Abstract
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 uniform in terms of the discretization parameters for \(\gamma > d/2\). All results are 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.
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References
Ball, J.M.: A Version of the Fundamental Theorem for Young Measures. Lecture Notes in Physics, vol. 344, pp. 207–215. Springer, Berlin (1989)
Cho, Y., Choe, H.J., Kim, H.: Unique solvability of the initial boundary value problems for compressible viscous fluids. J. Math. Pures Appl. 83, 243–275 (2004)
Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001)
Dolejší, V., Feistauer, M.: Discontinuous Galerkin Method. Volume 48 of Springer Series in Computational Mathematics, xiv+572 pp. Springer, Cham. Analysis and Applications to Compressible Flow (2015)
Chainais-Hillairet, C., Droniou, J.: Finite-volume schemes for noncoercive elliptic problems with Neumann boundary conditions. IMA J. Numer. Anal. 31(1), 61–85 (2011)
Feireisl, E., Gwiazda, P., Świerczewska-Gwiazda, A., Wiedemann, E.: Dissipative measure-valued solutions to the compressible Navier–Stokes system. Calc. Var. Partial Dif. 55(6), 55–141 (2016)
Feireisl, E., Jin, B.J., Novotný, A.: Relative entropies, suitable weak solutions, and weak strong uniqueness for the compressible Navier–Stokes system. J. Math. Fluid Mech. 14(4), 717–730 (2012)
Feireisl, E., Lukáčová-Medvid’ová, M.: Convergence of a mixed finite element-finite volume scheme for the isentropic Navier–Stokes system via the dissipative measure-valued solutions. Found. Comput. Math. 18(3), 703–730 (2018)
Feireisl, E., Lukáčová-Medvid’ová, M., Mizerová, H.: A finite volume scheme for the Euler system inspired by the two velocities approach. Numer. Math. 144, 89–132 (2020)
Feireisl, E., Lukáčová-Medvid’ová, M., She, B.: Convergence of a finite volume scheme for the compressible Navier–Stokes system. ESAIM: M2AN 53(6), 1957–1979 (2019)
Feireisl, E., Lukáčová-Medvid’ová, M., Mizerová, H., She, B.: On the convergence of a finite volume scheme for the compressible Navier–Stokes–Fourier system. arXiv:1903.08526
Feireisl, E., Lukáčová-Medvid’ová, M., Nečasová, Š., Novotný, A., She, B.: Asymptotic preserving error estimates for numerical solutions of compressible Navier–Stokes equations in the low Mach number regime. Multiscale Model. Simul. 16(1), 150–183 (2018)
Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids, 2nd edn. Birkhäuser, Basel (2017)
Feireisl, E., Novotný, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier–Stokes equations. J. Math. Fluid Mech. 3(4), 358–392 (2001)
Gallouët, T., Gastaldo, L., Herbin, R., Latché, J.C.: An unconditionally stable pressure correction scheme for the compressible barotropic Navier–Stokes equations. ESAIM: M2AN 42(2), 303–331 (2008)
Gallouët, T., Maltese, D., Novotný, A.: Error estimates for the implicit MAC scheme for the compressible Navier–Stokes equations. Numer. Math. 141, 495–567 (2019)
Gallouët, T., Herbin, R., Maltese, D., Novotný, A.: Error estimates for a numerical approximation to the compressible barotropic Navier–Stokes equations. IMA J. Numer. Anal. 36(2), 543–592 (2016)
Haack, J., Jin, S., Liu, J.G.: An all-speed asymptotic preserving method for the isentropic Euler and Navier–Stokes equations. Commun. Comput. Phys. 12(4), 955–980 (2012)
Hošek, R., She, B.: Stability and consistency of a finite difference scheme for compressible viscous isentropic flow in multi-dimension. J. Numer. Math. 26(3), 111–140 (2018)
Jovanović, V.: An error estimate for a numerical scheme for the compressible Navier–Stokes system. Kragujevac J. Math. 30, 263–275 (2007)
Karper, T.: A convergent FEM-DG method for the compressible Navier–Stokes equations. Numer. Math. 125(3), 441–510 (2013)
Lions, P.L.: Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Models. xiv+348pp. Oxford (1998)
Liu, B.: The analysis of a finite element method with streamline diffusion for the compressible Navier–Stokes equations. SIAM J. Numer. Anal. 38, 1–16 (2000)
Liu, B.: On a finite element method for three-dimensional unsteady compressible viscous flows. Numer. Methods Partial Differ. Equ. 20, 432–449 (2004)
Pedregal, P.: Parametrized Measures and Variational Principles. Birkhäuser, Basel (1997)
Sun, Y., Wang, C., Zhang, Z.: A Beale–Kato–Majda blow-up criterion for the 3-D compressible Navier–Stokes equations. J. Math. Pures Appl. 95(1), 36–47 (2011)
Valli, A.: An existence theorem for compressible viscous fluids. Ann. Mat. Pura Appl. 130, 197–213 (1982)
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Hana Mizerová and Bangwei She thank Czech Sciences Foundation (GAČR), Grant Agreement 18-05974S for supporting the research. The Institute of Mathematics of the Czech Academy of Sciences is supported by RVO:67985840.
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Mizerová, H., She, B. Convergence and Error Estimates for a Finite Difference Scheme for the Multi-dimensional Compressible Navier–Stokes System. J Sci Comput 84, 25 (2020). https://doi.org/10.1007/s10915-020-01278-x
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DOI: https://doi.org/10.1007/s10915-020-01278-x