Variational discretization of the Navier-Stokes-Fourier system

TL;DR

This paper develops a variational discretization method for the compressible Navier-Stokes-Fourier system, combining discrete exterior calculus and nonholonomic variational integrators to improve numerical simulations of thermodynamic fluid flows.

Contribution

It introduces a novel variational discretization framework for the NSF system using discrete exterior calculus and nonholonomic variational integrators, advancing numerical methods for thermodynamic fluid dynamics.

Findings

Established a semi-discrete variational principle for NSF.

Developed a fully discrete variational integrator for NSF.

Provided a new approach to numerically simulate thermodynamic flows.

Abstract

This paper presents the variational discretization of the compressible Navier-Stokes-Fourier system, in which the viscosity and the heat conduction terms are handled within the variational approach to nonequilibrium thermodynamics as developed by one of the authors. In a first part, we review the variational framework for the Navier-Stokes-Fourier (NSF) system in the smooth setting. In a second part, we review a discrete exterior calculus based on discrete diffeomorphisms then proceed to establish the spatially discretized variational principle for the NSF system through the use of this discrete exterior calculus, which yields a semi-discrete nonholonomic variational principle, as well as semi-discrete evolution equations. In order to avoid important technical difficulties, further treatment of the phenomenological constraint is needed. In a third part we discretize in time the spatial variational principle underlying the NSF system by extending previous work of the authors, which at last yields a nonholonomic variational integrator for the NSF system, as well as fully discrete evolution equations.