Stability and Convergence of Stepsize-Dependent Linear Multistep Methods for Nonlinear Dissipative Evolution Equations in Banach Space (Journal of Computational Mathematics, 2024)
Stability and global error bounds are studied for a class of stepsize-dependent linearmultistep methods for nonlinear evolution equations governed by $ω$-dissipative vector fieldsin Banach space. To break through the order barrier $p ≤ 1$ of unconditionally contractivelinear multistep methods for dissipative systems, strongly dissipative systems are introduced. By employing the error growth function of the methods, new contractivity andconvergence results of stepsize-dependent linear multistep methods on infinite integrationintervals are provided for strictly dissipative systems $(ω < 0)$ and strongly dissipative systems. Some applications of the main results to several linear multistep methods, includingthe trapezoidal rule, are supplied. The theoretical results are also illustrated by a set ofnumerical experiments.
Keywords: Nonlinear evolution equation; Linear multistep methods; ω-dissipative operators; Stability; Convergence; Banach space.