Stability of nonlocal difference schemes in a subspace

Abstract

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 λ 0 in the left complex half-plane, while the remaining eigenvalues λ 1, λ 2, …, λ N−1 lie in the right half-plane. The corresponding grid space H N is represented as the direct sum H N = H 0⊕H N−1 of a one-dimensional subspace and the subspace H N−1 that is the linear span of eigenvectors µ(1), µ(2), …, µ(N−1). We introduce the notion of stability in the subspace H N−1 and derive a stability criterion.