximum norm estimates for both schemes. The results can be generalized to the multidimensional case and for rectangular domains. In special cases of the potentials and mobilities considered they are able to explicitly give (sufficient) time step bounds.NEWLINENEWLINEAssuming the solution of the original problem to be sufficiently smooth (which is not specified concretely), they then show an error estimate for the fully discrete scheme in the maximum norm guaranteeing first order in time and space -- provided one starts from the values of the initial condition.NEWLINENEWLINEIn their numerical experiments verifying their theoretical results they consider 1D and 2D cases and add an adaptive time step control. Let us remark that two figures (6.3 and 6.4) probably were in colors but are not distinguishable here.
scientific article; zbMATH DE number 6655257