Convergence of Multistep Time Discretizations of Nonlinear Dissipative Evolution Equations

Global error bounds are derived for multistep time discretizations of fully nonlinear evolution equations on infinite dimensional spaces. In contrast to earlier studies, the analysis presented here is not based on linearization procedures but on the fully nonlinear framework of logarithmic Lipschitz constants and nonlinear semigroups. The error bounds reveal how the contractive or dissipative behavior of the vector field, governing the evolution, and the properties of the multistep method influence the convergence. A multistep method which is consistent of order $p$ is proven to be convergent of the same order when the vector field is contractive or strictly dissipative, i.e., of the same order as in the ODE-setting. In the contractive context it is sufficient to require strong zero-stability of the method, whereas strong $A$-stability is sufficient in the dissipative case.

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