On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations

Le Minh Hieu; Nguyen Huu Nguyen Xuan; Dang Ngoc Hoang Thanh

doi:10.3842/umzh.v76i1.7273

Le Minh Hieu Department of Economics, University of Economics – The University of Danang, Vietnam
Nguyen Huu Nguyen Xuan Department of Research and International Cooperation, University of Economics – The University of Danang, Vietnam
Dang Ngoc Hoang Thanh Department of Information Technology, School of Business Information Technology, University of Economics, Ho Chi Minh city, Vietnam

DOI: https://doi.org/10.3842/umzh.v76i1.7273

Keywords: Maximum principle, two-side estimates, monotone method, finite-difference scheme, multidimensional quasilinear parabolic equation, convergence, weakly couple system, scientific computing, regularization principle, convection-diffusion problem, third boundary value problem.

Abstract

UDC 517.9

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). Namely, we consider the Dirichlet initial-boundary value problem for multidimensional quasilinear parabolic equation with an unbounded nonlinearity. Unconditionally monotone linearized finite-difference schemes of the second-order of accuracy are constructed on uniform grids. A two-sided estimate for the grid solution, which is completely consistent with similar estimates for the exact solution, is obtained. These estimates are used to prove the convergence of FDSs in the grid $L_2$-norm. 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. The goal is a combination of the assumption of existence and uniqueness of a smooth solution and the regularization principle. In this case, the boundary conditions are directly approximated on a two-point stencil of the second order.

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