Stability and convergence of stepsize-dependent linear multistep methods for nonlinear dissipative evolution equations in Banach space. (English) Zbl 07806675
Summary: Stability and global error bounds are studied for a class of stepsize-dependent linear multistep methods for nonlinear evolution equations governed by \(\omega\)-dissipative vector fields in Banach space. To break through the order barrier \(p\leq 1\) of unconditionally contractive linear multistep methods for dissipative systems, strongly dissipative systems are introduced. By employing the error growth function of the methods, new contractivity and convergence results of stepsize-dependent linear multistep methods on infinite integration intervals are provided for strictly dissipative systems \((\omega <0)\) and strongly dissipative systems. Some applications of the main results to several linear multistep methods, including the trapezoidal rule, are supplied. The theoretical results are also illustrated by a set of numerical experiments.
MSC:
| 65J15 | Numerical solutions to equations with nonlinear operators |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65J08 | Numerical solutions to abstract evolution equations |
Keywords:
nonlinear evolution equation; linear multistep methods; \(\omega\)-dissipative operators; stability; convergence; Banach spaceSoftware:
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