Approximate Linearization of Fixed Point Iterations: Error Analysis of Tangent and Adjoint Problems Linearized about Non-Stationary Points

TL;DR

This paper analyzes how the accuracy of tangent and adjoint linearizations is affected by the convergence of nonlinear fixed point iterations, providing error estimates for various solution schemes.

Contribution

It offers a comprehensive error analysis of approximate linearizations based on fixed point iterations, including exact and inexact Newton methods and Runge-Kutta schemes.

Findings

Error bounds depend on nonlinear convergence properties

Approximate linearizations can be accurate with sufficient convergence

Results validated for multiple solution algorithms

Abstract

Previous papers have shown the impact of partial convergence of discretized PDE on the accuracy of tangent and adjoint linearizations. A series of papers suggested linearization of the fixed point iteration used in the solution process as a means of computing the sensitivities rather than linearizing the discretized PDE, as the lack of convergence of the nonlinear problem indicates that the discretized form of the governing equations has not been satisfied. These works showed that the accuracy of an approximate linearization depends in part on the convergence of the nonlinear system. This work shows an error analysis of the impact of the approximate linearization and the convergence of the nonlinear problem for both the tangent and adjoint modes and provides a series of results for an exact Newton solver, an inexact Newton solver, and a low storage explicit Runge-Kutta scheme to confirm the error analyses.