Computational Methods in Applied Mathematics· 2026Q2
Difference Analogues of Two-Sided Estimates for Quasilinear Parabolic Equations with Mixed Derivatives and Arbitrary-Sign Reaction Coefficients
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- 2026year
Short summary
New finite-difference schemes for quasilinear parabolic equations with mixed derivatives and arbitrary-sign reaction coefficients satisfy a nonstandard maximum principle unconditionally, ensuring stability and convergence.
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Key points
- Finite-difference schemes constructed for quasilinear parabolic equations with mixed derivatives and arbitrary-sign reaction coefficients.
- Schemes satisfy a nonstandard maximum principle unconditionally, ensuring monotonicity.
- Discrete two-sided estimates derived using the Ladyzhenskaya transformation are consistent with continuous problem estimates.
- Convergence and stability of the schemes are rigorously justified in the grid L2-norm.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract This paper deals with initial-boundary value problems (IBVPs) for quasilinear parabolic equations with unbounded nonlinearity, mixed derivatives, and reaction coefficients of arbitrary sign. Finite-difference schemes (FDSs) with second-order spatial approximation are constructed for these problems. It is proved that the proposed schemes satisfy a nonstandard maximum principle under separate conditions on the temporal and spatial grid steps, without any coupling restriction between them, thereby ensuring unconditional monotonicity. Moreover, using the Ladyzhenskaya transformation, discrete two-sided estimates for the numerical solution are derived and shown to be fully consistent with the corresponding estimates for the continuous problem. The convergence and stability of the proposed schemes are rigorously justified in the grid L 2 {L_{2}} -norm.
The authors' abstract, as published at the source. Computational Methods in Applied Mathematics, 2026 · DOI ↗
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Field: Numerical Analysis
Numerical AnalysisMathematics