A. V. Gulin

Stability of a Nonlocal Two-Dimensional Finite-Difference Problem

We consider a two-dimensional problem for the heat equation with boundary conditions of the first kind with respect to one of the space variables and nonlocal conditions with respect to the other variable. For an explicit finite-difference scheme, we obtain a criterion for stability with respect to the initial data in a specially constructed energy norm. The analysis is carried out on the basis of the general stability theory of operator-difference schemes [1], and here we substantially use the results of [2], where finite-difference schemes approximating the spatially one-dimensional heat equation with nonlocal boundary conditions were considered. Consider the explicit finite-difference scheme ( y i − y i ) /τ = y xx,i, i = 1, . . . , N1 − 1, N1h1 = 1, y 0 = 0, y 0 i = u0 (xi) , ( y N1 − y n N1 ) /τ = (2/h) ( y x,0 − y x,N1 ) , (1.1)

Stability criterion for a family of nonlocal difference schemes

We consider finite-difference schemes for the heat equation with nonlocal boundary conditions that contain a real parameter γ. A stability criterion for finite-difference schemes with respect to the initial data was earlier obtained for |γ| ≤ 1. In the present paper, we consider the case in which γ ∈ (−cosh π,−1) and the original differential problem is stable, while the stability conditions for the finite-difference schemes substantially depend on γ. We obtain estimates for the energy norm of the solution of the finite-difference problem via the same norm of the initial data and prove the equivalence of the energy norm and the grid L2-norm.

On a family of nonlocal difference schemes

We consider a weighted difference scheme approximating the heat equation with nonlocal boundary conditions. We analyze the behavior of the spectrum of the main finite-difference operator depending on the parameters occurring in the boundary conditions. We state inequalities whose validity is necessary and sufficient for the stability of the difference scheme with respect to the initial data.

Stability of a nonlocal difference problem with a complex parameter

We consider a difference scheme for the heat equation with nonlocal boundary conditions containing a complex parameter γ. We single out stability domains in the complex γ-plane and derive a criterion for stability with respect to the initial data in the form of a condition on γ and the grid increments.

Stability of nonlocal difference schemes in a subspace

We consider a family of two-layer difference schemes for the heat equation with nonlocal boundary conditions containing the parameter γ. In some interval γ ∈ (1, γ+), the spectrum of the main difference operator contains a unique eigenvalue λ0 in the left complex half-plane, while the remaining eigenvalues λ1, λ2, …, λN−1 lie in the right half-plane. The corresponding grid space HN is represented as the direct sum HN = H0⊕HN−1 of a one-dimensional subspace and the subspace HN−1 that is the linear span of eigenvectors µ(1), µ(2), …, µ(N−1). We introduce the notion of stability in the subspace HN−1 and derive a stability criterion.

On the spectral stability in subspaces for difference schemes with nonlocal boundary conditions

We consider an initial-boundary value problem for the heat equation with nonlocal boundary conditions containing a parameter γ > 1. The spectrum of the main differential operator contains some number (depending on γ) of eigenvalues lying in the left complex half-plane, which results in the instability of the problem with respect to the initial data. For difference schemes approximating the original problem, we obtain a criterion for stability in the subspaces generated by stable harmonics.

Difference scheme for the Samarskii-Ionkin problem with a parameter

AbstractWe consider a weighted difference scheme approximating the heat equation with the nonlocal boundary conditions

On the stability of nonlocal difference schemes in subspaces

u(0,t) = 0, \frac{{\partial u}} {{\partial x}}(0,t) + \frac{{\partial u}} {{\partial x}}(1,t) = 0