Penalization method for the Navier-Stokes-Fourier system

TL;DR

This paper introduces a penalization approach for solving the Navier-Stokes-Fourier system with Dirichlet boundary conditions, demonstrating convergence and efficiency through numerical experiments.

Contribution

The paper develops a penalization method for the Navier-Stokes-Fourier system, enabling boundary condition enforcement via singular terms and proving convergence to the true solution.

Findings

Solutions of the penalized problem converge to the original problem's solution.

Numerical experiments confirm the method's efficiency.

The approach effectively handles complex boundary conditions.

Abstract

We apply the method of penalization to the Dirichlet problem for the Navier-Stokes-Fourier system governing the motion of a general viscous compressible fluid confined to a bounded Lipschitz domain. The physical domain is embedded into a large cube on which the periodic boundary conditions are imposed. The original boundary conditions are enforced through a singular friction term in the momentum equation and a heat source/sink term in the internal energy balance. The solutions of the penalized problem are shown to converge to the solution of the limit problem. Numerical experiments are performed to illustrate the efficiency of the method.