@article{Hieu_Xuan_Thanh_2024, title={On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations}, volume={76}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/7273}, DOI={10.3842/umzh.v76i1.7273}, abstractNote={<p>UDC 517.9</p> <p>We consider the difference maximum principle with input data of variable sign and its application to the investigation of the monotonicity and convergence of finite-difference schemes (FDSs).<span class="Apple-converted-space"> </span>Namely, we consider the Dirichlet initial-boundary value problem for multidimensional quasilinear parabolic equation with an unbounded nonlinearity.<span class="Apple-converted-space"> </span>Unconditionally monotone linearized finite-difference schemes of the second-order of accuracy are constructed on uniform grids.<span class="Apple-converted-space"> </span>A two-sided estimate for the grid solution, which is completely consistent with similar estimates for the exact solution, is obtained.<span class="Apple-converted-space"> </span>These estimates are used to prove the convergence of FDSs in the grid $L_2$-norm.<span class="Apple-converted-space"> </span>We also present a study aimed at constructing second-order monotone difference schemes for the parabolic convection-diffusion equation with boundary conditions of the third kind and unlimited nonlinearity without using the initial differential equation on the domain boundaries.<span class="Apple-converted-space"> </span>The goal is a combination of the assumption of existence and<span class="Apple-converted-space"> </span>uniqueness of a smooth solution and the regularization principle.<span class="Apple-converted-space"> </span>In this case, the boundary conditions<span class="Apple-converted-space"> </span>are directly approximated on a two-point stencil of the second order.</p>}, number={1}, journal={Ukrainsâ€™kyi Matematychnyi Zhurnal}, author={Hieu, Le Minh and Xuan, Nguyen Huu Nguyen and Thanh, Dang Ngoc Hoang}, year={2024}, month={Feb.}, pages={132 - 146} }