Stability of nonlocal difference schemes in a subspace (Q1760454)

The author considers a family of two-layer difference schemes for the heat equation with nonlocal boundary conditions containing a parameter \(\gamma\). In some interval \(\gamma \in (1, \gamma_+)\), the spectrum of the main difference operator contains a unique eigenvalue \(\lambda_0\) in the left complex half-plane, while the remaining eigenvalues \(\lambda_1, \lambda_2, \ldots , \lambda_{N-1}\) lie in the right half-plane. The author represents the corresponding grid space \(H_N\) as the direct sum \(H_N = H_0\oplus H_{N-1}\) of a one-dimensional subspace and the subspace \(H_{N-1}\) that is the linear span of eigenvectors \(\mu^{(1)}, \mu^{(2)}, \dots, \mu^{(N-1)}\). The author introduces the notion of stability in the subspace \(H_{N-1}\) and derive a stability criterion.