Comparison of different nonlinear solvers for 2D time-implicit stellar hydrodynamics

TL;DR

This study compares various nonlinear solvers for 2D time-implicit hydrodynamics in stellar models, highlighting Broyden's quasi-Newton method as the most efficient, and discusses accuracy considerations for implicit schemes in stellar interior simulations.

Contribution

It provides a benchmark of nonlinear solvers for 2D implicit hydrodynamics and demonstrates the effectiveness of Broyden's method in stellar interior modeling.

Findings

Broyden's quasi-Newton method outperforms other solvers in speed.

Implicit schemes can match explicit scheme accuracy with proper conditions.

Implicit methods are promising for 3D stellar interior simulations.

Abstract

Time-implicit schemes are attractive since they allow numerical time steps that are much larger than those permitted by the Courant-Friedrich-Lewy criterion characterizing time-explicit methods. This advantage comes, however, with a cost: the solution of a system of nonlinear equations is required at each time step. In this work, the nonlinear system results from the discretization of the hydrodynamical equations with the Crank-Nicholson scheme. We compare the cost of different methods, based on Newton-Raphson iterations, to solve this nonlinear system, and benchmark their performances against time-explicit schemes. Since our general scientific objective is to model stellar interiors, we use as test cases two realistic models for the convective envelope of a red giant and a young Sun. Focusing on 2D simulations, we show that the best performances are obtained with the quasi-Newton method proposed by Broyden. Another important concern is the accuracy of implicit calculations. Based on the study of an idealized problem, namely the advection of a single vortex by a uniform flow, we show that there are two aspects: i) the nonlinear solver has to be accurate enough to resolve the truncation error of the numerical discretization, and ii) the time step has be small enough to resolve the advection of eddies. We show that with these two conditions fulfilled, our implicit methods exhibit similar accuracy to time-explicit schemes, which have lower values for the time step and higher computational costs. Finally, we discuss in the conclusion the applicability of these methods to fully implicit 3D calculations.