Stability of numerical methods for ordinary differential equations
This paper is concerned with the stability properties of three multivalue methods on non uniform grids. In the first case the authors consider the \(A(\alpha)\)-stability of the two step backward differentiation formula (BDF) method with variable coefficients. Here the \( 2 \times 2 \) stability matrix \( A = A(r,z)\) depends on two parameters: the step size ratio \( r= h_n/h_{n-1}\) between two consecutive steps and the complex variable \(z\). By choosing a suitable real matrix \(T\) the authors prove that the norm \( \|A \|= \|T^{-1} A T \|_1 \leq 0\) for all \( r \in [0, ( 1 + \sqrt 3)/ 2 ]\) and \( |\arg z - \pi |\leq 70^0\) which implies \(A(\alpha)\)-stability for arbitrary step size ratios in the range \([0, ( 1 + \sqrt 3)/ 2 ]\). The second multivalue method is a first order two step general linear method whose stability matrix \( M(r,z)\) is a \(( 2 \times 2)\) matrix of rank one. Here by using the factorization \( M(r,z)= v u^T \) the authors show that any product of stability matrices is \(A(0)\)-stable for all \( r \in [0, 2.94].\) Finally the zero stability of the three step BDF method with variable coefficients is considered. Now the propagation matrix can be written as a \((2 \times 2)\) matrix depending on two consecutive step size ratios \( C (r_n, r_{n-1})\) and using a discrete Lyapunov type function the authors show that zero stability holds for arbitrary \( r \leq ( 1 + \sqrt 5)/2 \) which is an optimal step size bound for arbitrary step sizes. It must be noticed that similar results for BDF methods (not mentioned in the present paper) have been obtained, among others, by Brayton and Conley, Dahlquist-Liniger-Nevanlinna, Grigorieff, Calvo-Grande-Grigorieff, and Sand.
scientific article; zbMATH DE number 1879359