Penalty method for the Navier-Stokes-Fourier system with Dirichlet boundary conditions: convergence and error estimates

TL;DR

This paper analyzes a finite volume method for the compressible Navier-Stokes-Fourier system with Dirichlet boundary conditions, establishing convergence, error estimates, and rates under various assumptions.

Contribution

It introduces a domain-penalized finite volume scheme for the system, providing convergence proofs and error rates, including strong convergence to strong solutions.

Findings

Weak convergence to dissipative measure-valued solutions.

Strong convergence with rate 1/4 when a strong solution exists.

Optimal strong convergence rate 1/2 under bounded velocities.

Abstract

We study the convergence and error estimates of a finite volume method for the compressible Navier-Stokes-Fourier system with Dirichlet boundary conditions. Physical fluid domain is typically smooth and needs to be approximated by a polygonal computational domain. This leads to domain-related discretization errors, the so-called variational crimes. To treat them efficiently we embed the fluid domain into a large enough cubed domain, and propose a finite volume scheme for the corresponding domain-penalized problem. Under the assumption that the numerical density and temperature are uniformly bounded, we derive the ballistic energy inequality, yielding a priori estimates and the consistency of the penalization finite volume approximations. Further, we show that the numerical solutions converge weakly to a generalized, the so-called dissipative measure-valued, solution of the corresponding Dirichlet problem. If a strong solution exists, we prove that our numerical approximations converge strongly with the rate 1/4. Additionally, assuming uniform boundedness of the approximate velocities, we obtain global existence of the strong solution. In this case we prove that the numerical solutions converge strongly to the strong solution with the optimal rate 1/2.