Convergence and error estimates of a penalization finite volume method for the compressible Navier-Stokes system

TL;DR

This paper introduces a penalization finite volume method for the compressible Navier-Stokes system, providing convergence and error estimates by embedding the physical domain into a larger computational domain and analyzing the penalized problem.

Contribution

It offers a novel penalty approach to analyze domain approximation errors and derives convergence and error estimates for the finite volume method in compressible fluid simulations.

Findings

Numerical solutions converge to dissipative weak solutions.

Error estimates are derived when strong solutions exist.

Numerical experiments confirm theoretical convergence and error bounds.

Abstract

In numerical simulations a smooth domain occupied by a fluid has to be approximated by a computational domain that typically does not coincide with a physical domain. Consequently, in order to study convergence and error estimates of a numerical method domain-related discretization errors, the so-called variational crimes, need to be taken into account. In this paper we present an elegant alternative to a direct, but rather technical, analysis of variational crimes by means of the penalty approach. We embed the physical domain into a large enough cubed domain and study the convergence of a finite volume method for the corresponding domain-penalized problem. We show that numerical solutions of the penalized problem converge to a generalized, the so-called dissipative weak, solution of the original problem. If a strong solution exists, the dissipative weak solution emanating from the same initial data coincides with the strong solution. In this case, we apply a novel tool of the relative energy and derive the error estimates between the numerical solution and the strong solution. Extensive numerical experiments that confirm theoretical results are presented.