Convergence of a finite volume method to weak solutions for the compressible Navier-Stokes-Fourier system
This work addresses the theoretical foundation for numerical methods in fluid dynamics, specifically for compressible flows, but is incremental as it extends existing convergence proofs to a more complex system.
The authors tackled the problem of proving strong convergence of an upwind-type finite volume method to weak solutions for the compressible Navier-Stokes-Fourier system with Dirichlet boundary conditions, achieving a result where the limit solution satisfies weak forms of mass, momentum, entropy, and ballistic energy inequalities and complies with the weak-strong uniqueness principle.
We prove strong convergence of an upwind-type finite volume method to a weak solution of the Navier-Stokes-Fourier system with the Dirichlet boundary conditions. The limit solution satisfies a weak form of the mass and momentum equations, together with a weak form of the entropy and ballistic energy inequalities, and complies with the weak-strong uniqueness principle. The finite volume method uses piecewise-constant spatial approximations. The convergence proof is based on a combination of delicate consistency estimates with a careful analysis of the oscillations of numerical densities via renormalisation of the continuity equation.