Consistency of finite volume approximations to nonlinear hyperbolic balance laws
HTML articles powered by AMS MathViewer
- by Matania Ben-Artzi and Jiequan Li;
- Math. Comp. 90 (2021), 141-169
- DOI: https://doi.org/10.1090/mcom/3569
- Published electronically: October 6, 2020
- HTML | PDF | Request permission
Abstract:
This paper addresses the three concepts of consistency, stability and convergence in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of “balance laws”. Such laws express the relevant physical conservation laws in the presence of discontinuities. Finite volume approximations employ this viewpoint, and the present paper can be regarded as being in this category. It is first shown that under very mild conditions a weak solution is indeed a solution to the balance law. The schemes considered here allow the computation of several quantities per mesh cell (e.g., slopes) and the notion of consistency must be extended to this framework. Then a suitable convergence theorem is established, generalizing the classical convergence theorem of Lax and Wendroff. Finally, the limit functions are shown to be entropy solutions by using a notion of “Godunov compatibility”, which serves as a substitute to the entropy condition.References
- Matania Ben-Artzi and Joseph Falcovitz, A second-order Godunov-type scheme for compressible fluid dynamics, J. Comput. Phys. 55 (1984), no. 1, 1–32. MR 757422, DOI 10.1016/0021-9991(84)90013-5
- Matania Ben-Artzi and Joseph Falcovitz, Generalized Riemann problems in computational fluid dynamics, Cambridge Monographs on Applied and Computational Mathematics, vol. 11, Cambridge University Press, Cambridge, 2003. MR 1978164, DOI 10.1017/CBO9780511546785
- Matania Ben-Artzi, Joseph Falcovitz, and Jiequan Li, The convergence of the GRP scheme, Discrete Contin. Dyn. Syst. 23 (2009), no. 1-2, 1–27. MR 2449066, DOI 10.3934/dcds.2009.23.1
- Matania Ben-Artzi and Jiequan Li, Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem, Numer. Math. 106 (2007), no. 3, 369–425. MR 2302058, DOI 10.1007/s00211-007-0069-y
- F. Bouchut, Ch. Bourdarias, and B. Perthame, A MUSCL method satisfying all the numerical entropy inequalities, Math. Comp. 65 (1996), no. 216, 1439–1461. MR 1348038, DOI 10.1090/S0025-5718-96-00752-1
- Alberto Bressan, The unique limit of the Glimm scheme, Arch. Rational Mech. Anal. 130 (1995), no. 3, 205–230. MR 1337114, DOI 10.1007/BF00392027
- Claire Chainais-Hillairet, Second-order finite-volume schemes for a non-linear hyperbolic equation: error estimate, Math. Methods Appl. Sci. 23 (2000), no. 5, 467–490. MR 1746245, DOI 10.1002/(SICI)1099-1476(20000325)23:5<467::AID-MMA124>3.3.CO;2-Z
- Gui-Qiang Chen, Monica Torres, and William P. Ziemer, Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, Comm. Pure Appl. Math. 62 (2009), no. 2, 242–304. MR 2468610, DOI 10.1002/cpa.20262
- P. Colella and P. R. Woodward, The Piecewise Parabolic Method (PPM) for gas dynamical simulations, J. Comput. Phys., 54 (1984), 174–201.
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 4th ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2016. MR 3468916, DOI 10.1007/978-3-662-49451-6
- Bruno Despres, Lax theorem and finite volume schemes, Math. Comp. 73 (2004), no. 247, 1203–1234. MR 2047085, DOI 10.1090/S0025-5718-03-01618-1
- Xia Xi Ding, Gui Qiang Chen, and Pei Zhu Luo, Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics, Comm. Math. Phys. 121 (1989), no. 1, 63–84. MR 985615, DOI 10.1007/BF01218624
- R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), no. 1, 27–70. MR 684413, DOI 10.1007/BF00251724
- Volker Elling, A Lax-Wendroff type theorem for unstructured quasi-uniform grids, Math. Comp. 76 (2007), no. 257, 251–272. MR 2261020, DOI 10.1090/S0025-5718-06-01881-3
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845, DOI 10.1090/gsm/019
- Robert Eymard, Thierry Gallouët, and Raphaèle Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 713–1020. MR 1804748, DOI 10.1016/S1570-8659(00)07005-8
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 257325
- Ulrik S. Fjordholm, Roger Käppeli, Siddhartha Mishra, and Eitan Tadmor, Construction of approximate entropy measure-valued solutions for hyperbolic systems of conservation laws, Found. Comput. Math. 17 (2017), no. 3, 763–827. MR 3648106, DOI 10.1007/s10208-015-9299-z
- T. Gallouët, R. Herbin, and J.-C. Latché, On the weak consistency of finite volumes schemes for conservation laws on general meshes, SeMA J. 76 (2019), no. 4, 581–594. MR 4025828, DOI 10.1007/s40324-019-00194-x
- Edwige Godlewski and Pierre-Arnaud Raviart, Hyperbolic systems of conservation laws, Mathématiques & Applications (Paris) [Mathematics and Applications], vol. 3/4, Ellipses, Paris, 1991. MR 1304494
- S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Mat. Sb. (N.S.) 47(89) (1959), 271–306 (Russian). MR 119433
- Claus R. Goetz and Michael Dumbser, A novel solver for the generalized Riemann problem based on a simplified LeFloch-Raviart expansion and a local space-time discontinuous Galerkin formulation, J. Sci. Comput. 69 (2016), no. 2, 805–840. MR 3551345, DOI 10.1007/s10915-016-0218-5
- Jonathan B. Goodman and Randall J. LeVeque, A geometric approach to high resolution TVD schemes, SIAM J. Numer. Anal. 25 (1988), no. 2, 268–284. MR 933724, DOI 10.1137/0725019
- D. Kröner, M. Rokyta, and M. Wierse, A Lax-Wendroff type theorem for upwind finite volume schemes in $2$-D, East-West J. Numer. Math. 4 (1996), no. 4, 279–292. MR 1430241
- S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR Sbornik, 10(1970), 217–243.
- Peter Lax, Shock waves and entropy, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York-London, 1971, pp. 603–634. MR 393870
- Peter Lax and Burton Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217–237. MR 120774, DOI 10.1002/cpa.3160130205
- Philippe G. LeFloch and Jian-Guo Liu, Generalized monotone schemes, discrete paths of extrema, and discrete entropy conditions, Math. Comp. 68 (1999), no. 227, 1025–1055. MR 1627801, DOI 10.1090/S0025-5718-99-01062-5
- Jiequan Li and Yue Wang, Thermodynamical effects and high resolution methods for compressible fluid flows, J. Comput. Phys. 343 (2017), 340–354. MR 3654063, DOI 10.1016/j.jcp.2017.04.048
- Ta Tsien Li and Wen Ci Yu, Boundary value problems for quasilinear hyperbolic systems, Duke University Mathematics Series, V, Duke University, Mathematics Department, Durham, NC, 1985. MR 823237
- P.-L. Lions and P. E. Souganidis, Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton-Jacobi equations, Numer. Math. 69 (1995), no. 4, 441–470. MR 1314597, DOI 10.1007/s002110050102
- Tai Ping Liu, Uniqueness of weak solutions of the Cauchy problem for general $2\times 2$ conservation laws, J. Differential Equations 20 (1976), no. 2, 369–388. MR 393871, DOI 10.1016/0022-0396(76)90114-5
- Xu-Dong Liu, Stanley Osher, and Tony Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys. 115 (1994), no. 1, 200–212. MR 1300340, DOI 10.1006/jcph.1994.1187
- K. W. Morton and D. F. Mayers, Numerical solution of partial differential equations, 2nd ed., Cambridge University Press, Cambridge, 2005. An introduction. MR 2153063, DOI 10.1017/CBO9780511812248
- Stanley Osher, Convergence of generalized MUSCL schemes, SIAM J. Numer. Anal. 22 (1985), no. 5, 947–961. MR 799122, DOI 10.1137/0722057
- Jianzhen Qian, Jiequan Li, and Shuanghu Wang, The generalized Riemann problems for compressible fluid flows: towards high order, J. Comput. Phys. 259 (2014), 358–389. MR 3148575, DOI 10.1016/j.jcp.2013.12.002
- Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, 2nd ed., Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 220455
- Nicolas Seguin and Julien Vovelle, Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients, Math. Models Methods Appl. Sci. 13 (2003), no. 2, 221–257. MR 1961002, DOI 10.1142/S0218202503002477
- Chi-Wang Shu, High order WENO and DG methods for time-dependent convection-dominated PDEs: a brief survey of several recent developments, J. Comput. Phys. 316 (2016), 598–613. MR 3494371, DOI 10.1016/j.jcp.2016.04.030
- Eitan Tadmor, A review of numerical methods for nonlinear partial differential equations, Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 4, 507–554. MR 2958929, DOI 10.1090/S0273-0979-2012-01379-4
- E. F. Toro and V. A. Titarev, Derivative Riemann solvers for systems of conservation laws and ADER methods, J. Comput. Phys. 212 (2006), no. 1, 150–165. MR 2183608, DOI 10.1016/j.jcp.2005.06.018
- Bram van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method [J. Comput. Phys. 32 (1979), no. 1, 101–136], J. Comput. Phys. 135 (1997), no. 2, 227–248. With an introduction by Ch. Hirsch; Commemoration of the 30th anniversary {of J. Comput. Phys.}. MR 1486274, DOI 10.1006/jcph.1997.5757
- J.-P. Vila, An analysis of a class of second-order accurate Godunov-type schemes, SIAM J. Numer. Anal. 26 (1989), no. 4, 830–853. MR 1005512, DOI 10.1137/0726046
Bibliographic Information
- Matania Ben-Artzi
- Affiliation: Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel
- MR Author ID: 34290
- ORCID: 0000-0002-6782-4085
- Email: mbartzi@math.huji.ac.il
- Jiequan Li
- Affiliation: Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, People’s Republic of China; Center for Applied Physics and Technology, Peking University, People’s Republic of China; and State Key Laboratory for Turbulence Research and Complex System, Peking University, People’s Republic of China
- Email: li_jiequan@iapcm.ac.cn
- Received by editor(s): September 23, 2019
- Received by editor(s) in revised form: March 16, 2020, May 9, 2020, and May 30, 2020
- Published electronically: October 6, 2020
- Additional Notes: The first author thanks the Institute of Applied Physics and Computational Mathematics, Beijing, for the hospitality and support.
The second author was supported by NSFC (Nos. 11771054, 91852207), the Sino-German Research Group Project (No. GZ1465) and Foundation of LCP. \thanks{The second author is the corresponding author.} - © Copyright 2020 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 141-169
- MSC (2010): Primary 65M12; Secondary 35L65, 65M08
- DOI: https://doi.org/10.1090/mcom/3569
- MathSciNet review: 4166456