Title: On convergence of difference schemes for Dirichlet IBVP for two-dimensional quasilinear parabolic equations

Authors: Piotr Matus; Dmitriy Poliakov; Dorota Pylak

Addresses: Institute of Mathematics and Computer Science, The John Paul II Catholic University of Lublin, Al. Raclawickie 14, 20-950 Lublin, Poland; Institute of Mathematics, NAS of Belarus, 11 Surganov St., 220072 Minsk, Belarus ' Institute of Mathematics, NAS of Belarus, 11 Surganov St., 220072 Minsk, Belarus ' Institute of Mathematics and Computer Science, The John Paul II Catholic University of Lublin, Al. Raclawickie 14, 20-950 Lublin, Poland

Abstract: For Dirichlet initial boundary value problem (IBVP) for two-dimensional quasilinear parabolic equations, a monotone linearised difference scheme is constructed. The uniform parabolicity condition 0 < k₁ ≤ k_α(u) ≤ k₂, α = 1, 2 is assumed to be fulfilled for the sign alternating solution u(x, t) ∈ D (u) only in the domain of exact solution values (unbounded nonlinearity). On the basis of the proved new corollaries of the maximum principle not only two-sided estimates for the approximate solution y but its belonging to the domain of exact solution values are established. We assume that the solution is continuous and its first derivatives  ∂u /∂x_ihave discontinuities of the first kind in the neighbourhood of the finite number of discontinuity lines. No smoothness of the time derivative is assumed. Convergence of approximate solution to generalised solution of differential problem in the grid norm L₂ is proved.

Keywords: Convergence in the grid norm L2; Dirichlet IBVP; monotone linearised difference scheme; sign alternating solution; uniform parabolicity condition; domain of exact solution values; corollaries of the maximum principle; discontinuities of the first kind; no smoothness of the time derivative; generalised solution; initial boundary value problem; 2D quasilinear parabolic equation; unbounded nonlinearity; two-sided estimates.

DOI: 10.1504/IJEP.2019.104515

International Journal of Environment and Pollution, 2019 Vol.66 No.1/2/3, pp.63 - 79

Received: 15 Mar 2018
Accepted: 23 Sep 2018
Published online: 17 Jan 2020 *

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