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Monotone Finite-Difference Schemes of Second-Order Accuracy for Quasilinear Parabolic Equations with Mixed Derivatives

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Abstract

We consider the initial-boundary value problem for quasilinear parabolic equation with mixed derivatives and an unbounded nonlinearity. We construct unconditionally monotone and conservative finite-difference schemes of the second-order accuracy for arbitrary sign alternating coefficients of the equation. For the finite-difference solution, we obtain a two-sided estimate completely consistent with similar estimates for the solution of the differential problem, and also obtain an important a priori estimate in the uniform C-norm. These estimates are used to prove the convergence of finite-difference schemes in the grid L2-norm. All theoretical results are obtained under the assumption that some conditions imposed only on the input data of the differential problem are satisfied.

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Belarus, Minsk, 220072, Belarus

    P. P. Matus

  2. John Paul II Catholic University of Lublin, Lublin, 20-950, Poland

    P. P. Matus & D. Pylak

  3. University of Economics - The University of Danang, Danang, Vietnam

    L. M. Hieu

Authors
  1. P. P. Matus
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  2. L. M. Hieu
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  3. D. Pylak
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Correspondence to P. P. Matus, L. M. Hieu or D. Pylak.

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Matus, P.P., Hieu, L.M. & Pylak, D. Monotone Finite-Difference Schemes of Second-Order Accuracy for Quasilinear Parabolic Equations with Mixed Derivatives. Diff Equat 55, 424–436 (2019). https://doi.org/10.1134/S0012266119030157

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  • DOI: https://doi.org/10.1134/S0012266119030157

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