Study of the norm in stability problems for nonlocal difference schemes

Semi-Implicit Difference Scheme for a Two-Dimensional Parabolic Equation With an Integral Boundary Condition

“…In [8,9,11,12,16,17,23,24,30] the conditions of stability of difference schemes for one-dimensional parabolic equations with integral and other types of nonlocal boundary conditions have been obtained. In all papers the investigation of stability was based on a priori estimations, the maximum principle, structure of the spectrum of the transmission matrix or some modifications of these methods.…”

“…Remark 2 allows us to explain why, when approximating differential problem (1.1)-(1.3) by semi-implicit difference scheme (2.2)-(2.4), we use not the classical but new type formula of numerical integration (2.7). Let us take any other standard formula of numerical integration instead of formula (2.7), for example, the trapezoid formula 16) in which ρ ml are weight coefficients. Let us now insert the values of u n ij , (i, j) ∈ Γ h from formula (2.16) into that equations of (2.2), in which i = 1, N − 1 or j = 1, N − 1.…”

On the stability of a weighted finite difference scheme for wave equation with nonlocal boundary conditions

“…According to [20,21], one can study the stability conditions for the two-layer difference scheme (21) by analyzing the spectrum of the matrix S. Note that the matrices S and Λ are nonsymmetric (matrix Λ is nonsymmetric except the classical case γ 1 " 0 and γ 2 " 0).…”

On The Stability Of Finite-Difference Schemes For Parabolic Equations Subject To Integral Conditions With Applications To Thermoelasticity

“…The stability of the finite difference schemes for parabolic equations subject to various other types of nonlocal conditions is dealt in the articles [2,4,8] (also, see [9] and the references therein). Article [13] uses the method of lines to solve the quasi-linear parabolic equation.…”