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Stability with respect to the input data and monotonicity of an implicit finite-difference scheme for a quasilinear parabolic equation

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Abstract

We analyze the stability and monotonicity of a conservative difference scheme approximating an initial-boundary value problem for a quasilinear parabolic equation under specific conditions imposed solely on the problem input data. We prove some kinds of the maximum principle for the nonlinear equations that are used in the derivation of a priori estimates for the solution; we also prove estimates for some kinds of recursion inequalities that are used in the derivation of a priori estimates for higher-order derivatives, these estimates being necessary for proving the continuous dependence of the solution on small perturbations of the input data and for analyzing monotonicity in the nonlinear case. We show that, depending on the properties of the input data, higher derivatives can become infinite in finite critical time. We obtain conditions on the input data guaranteeing the stability of the difference scheme on the entire time interval.

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Authors and Affiliations

  1. Belarus State University, Minsk, Belarus

    R. M. Yakubuk

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  1. R. M. Yakubuk
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Original Russian Text © R.M. Yakubuk, 2012, published in Differentsial’nye Uravneniya, 2012, Vol. 48, No. 2, pp. 274–285.

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Yakubuk, R.M. Stability with respect to the input data and monotonicity of an implicit finite-difference scheme for a quasilinear parabolic equation. Diff Equat 48, 283–295 (2012). https://doi.org/10.1134/S0012266112020127

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  • DOI: https://doi.org/10.1134/S0012266112020127

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