On monotonic differential schemes

Method of construction of monotonic differential schemes for solving the simplest partial differential equations of elliptic and parabolic types with first derivatives and a small parameter at highest derivative is suggested. For this, the concept of adaptive artificial viscosity (AAV) is introduced. The AAV was used for construction of monotonic differential schemes of the approximation order $O(h^4)$ for the problem with boundary layer and $O(\tau^2+h^2)$ for Burgers equation, where $h$ and $\tau$ are mesh steps in space and time correspondingly. Samarsky–Golant approximation schemes (or schemes with ordered differences) are used out of the domains of large gradients. Importance of usage of second order time schemes is outlined. Numerical results are presented.