# Resolving Entropy Growth from Iterative Methods

## TL;DR

This paper investigates how iterative solvers affect entropy properties in discretized nonlinear conservation laws, revealing that relaxation techniques can preserve entropy dissipation and improve solution accuracy.

## Contribution

It demonstrates that Newton's method can reverse entropy dissipation in implicit schemes and proposes relaxation as an effective remedy to maintain entropy properties.

## Key findings

- Newton's method can cause anti-dissipative behavior in entropy schemes
- Relaxation techniques effectively preserve entropy dissipation in iterative solvers
- Entropy conservative schemes yield more accurate solutions even with larger tolerances

## Abstract

We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show that Newton's method can turn an entropy dissipative scheme into an anti-dissipative one, even when the iteration error is smaller than the time integration error. We explore several remedies, of which the most performant is a relaxation technique, originally designed to fix entropy errors in time integration methods. Thus, relaxation works well in consort with iterative solvers, provided that the iteration errors are on the order of the time integration method. To corroborate our findings, we consider Burgers' equation and nonlinear dispersive wave equations. We find that entropy conservation results in more accurate numerical solutions than non-conservative schemes, even when the tolerance is an order of magnitude larger.