Gap in Nonlinear Equivalence for Numerical Methods for PDEs

TL;DR

This paper investigates the relationship between stability and convergence in numerical methods for nonlinear PDEs, establishing conditions under which they are equivalent and analyzing the gap between these conditions.

Contribution

It introduces a general nonlinear equivalence framework for stability and convergence in numerical PDE methods, clarifying the conditions needed for their equivalence.

Findings

Identifies necessary and sufficient stability conditions for nonlinear PDE approximations.

Analyzes the gap between stability conditions, leading to a unified understanding.

Establishes a nonlinear equivalence between stability and convergence.

Abstract

For a large class of nonlinear evolution PDEs, and more generally, of nonlinear semigroups, as well as their approximating numerical methods, two rather natural stability type convergence conditions are given, one being necessary, while the other is sufficient. The gap between these two stability conditions is analyzed, thus leading to a general nonlinear equivalence between stability and convergence.