The Stability Boundary of Certain Two-Layer and Three-Layer Difference Schemes

The stability with respect to initial data of difference schemes with operator weights is investigated in the frameworks of the general stability theory of operator-difference schemes. The stability is defined as the existence of a selfadjoint positive operator which determines the time-nonincreasing norm of the difference solution. The norm-independent stability criterions are obtained in the form of operator inequalities. The notation of stability boundary in the plane of two grid parameters is introduced for multi-parameter difference schemes which approximate two-dimensional parabolic and hyperbolic differential equations. The noniterative numerical algorithm is suggested for the construction of the stability boundaries of difference schemes with variable weighting factors. The approach is based on finding the smallest eigenvalue of an auxiliary selfadjoint eigenvalue problem.