Unconditional convergence and error estimates for bounded numerical solutions of the barotropic Navier–Stokes system

Feireisl, Eduard; Hošek, Radim; Maltese, David; Novotný, Antonín

doi:10.1002/num.22140

Cited by 11 publications

(5 citation statements)

References 19 publications

(48 reference statements)

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“…In accordance with the conclusion of Theorem 4.1, the numerical solutions will converge to a classical exact solution as soon as the latter exists. In fact this has been shown in [9] by means of a discrete analogue of the relative energy inequality.…”

Section: Numerical Schemesmentioning

confidence: 87%

Dissipative measure-valued solutions to the compressible Navier–Stokes system

et al. 2016

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We introduce a new concept of dissipative measure-valued solution to the compressible Navier-Stokes system satisfying, in addition, a relevant form of the total energy balance. Then we show that a dissipative measure-valued and a standard smooth classical solution originating from the same initial data coincide (weak-strong uniqueness principle) as long as the latter exists. Such a result facilitates considerably the proof of convergence of solutions to various approximations including certain numerical schemes that are known to generate a measure-valued solution. As a byproduct we show that any measure-valued solution with bounded density component that starts from smooth initial data is necessarily a classical one.

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Section: Numerical Schemesmentioning

confidence: 87%

Dissipative measure-valued solutions to the compressible Navier–Stokes system

et al. 2016

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“…This technique has been largely used in the analysis of the weak-strong uniqueness and singular limit of the compressible fluid flows, see the monograph of Feireisl and Novotný [13], Březina and Feireisl [1], and Feireisl et al [11,12]. Recently, this technique has also been successfully applied to the convergence analysis of numerical solutions of compressible viscous fluids, see Feireisl et al [7] and Mizerová and She [23]. Here we adapt the technique to the Euler system and estimate the corresponding relative energy, which yields the L 2 -error estimates of density, momentum and entropy.…”

Section: Introductionmentioning

confidence: 99%

Error Estimates of the Godunov Method for the Multidimensional Compressible Euler System

2022

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We derive a priori error estimates of the Godunov method for the multidimensional compressible 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 $$L^2$$ L 2 -norms of the errors in density, momentum and entropy. Under the assumption, that the numerical density is uniformly bounded from below by a positive constant and that the energy is uniformly bounded from above and stays positive, we obtain a convergence rate of 1/2 for the relative energy in the $$L^1$$ L 1 -norm, that is to say, a convergence rate of 1/4 for the $$L^2$$ L 2 -error of the numerical solution. Further, under the assumption—the total variation of the numerical solution is uniformly bounded, we obtain the first order convergence rate for the relative energy in the $$L^1$$ L 1 -norm, consequently, the numerical solution converges in the $$L^2$$ L 2 -norm with the convergence rate of 1/2. The numerical results presented are consistent with our theoretical analysis.

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“…The numerical method must be therefore adapted to approximate not only the exact solutions but also the physical space Q, see e.g. [7], [8].…”

Section: Periodic Boundary Conditionsmentioning

confidence: 99%

Compressible fluid motion with uncertain data

Feireisl

2022

Preprint

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We propose a suitable analytical framework to perform numerical analysis of problems arising in compressible fluid models with uncertain data. We discuss both weak and strong stochastic approach, where the former is based on the knowledge of the mere distribution (law) of the random data typical for the Monte-Carlo and related methods, while the latter assumes the data to be known as a random variable on a given probability space aiming at obtaining the associated solution in the same form. As an example of the strong approach, we discuss the stochastic collocation method based on a piecewise constant approximation of the random data.

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Unconditional convergence and error estimates for bounded numerical solutions of the barotropic Navier–Stokes system

Cited by 11 publications

References 19 publications

Dissipative measure-valued solutions to the compressible Navier–Stokes system

Dissipative measure-valued solutions to the compressible Navier–Stokes system

Error Estimates of the Godunov Method for the Multidimensional Compressible Euler System

Compressible fluid motion with uncertain data

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