Compact and monotone difference schemes for parabolic equations

In this paper, we consider compact and monotone difference schemes of the fourth order of approximation for linear, semilinear, and quasilinear equations of parabolic type. For the Fisher equation, the monotonicity, stability and convergence of the proposed methods are proved in the uniform norm $L_\infty$ or $C$. The results obtained are generalized to quasilinear parabolic equations with nonlinearities such as a porous medium. The work an abstract level defines the monotonicity of a difference scheme in the nonlinear case. The performed computational experiment illustrates the effectiveness of the considered methods. A way of determining the order of convergence of the proposed methods based on the Runge method in the case of the presence of several variables and different orders in different variables is indicated in the article.