A Central Scheme for Two Coupled Hyperbolic Systems

Abstract

A novel numerical scheme to solve two coupled systems of conservation laws is introduced. The scheme is derived based on a relaxation approach and does not require information on the Lax curves of the coupled systems, which simplifies the computation of suitable coupling data. The coupling condition for the underlying relaxation system plays a crucial role as it determines the behaviour of the scheme in the zero relaxation limit. The role of this condition is discussed, a consistency concept with respect to the original problem is introduced, the well-posedness is analyzed and explicit, nodal Riemann solvers are provided. Based on a case study considering the p-system of gas dynamics, a strategy for the design of the relaxation coupling condition within the new scheme is provided.

Keywords

Coupled conservation laws / Hyperbolic systems / Finite-volume schemes / Coupling conditions / Relaxation system

References

[1.] Adimurthi MS, Veerappa Gowda GD. Optimal entropy solutions for conservation laws with discontinuous flux-functions. J. Hyperbolic Differ. Equ., 2005, 2(4): 783-837,

[2.] Ambroso A, Chalons C, Coquel F, Godlewski E, Lagoutière F, Raviart P-A, Seguin N. Coupling of general Lagrangian systems. Math. Comput., 2008, 77(262): 909-941,

[3.] Andreianov B, Karlsen KH, Risebro NH. A theory of L 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}-dissipative solvers for scalar conservation laws with discontinuous flux. Arch. Ration. Mech. Anal., 2011, 201(1): 27-86,

[4.] Banda MK, Häck A-S, Herty M. Numerical discretization of coupling conditions by high-order schemes. J. Sci. Comput., 2016, 69(1): 122-145,

[5.] Banda MK, Herty M, Klar A. Gas flow in pipeline networks. Netw. Heterog. Media, 2006, 1(1): 41-56,

[6.] Banda MK, Herty M, Klar A. Coupling conditions for gas networks governed by the isothermal Euler equations. Netw. Heterog. Media, 2006, 1(2): 295-314,

[7.] Banda MK, Herty M, Ngnotchouye JMT. On linearized coupling conditions for a class of isentropic multiphase drift-flux models at pipe-to-pipe intersections. J. Comput. Appl. Math., 2015, 276: 81-97,

[8.] Borsche R. Numerical schemes for networks of hyperbolic conservation laws. Appl. Numer. Math., 2016, 108: 157-170,

[9.] Borsche R, Kall J. ADER schemes and high order coupling on networks of hyperbolic conservation laws. J. Comput. Phys., 2014, 273: 658-670,

[10.] Borsche R, Klar A. Kinetic layers and coupling conditions for scalar equations on networks. Nonlinearity, 2018, 31(7): 3512-3541,

[11.] Bressan A. . Hyperbolic Systems of Conservation Laws, 2000 Oxford Oxford University Press 250, The one-dimensional Cauchy problem

[12.] Bressan A, Čanić S, Garavello M, Herty M, Piccoli B. Flows on networks: recent results and perspectives. EMS Surv. Math. Sci., 2014, 1(1): 47-111,

[13.] Bretti G, Natalini R, Piccoli B. Fast algorithms for the approximation of a traffic flow model on networks. Discret. Contin. Dyn. Syst. Ser. B, 2006, 6(3): 427-448,

[14.] Brouwer J, Gasser I, Herty M. Gas pipeline models revisited: model hierarchies, nonisothermal models, and simulations of networks. Multiscale Model. Simul., 2011, 9(2): 601-623,

[15.] Bürger R, Karlsen KH, Klingenberg C, Risebro NH. A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units. Nonlinear Anal. Real World Appl., 2003, 4(3): 457-481,

[16.] Chalons C, Raviart P-A, Seguin N. The interface coupling of the gas dynamics equations. Q. Appl. Math., 2008, 66(4): 659-705,

[17.] Chapman S, Cowling TG. . The Mathematical Theory of Non-Uniform Gases: an Account of the Kinetic Theory of Viscosity, Thermal Conduction, and Diffusion in Gases, 1990 3 Cambridge, New York Cambridge University Press

[18.] Chertkov, M., Fisher, M., Backhaus, S., Bent, R., Misra, S.: Pressure fluctuations in natural gas networks caused by gas-electric coupling. In: 2015 48th Hawaii International Conference on System Sciences, pp. 2738–2747 (2015). https://doi.org/10.1109/hicss.2015.330

[19.] Coclite GM, Garavello M. Vanishing viscosity for traffic on networks. SIAM J. Math. Anal., 2010, 42(4): 1761-1783,

[20.] Colombo RM, Herty M, Sachers V. On 2 × 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2$$\end{document} conservation laws at a junction. SIAM J. Math. Anal., 2008, 40(2): 605-622,

[21.] Colombo RM, Mauri C. Euler system for compressible fluids at a junction. J. Hyperbolic Differ. Equ., 2008, 5(3): 547-568,

[22.] Coquel F, Jin S, Liu J-G, Wang L. Well-posedness and singular limit of a semilinear hyperbolic relaxation system with a two-scale discontinuous relaxation rate. Arch. Ration. Mech. Anal., 2014, 214(3): 1051-1084,

[23.] D’Apice C, Göttlich S, Herty M, Piccoli B. . Modeling, Simulation, and Optimization of Supply Chains, 2010 Philadelphia Society for Industrial and Applied Mathematics (SIAM) 206, A continuous approach

[24.] Diehl S. On scalar conservation laws with point source and discontinuous flux function. SIAM J. Math. Anal., 1995, 26(6): 1425-1451,

[25.] Dubois F, Le Floch P. Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differential Equations, 1988, 71(1): 93-122,

[26.] Egger H. A robust conservative mixed finite element method for isentropic compressible flow on pipe networks. SIAM J. Sci. Comput., 2018, 40(1): 108-129,

[27.] Formaggia L, Nobile F, Quarteroni A, Veneziani A. Multiscale modelling of the circulatory system: a preliminary analysis. Comput. Vis. Sci., 1999, 2(2/3): 75-83,

[28.] Garavello M, Han K, Piccoli B. . Models for vehicular traffic on networks, 2016 American Institute of Mathematical Sciences (AIMS) Springfield 474

[29.] Garavello M, Piccoli B. . Traffic Flow on Networks: Conservation Law Models, 2006 Springfield American Inst. of Mathematical Sciences

[30.] Gimse T, Risebro NH. Solution of the Cauchy problem for a conservation law with a discontinuous flux function. SIAM J. Math. Anal., 1992, 23(3): 635-648,

[31.] Godlewski E, Raviart P-A. The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: I. The scalar case. Numer. Math., 2004, 97(1): 81-130,

[32.] Godlewski E, Raviart PA. A method of coupling non-linear hyperbolic systems: examples in CFD and plasma physics. Int. J. Numer. Methods Fluids, 2005, 47(10/11): 1035-1041, 8th ICFD Conference on Numerical Methods for Fluid Dynamics. Part 2

[33.] Hantke M, Müller S. Analysis and simulation of a new multi-component two-phase flow model with phase transitions and chemical reactions. Q. Appl. Math., 2018, 76(2): 253-287,

[34.] Hantke M, Müller S. Closure conditions for a one temperature non-equilibrium multi-component model of Baer-Nunziato type. ESAIM Proc. Surv., 2019, 66: 42-60,

[35.] Herty M, Jörres C, Piccoli B. Existence of solution to supply chain models based on partial differential equation with discontinuous flux function. J. Math. Anal. Appl., 2013, 401(2): 510-517,

[36.] Herty M, Kolbe N, Müller S. Central schemes for networked scalar conservation laws. Netw. Heterog. Media, 2023, 18(1): 310-340,

[37.] Herty M, Mohring J, Sachers V. A new model for gas flow in pipe networks. Math. Methods Appl. Sci., 2010, 33(7): 845-855,

[38.] Herty M, Müller S, Sikstel A. Coupling of compressible Euler equations. Vietnam J. Math., 2019, 47(4): 769-792,

[39.] Herty M, Rascle M. Coupling conditions for a class of second-order models for traffic flow. SIAM J. Math. Anal., 2006, 38(2): 595-616,

[40.] Holden H, Risebro NH. A mathematical model of traffic flow on a network of unidirectional roads. SIAM J. Math. Anal., 1995, 26(4): 999-1017,

[41.] Holle Y, Herty M, Westdickenberg M. New coupling conditions for isentropic flow on networks. Netw. Heterog. Media, 2020, 15(4): 605-631,

[42.] Hu J, Jin S, Li Q. Asymptotic-preserving schemes for multiscale hyperbolic and kinetic equations. Handbook of Numerical Analysis, 2017 Amsterdam Elsevier 103-129,

[43.] Jin S, Liu J-G, Wang L. A domain decomposition method for semilinear hyperbolic systems with two-scale relaxations. Math. Comput., 2013, 82(282): 749-779,

[44.] Jin S, Xin Z. The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math., 1995, 48(3): 235-276,

[45.] Karlsen KH, Klingenberg C, Risebro NH. A relaxation scheme for conservation laws with a discontinuous coefficient. Math. Comput., 2003, 73(247): 1235-1260,

[46.] Karlsen KH, Towers JD. Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition. J. Hyperbolic Differ. Equ., 2017, 14(04): 671-701,

[47.] Kolb O, Lang J, Bales P. An implicit box scheme for subsonic compressible flow with dissipative source term. Numer. Algorithms, 2010, 53(2/3): 293-307,

[48.] Kolbe N. Numerical relaxation limit and outgoing edges in a central scheme for networked conservation laws. PAMM, 2023, 23(1): 202200150,

[49.] Liu T-P. Hyperbolic conservation laws with relaxation. Commun. Math. Phys., 1987, 108(1): 153-175,

[50.] Mishra S. Numerical methods for conservation laws with discontinuous coefficients. Handbook of Numerical Analysis, 2017 Amsterdam Elsevier 479-506,

[51.] Müller LO, Blanco PJ. A high order approximation of hyperbolic conservation laws in networks: application to one-dimensional blood flow. J. Comput. Phys., 2015, 300: 423-437,

[52.] Müller S, Voss A. The Riemann problem for the Euler equations with nonconvex and nonsmooth equation of state: construction of wave curves. SIAM J. Sci. Comput., 2006, 28(2): 651-681,

[53.] Reigstad GA, Flåtten T, Erland Haugen N, Ytrehus T. Coupling constants and the generalized Riemann problem for isothermal junction flow. J. Hyperbolic Differ. Equ., 2015, 12(1): 37-59,

[54.] Sikstel, A.: Analysis and numerical methods for coupled hyperbolic conservation laws. Dissertation, RWTH Aachen University, Aachen (2020). https://doi.org/10.18154/RWTH-2020-07821

[55.] Towers JD. An explicit finite volume algorithm for vanishing viscosity solutions on a network. Netw. Heterog. Media, 2022, 17(1): 1,

[56.] Zlotnik, A., Roald, L., Backhaus, S., Chertkov, M., Andersson, G.: Control policies for operational coordination of electric power and natural gas transmission systems. In: 2016 American Control Conference (ACC), pp. 7478–7483 (2016). https://doi.org/10.1109/ACC.2016.7526854

Funding

Deutsche Forschungsgemeinschaft(HE5386/18-1,19-2,22-1,23-1,25-1); Deutsche Forschungsgemeinschaft(ERS SFDdM035); RWTH Aachen University (3131)