Bangwei She

Eduard Feireisl, Maria Lukacova-Medvidova, Hana Mizerova et al.

We study convergence of a finite volume scheme for the Navier-Stokes-Fourier system describing the motion of compressible viscous and heat conducting fluids. The numerical flux uses upwinding with an additional numerical diffusion of order $\mathcal{O} (h^{ \varepsilon+1})$, $0<\varepsilon<1$. The approximate solutions are piecewise constant functions with respect to the underlying mesh. We show that any uniformly bounded sequence of numerical solutions converges unconditionally to the solution of the Navier-Stokes-Fourier system. In particular, the existence of the solution to the Navier-Stokes-Fourier system is not a priori assumed.

Eduard Feireisl, Maria Lukacova-Medvidova, Bangwei She et al.

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.

Eduard Feireisl, Maria Lukacova-Medvidova, Bangwei She et al.

We study the long-time behaviour of the temperature-driven compressible flows. We show that numerical solutions of a structure-preserving finite volume method generate a discrete attractor that consists of entire discrete trajectories. Further, we prove the convergence of discrete attractors to their continuous counterparts. Theoretical results are illustrated by extensive numerical simulations of the well-known Rayleigh-Benard problem. The numerical results also indicate the validity of the ergodic hypothesis and imply that a non-zero Reynolds stress persist for long time. Finally, we also observe that any invariant measure is of Gaussian type in sharp contrast with the conjecture proposed by [Glimm et al., SN Applied Sciences 2, 2160 (2020)].

Mária Lukáčová-Medviďová, Bangwei She

In this paper, we present convergence theorems for numerical solutions of the incompressible Euler equations. The first result is the Lax-Wendroff-type theorem, while the second can be formulated in the framework of the Lax equivalence theorem. To illustrate their application, we study a finite element method that uses a pair of $RT_0/P_0$ elements to approximate the velocity and pressure, respectively. Applying the concept of the relative energy, we derive the convergence rates of our numerical method using two different approaches. Finally, we validate the theoretical convergence results through numerical experiments.

Eduard Feireisl, Maria Lukacova-Medvidova, Hana Mizerova et al.

We study convergence of a finite volume scheme for the compressible (barotropic) Navier--Stokes system. First we prove the energy stability and consistency of the scheme and show that the numerical solutions generate a dissipative measure-valued solution of the system. Then by the dissipative measure-valued-strong uniqueness principle, we conclude the convergence of the numerical solution to the strong solution as long as the latter exists. Numerical experiments for standard benchmark tests support our theoretical results.