Abstract
We propose an approach to construct monotonic finite difference schemes for solving the simplest elliptic and parabolic equations with the first derivatives and a small parameter at the highest derivative. For this, the notion of adaptive artificial viscosity is introduced. The adaptive artificial viscosity is used to construct monotonic difference schemes with the flow approximation of order \(O({{h}^{4}})\) for the boundary layer problem and \(O({{\tau }^{2}} + {{h}^{2}})\) for Burgers’ equation, where \(h\) and \(\tau \) are mesh steps in space and time, respectively. The Samarskii–Golant approximation (or upwind difference schemes) is used outside the region of high gradients. The importance of using schemes of second-order accuracy in time is outlined. The computational results are presented.
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In Memory of I.V. Fryazinov
Translated by I. Tselishcheva
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Popov, I.V. On Monotonic Finite Difference Schemes. Math Models Comput Simul 12, 195–209 (2020). https://doi.org/10.1134/S207004822002012X
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DOI: https://doi.org/10.1134/S207004822002012X