|
|
Pure Mathematics 2025
可压欧拉方程高精度数值格式耗散测度值解的收敛性
|
Abstract:
本文主要考虑可压欧拉方程组的初边值问题。研究了两类具有高阶精度的熵稳定有限体积格式的收敛性，通过对数值解建立合适的一致性估计，证明随着步长 h → 0，若数值解的密度是远离真空且有界的，则由这两类熵稳定数值格式构造的解可以生成耗散测度值解。
In this paper, we primarily consider the initial boundary value problem for compress ible Euler equations. We study the convergence of two classes of high-order accurate entropy stable finite volume schemes. By establishing appropriate the priori esti mates for the numerical solutions, we prove that as the step size h → 0, the solutions
constructed by these two types of entropy stable numerical schemes can generate dissipative measure-valued solutions, provided that the density of the approximate solutions is bounded away from vacuum and bounded above.
| [1] | Evans, L. C. (1998) Partial Differential Equations (Graduate Studies in Mathematics). 2nd
edition, Americam Mathematical Society. |
| [2] | de Lellis, C. and Székelyhidi, L. (2008) On Admissibility Criteria for Wea k Solutions of the
Euler Equations. Archive for Rational Mechanics and Analysis, 195, 225-260.
https://doi.org/10.1007/s00205-008-0201-x |
| [3] | de Lellis, C. and Székelyhidi, L. (2009) The Euler Equations as a Differential Inclusion. Annals
of Mathematics, 170, 1417-1436. https://doi.org/10.4007/annals.2009.170.1417 |
| [4] | Chiodaroli, E., De Lellis, C. and Kreml, O. (2014) Global Ill‐Posedness of the Isentropic
System of Gas Dynamics. Communications on Pure and Applied Mathematics, 68, 1157-1190.
https://doi.org/10.1002/cpa.21537 |
| [5] | DiPerna, R.J. (1985) Measure-Valued Solutions to Conservation Laws. Archive for Rational
Mechanics and Analysis, 88, 223-270. https://doi.org/10.1007/bf00752112 |
| [6] | Fjordholm, U. (2013) High-Order Accurate Entropy Stable Numerical Schemes for Hyperbolic
Conservation Laws. ETH Zürich Dissertation Nr. 21025. |
| [7] | Fjordholm, U.S., K？ppeli, R., Mishra, S. and Tadmor, E. (2015) Construction of Approximate
Entropy Measure-Valued Solutions for Hyperbolic Systems of Conservation Laws. Foundations
of Computational Mathematics, 17, 763-827. https://doi.org/10.1007/s10208-015-9299-z |
| [8] | Feireisl, E., Gwiazda, P., ？wierczewska-Gwiazda, A. and Wiedemann, E. (2016) Dissipative
Measure-Valued Solutions to the Compressible Navier-Stokes System. Calculus of Variations
and Partial Differential Equations, 55, Article No. 141.
https://doi.org/10.1007/s00526-016-1089-1 |
| [9] | Prodi, G. (1959) Un teorema di unicità per le equazioni di Navier-Stokes. Annali di Matem atica Pura ed Applicata, 48, 173-182. https://doi.org/10.1007/bf02410664 |
| [10] | Serrin, J. (1963) The Initial Value Problem for the Navier-Stokes Equations. In: Langer, R.E.,
Ed., Nonlinear Problems, University of Wisconsin Press, 69-98. |
| [11] | Feireisl, E., Luká？ová-Medvid’ová, M. and Mizerová, H. (2019) Convergence of Finite Volume
Schemes for the Euler Equations via Dissipative Measure-Valued Solutions. Foundations of
Computational Mathematics, 20, 923-966. https://doi.org/10.1007/s10208-019-09433-z |
| [12] | Feireisl, E. and Luká？ová-Medvid’ová, M. (2017) Convergence of a Mixed Finite Element Finite Volume Scheme for the Isentropic Navier-Stokes System via Dissipative Measure-Valued
Solutions. Foundations of Computational Mathematics, 18, 703-730.
https://doi.org/10.1007/s10208-017-9351-2 |
| [13] | Tadmor, E. (1987) The Numerical Viscosity of Entropy Stable Schemes for Systems of Con servation Laws. I. Mathematics of Computation, 49, 91-103. https://doi.org/10.2307/2008251 |
| [14] | LeFloch, P.G., Mercier, J.M. and Rohde, C. (2002) Fully Discrete, Entropy Conservative
Schemes of Arbitraryorder. SIAM Journal on Numerical Analysis, 40, 1968-1992.
https://doi.org/10.1137/s003614290240069x |
| [15] | Fjordholm, U.S., Mishra, S. and Tadmor, E. (2012) ENO Reconstruction and ENO Interpo lation Are Stable. Foundations of Computational Mathematics, 13, 139-159.
https://doi.org/10.1007/s10208-012-9117-9 |
| [16] | Pedregal, P. (1997) Parametrized Measures and Variational Principles. Birkh？user Basel. |
