A Residual Smoothing Strategy for Accelerating Newton Method Continuation

TL;DR

This paper introduces a residual smoothing technique to accelerate the convergence of Newton methods, effectively combining smoothing and Newton approaches for improved efficiency in solving nonlinear problems.

Contribution

It proposes a residual smoothing strategy that enhances Newton method convergence by integrating local nonlinear smoothers without altering the Jacobian matrix.

Findings

Significant efficiency gains in CFD problems

Method bridges local smoothing and Newton methods

Maintains quadratic convergence at large steps

Abstract

A technique for accelerating global convergence of pseudo-transient continuation Newton methods is proposed based on residual smoothing. The technique is motivated by the effectiveness of local nonlinear smoothers at overcoming strong nonlinear transients. In the limit of a small pseudo-time step, the method reduces to a local nonlinear smoothing technique, while in the limit of large pseudo-time steps, an exact Newton method is recovered along with its quadratic convergence properties. The formulation relies on the addition of a smoothing source term while leaving the Newton Jacobian matrix unchanged, thus simplifying implementations for existing Newton solvers. The proposed technique is demonstrated on a steady-state and an implicit time-dependent computational fluid dynamics problem, showing significant gains in overall solution efficiency.