Regularity of Fluxes in Nonlinear Hyperbolic Balance Laws

doi:10.1007/s42967-022-00224-y

Communications on Applied Mathematics and Computation ›› 2023, Vol. 5 ›› Issue (3): 1289-1298.doi: 10.1007/s42967-022-00224-y

• ORIGINAL PAPER • Previous Articles

Regularity of Fluxes in Nonlinear Hyperbolic Balance Laws

Matania Ben-Artzi¹, Jiequan Li^2,3

1. Institute of Mathematics, Hebrew University of Jerusalem, 91904, Jerusalem, Israel;
2. Academy of Multidisciplinary Studies, Capital Normal University, Beijing, 100048, China;
3. State Key Laboratory for Turbulence Research and Complex System, Peking University, Beijing, 100871, China

Received:2022-08-18 Revised:2022-08-18 Online:2023-09-20 Published:2023-08-29
Contact: Jiequan Li,E-mail:jiequan@cnu.edu.cn E-mail:mbartzi@math.huji.ac.il;jiequan@cnu.edu.cn
Supported by:
The first author thanks the Institute of Applied Physics and Computational Mathematics, Beijing, for the hospitality and support. The second author is supported by the NSFC (Nos. 11771054, 12072042,91852207), the Sino-German Research Group Project (No. GZ1465), and the National Key Project GJXM92579. It is a pleasure to thank C. Dafermos and M. Slemrod for many useful comments.

Abstract

Abstract: This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws. The basic idea is that the “meaningful objects” are the fluxes, evaluated across domain boundaries over time intervals. The fundamental result in this treatment is the regularity of the flux trace in the multi-dimensional setting. It implies that a weak solution indeed satisfies the balance law. In fact, it is shown that the flux is Lipschitz continuous with respect to suitable perturbations of the boundary. It should be emphasized that the weak solutions considered here need not be entropy solutions. Furthermore, the assumption imposed on the flux f(u) is quite minimal—just that it is locally bounded.

Key words: Balance laws, Hyperbolic conservation laws, Multi-dimensional, Discontinuous solutions, Finite-volume schemes, Flux, Trace on boundary

CLC Number:

Matania Ben-Artzi, Jiequan Li. Regularity of Fluxes in Nonlinear Hyperbolic Balance Laws[J]. Communications on Applied Mathematics and Computation, 2023, 5(3): 1289-1298.

TrendMD

References

[1] Ben-Artzi, M., Falcovitz, J.: Generalized Riemann Problems in Computational Fluid Dynamics. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2003)
[2] Ben-Artzi, M., Li, J.Q.: Consistency of finite volume approximations to nonlinear hyperbolic balance laws. Math. Comp. 90, 141–169 (2021)
[3] Chen, G.Q., Comi, G.E., Torres, M.: Cauchy fluxes and Gauss-Green formulas for divergence measure fields over general open sets. Arch. Rat. Mech. Anal. 233, 87–166 (2019)
[4] Chen, G.Q., Frid, H.: Divergence-measure fields and hyperbolic conservation laws. Arch. Rat. Mech. Anal. 147, 89–118 (1999)
[5] Chen, G.Q., Torres, M., Ziemer, W.: Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws. Comm. Pure Appl. Math. 62, 242–304 (2009)
[6] Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, 4th edn. Grundlehren der Mathematischen Wissenschaften, Springer, Heidelberg (2016)
[7] Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence, RI (1998)
[8] Eymard. R., Gallouët, T., Herbin, R.: Finite volume methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, Vol. VII, pp. 713–1020. North-Holland, New York (2000)
[9] Federer, H.: Geometric Measure Theory. Springer, Heidelberg (1969)
[10] Godlewski, E., Raviart, P.A.: Hyperbolic Systems of Conservation Laws. Ellipses, Paris (1991)
[11] Gurtin, M.E., Martins, L.C.: Cauchy’s theorem in classical physics. Arch. Rat. Mech. Anal. 60, 305–324 (1975/76)
[12] Šilhavý, M.: The existence of the flux vector and the divergence theorem for general Cauchy fluxes. Arch. Rat. Mech. Anal. 90, 195–212 (1985)
[13] Šilhavý, M.: Divergence-measure fields and Cauchy’s stress theorem. Rend. Sem. Mat. Padova 113, 15–45 (2005)
[14] Spivak, M.: A Comprehensive Introduction to Differential Geometry. Vol. I. Publish or Perish, Inc., Houston, Texas (1979)
[15] Van Leer, B.: On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher and Roe. SIAM J. Sci. Stat. Comput. 5, 1–20 (1984)

Regularity of Fluxes in Nonlinear Hyperbolic Balance Laws

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 9

Recommended Articles

Metrics

Comments

[1]	Gui-Qin Qiu, Gao-Wei Cao, Xiao-Zhou Yang, Yuan-An Zhao. Envelope Method and More General New Global Structures of Solutions for Multi-dimensional Conservation Law [J]. Communications on Applied Mathematics and Computation, 2023, 5(3): 1180-1234.
[2]	R. Abgrall, J. Nordström, P. Öffner, S. Tokareva. Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability [J]. Communications on Applied Mathematics and Computation, 2023, 5(2): 573-595.
[3]	Wasilij Barsukow. Stationarity Preservation Properties of the Active Flux Scheme on Cartesian Grids [J]. Communications on Applied Mathematics and Computation, 2023, 5(2): 638-652.
[4]	Jonas Zeifang, Andrea Beck. A Low Mach Number IMEX Flux Splitting for the Level Set Ghost Fluid Method [J]. Communications on Applied Mathematics and Computation, 2023, 5(2): 722-750.
[5]	R. Abgrall. A Combination of Residual Distribution and the Active Flux Formulations or a New Class of Schemes That Can Combine Several Writings of the Same Hyperbolic Problem: Application to the 1D Euler Equations [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 370-402.
[6]	Lukáš Vacek, Václav Kučera. Discontinuous Galerkin Method for Macroscopic Traffic Flow Models on Networks [J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 986-1010.
[7]	Jiawei Sun, Shusen Xie, Yulong Xing. Local Discontinuous Galerkin Methods for the abcd Nonlinear Boussinesq System [J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 381-416.
[8]	Zheng Sun, Chi-Wang Shu. Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity [J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 671-700.
[9]	Zachary Grant, Sigal Gottlieb, David C. Seal. A Strong Stability Preserving Analysis for Explicit Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions [J]. Communications on Applied Mathematics and Computation, 2019, 1(1): 21-59.