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We study random compressible viscous magnetohydrodynamic flows. Combining the Monte Carlo method with a deterministic finite volume method we solve the random system numerically. Quantitative error estimates including statistical and deterministic errors are analyzed up to a stopping time of the exact solution. On the life-span of an exact strong solution we prove the convergence of the numerical solutions. Numerical experiments illustrate rich dynamics of random viscous compressible magnetohydrodynamics., Comment: 39 pages, 13 figures

We present an overview of recent developments on the convergence analysis of numerical methods for inviscid multidimensional compressible flows that preserve underlying physical structures. We introduce the concept of generalized solutions, the so-called dissipative solutions, and explain their relationship to other commonly used solution concepts. In numerical experiments we apply K-convergence of numerical solutions and approximate turbulent solutions together with the Reynolds stress defect and the energy defect., Comment: 16 pages, 6 figures

We consider the Navier-Stokes-Fourier system governing the motion of a general compressible, heat conducting, Newtonian fluid driven by random initial/boundary data. Convergence of the stochastic collocation and Monte Carlo numerical methods is shown under the hypothesis that approximate solutions are bounded in probability. Abstract results are illustrated by numerical experiments for the Rayleigh-Benard convection problem., Comment: 53 pages, 10 figures

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., Comment: 45 pages, 3 figures

In the present paper we consider the initial data, external force, viscosity coefficients, and heat conductivity coefficient as random data for the compressible Navier--Stokes--Fourier system. The Monte Carlo method, which is frequently used for the approximation of statistical moments, is combined with a suitable deterministic discretisation method in physical space and time. Under the assumption that numerical densities and temperatures are bounded in probability, we prove the convergence of random finite volume solutions to a statistical strong solution by applying genuine stochastic compactness arguments. Further, we show the convergence and error estimates for the Monte Carlo estimators of the expectation and deviation. We present several numerical results to illustrate the theoretical results., Comment: 24 pages, 5 figures

In this paper, we propose a linear and monolithic finite element method for the approximation of an incompressible viscous fluid interacting with an elastic and deforming plate. We use the arbitrary Lagrangian-Eulerian (ALE) approach that works in the reference domain, meaning that no re-meshing is needed during the numerical simulation. For time discretization, we employ the backward Euler method. For space discretization, we respectively use P1-bubble, P1, and P1 finite elements for the approximation of the fluid velocity, pressure, and structure displacement. We show that our method fulfills the geometrical conservation law and dissipates the total energy on the discrete level. Moreover, we prove the (optimal) linear convergence with respect to the sizes of the time step $\tau$ and the mesh $h$. We present numerical experiments involving a substantially deforming fluid domain that do validate our theoretical results. A comparison with a fully implicit (thus nonlinear) scheme indicates that our semi-implicit linear scheme is faster and as accurate as the fully implicit one, at least in stable configurations., Comment: 42 pages, 5 figures

We study the motion of the coupled system, $\mathscr S$, constituted by a physical pendulum, $\mathscr B$, with an interior cavity entirely filled with a viscous, compressible fluid, $\mathscr F$. The presence of the fluid may strongly affect on the motion of $\mathscr B$. In fact, we prove that, under appropriate assumptions, the fluid acts as a damper, namely, $\mathscr S$ must eventually reach a rest-state. Such a state is characterized by a suitable time-independent density distribution of $\mathscr F$ and a corresponding equilibrium position of the center of mass of $\mathscr S$. These results are proved in the very general class of weak solutions and do not require any restriction on the initial data, other than having a finite energy. We complement our findings with some numerical tests. The latter show, among other things, the interesting property that ``large" compressibility favors the damping effect, since it drastically reduces the time that $\mathscr S$ takes to go to rest.

In this paper we study the convergence rate of a finite volume approximation of the compressible Navier--Stokes--Fourier system. To this end we first show the local existence of a highly regular unique strong solution and analyse its global extension in time as far as the density and temperature remain bounded. We make a physically reasonable assumption that the numerical density and temperature are uniformly bounded from above and below. The relative energy provides us an elegant way to derive a priori error estimates between finite volume solutions and the strong solution., Comment: 29 pages

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.

The goal of this paper is to study convergence and error estimates of the Monte Carlo method for the Navier-Stokes equations with random data. To discretize in space and time, the Monte Carlo method is combined with a suitable deterministic discretization scheme, such as a finite volume method. We assume that the initial data, force and the viscosity coefficients are random variables and study both, the statistical convergence rates as well as the approximation errors. Since the compressible Navier-Stokes equations are not known to be uniquely solvable in the class of global weak solutions, we cannot apply pathwise arguments to analyze the random Navier-Stokes equations. Instead we have to apply intrinsic stochastic compactness arguments via the Skorokhod representation theorem and the Gy\"{o}ngy-Krylov method. Assuming that the numerical solutions are bounded in probability, we prove that the Monte Carlo finite volume method converges to a statistical strong solution. The convergence rates are discussed as well. Numerical experiments illustrate theoretical results.

We present new error estimates for the finite volume and finite difference methods applied to the compressible Navier-Stokes equations. The main innovative ingredients of the improved error estimates are a refined consistency analysis combined with a continuous version of the relative energy inequality. Consequently, we obtain better convergence rates than those available in the literature so far. Moreover, the error estimates hold in the whole physically relevant range of the adiabatic coefficient.

We derive a priori error of the Godunov method for the multidimensional Euler system of gas dynamics. To this end we apply the relative energy principle and estimate the distance between the numerical solution and the strong solution. This yields also the estimates of the $L^2$-norm of errors in density, momentum and entropy. Under the assumption that the numerical density and energy are bounded, we obtain a convergence rate of $1/2$ for the relative energy in the $L^1$-norm. Further, under the assumption -- the total variation of numerical solution is bounded, we obtain the first order convergence rate for the relative energy in the $L^1$-norm. Consequently, numerical solutions (density, momentum and entropy) converge in the $L^2$-norm with the convergence rate of $1/2$. The numerical results presented for Riemann problems are consistent with our theoretical analysis., Comment: 31 pages

We introduce a simple model of the time evolution of a binary mixture of compressible fluids including the thermal effects. Despite its apparent simplicity, the model is thermodynamically consistent admitting an entropy balance equation. We introduce a suitable weak formulation of the problem based on a combination of the entropy inequality with the total energy conservation principle. Finally, we show compactness of any bounded family of weak solutions and establish a global existence result.

We study a general convergence theory for the numerical solutions of compressible viscous and electrically conducting fluids with a focus on numerical schemes that preserve the divergence free property of magnetic field exactly. Our strategy utilizes the recent concepts of dissipative weak solutions and consistent approximations. First, we show the dissipative weak--strong uniqueness principle, meaning a dissipative weak solution coincides with a classical solution as long as they emanate from the same initial data. Next, we show the convergence of consistent approximation towards the dissipative weak solution and thus the classical solution. Upon interpreting the consistent approximation as the stability and consistency of suitable numerical solutions we have established a generalized Lax equivalence theory: convergence $\Longleftrightarrow$ stability and consistency. Further, to illustrate the application of this theory, we propose two novel mixed finite volume-finite element methods with exact divergence-free magnetic field. Finally, by showing solutions of these two schemes are consistent approximations, we conclude their convergence towards the dissipative weak solution and the classical solution., Comment: arXiv admin note: text overlap with arXiv:2103.07253

We study a general convergence theory for the analysis of numerical solutions to the magnetohydrodynamic system describing the time evolution of compressible, viscous, electrically conducting fluids in space dimension d (= 2; 3). First, we introduce the concept of dissipative weak solutions and prove the weak-strong uniqueness property for dissipative weak solutions, meaning a dissipative weak solution coincides with a classical solution emanating from the same initial data on the lifespan of the latter. Next, we introduce the concept of consistent approximations and prove the convergence of consistent approximations towards the dissipative weak solution as well as the classical solution. Interpreting the consistent approximation as the energy stability and consistency of numerical solutions, we have built a nonlinear variant of the celebrated Lax-equivalence theorem. Finally, as an application of this theory, we show the convergence analysis for two numerical methods., Comment: 20 pages

Applying the concept of S-convergence, based on averaging in the spirit of Strong Law of Large Numbers, the vanishing viscosity solutions of the Euler system are studied. We show how to efficiently compute a viscosity solution of the Euler system as the S-limit of numerical solutions obtained by the Viscosity Finite Volume method. Theoretical results are illustrated by numerical simulations of the Kelvin--Helmholtz instability problem., Comment: 42 pages

We consider a model of a binary mixture of two immiscible compressible fluids. We propose a numerical scheme and discuss its basic properties: Stability, consistency, convergence. The convergence is established via the method of generalized weak solutions combined with the weak-strong uniqueness principle.

We prove convergence of a finite difference approximation of the compressible Navier--Stokes system towards the strong solution in $R^d,$ $d=2,3,$ for the adiabatic coefficient $\gamma>1$. Employing the relative energy functional, we find a convergence rate which is \emph{uniform} in terms of the discretization parameters for $\gamma \geq d/2$. All results are \emph{unconditional} in the sense that we have no assumptions on the regularity nor boundedness of the numerical solution. We also provide numerical experiments to validate the theoretical convergence rate. To the best of our knowledge this work contains the first unconditional result on the convergence of a finite difference scheme for the unsteady compressible Navier--Stokes system in multiple dimensions., Comment: 36pages

In this paper we introduce a numerical scheme for fluid-structure interaction problems in two or three space dimensions: A flexible elastic plate is interacting with a viscous, compressible barotropic fluid. Hence the physical domain of definition (the domain of Eulerian coordinates) is changing in time. We introduce a fully discrete scheme that is stable, satisfies geometric conservation, mass conservation and the positivity of the density. We also prove that the scheme is consistent with the definition of continuous weak solutions., Comment: 27 pages, 3 figures

We develop a method to compute effectively the Young measures associated to sequences of numerical solutions of the compressible Euler system. Our approach is based on the concept of $\mathcal{K}$-convergence adapted to sequences of parametrized measures. The convergence is strong in space and time (a.e.~pointwise or in certain $L^q$ spaces) whereas the measures converge narrowly or in the Wasserstein distance to the corresponding limit., Comment: 35 pages, 8 figures

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., Comment: 25 pages, 1 figure

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., Comment: 21 pages, 2 figures