In this paper, we consider compact and monotone difference schemes of the fourth order of approximation for linear, semilinear, and quasi-linear equations of the parabolic type. For the Fisher equation, the monotonicity, stability, and convergence of the proposed methods in the uniform norm L _( ∞ )or C are proved. The results obtained are generalized to quasi-linear parabolic equations with nonlinearities of the porous medium type. In this study, at an abstract level, a definition of the monotonicity of a difference scheme in the nonlinear case is given. The performed computational experiment illustrates the efficiency of the methods under consideration. This article indicates a method for determining the order of the rate 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.
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