On Estimating Machine-Zero Residual

TL;DR

This paper introduces two novel techniques to estimate the minimal residual achievable in solving discretized systems, providing accurate predictions of machine-zero residuals in fluid dynamics simulations.

Contribution

It presents two new methods for estimating machine-zero residuals using residual norms from perturbed solutions, applicable to complex fluid flow equations.

Findings

Accurately predicts machine-zero residuals in Euler and Navier-Stokes equations.

Effective for transonic and viscous flow simulations over various geometries.

Provides reliable residual estimates during iterative solutions.

Abstract

In this paper, we propose two techniques to estimate the magnitude of a machine-zero residual for a given problem, which is the smallest possible residual that can be achieved when we solve a system of discretized equations. We estimate the magnitude of the machine-zero residual by a norm of residuals computed with a randomly-perturbed approximate solution that is considered as close in magnitude to an exactly-converged solution. One method uses free-stream values as the approximate solution, and the other uses a current solution during an iterative solve as the approximate solution via the method of manufactured solutions. Numerical results show that these estimates predict the levels of machine-zero residuals very accurately for all equations of the Euler and Navier-Stokes equations in a transonic flow over an airfoil and viscous flows over a cylinder and a flat plate.