A finite volume method for undercompressive shock waves in two space dimensions

Chalons, Christophe; Rohde, Christian; Wiebe, Maria

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1987-2015.

Undercompressive shock waves arise in many physical processes which involve multiple phases. We propose a Finite Volume method in two space dimensions to approximate weak solutions of systems of hyperbolic or hyperbolic-elliptic conservation laws that contain undercompressive shock waves. The method can be seen as a generalization of the spatially one-dimensional and scalar approach in [C. Chalons, P. Engel and C. Rohde, SIAM J. Numer. Anal. 52 (2014) 554–579]. It relies on a moving mesh ansatz such that the undercompressive wave is represented as a sharp interface without any artificial smearing. It is proven that the method is locally conservative and recovers planar traveling wave solutions exactly. To demonstrate the efficiency and reliability of the new scheme we test it on scalar model problems and as an application on compressible liquid-vapour flow in two space dimensions.

R. Abeyaratne and J.K. Knowles, Kinetic relations and the propagation of phase boundaries in solids. Arch. Ration. Mech. Anal. 114 (1991) 119–154. | DOI | MR | Zbl

A.L. Bertozzi, A. Münch and M. Shearer, Undercompressive shocks in thin film flows. Phys. D 134 (1999) 431–464. | DOI | MR | Zbl

B. Boutin, C. Chalons, F. Lagoutière and P.G. Lefloch, Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I. Interfaces Free Bound. 10 (2008) 399–421. | DOI | MR | Zbl

C. Chalons, Transport-equilibrium schemes for computing nonclassical shocks. Scalar conservation laws. Numer. Methods Partial Differ. Equ. 24 (2008)1127–1147. | DOI | MR | Zbl

C. Chalons, F. Coquel, P. Engel and C. Rohde, Fast relaxation solvers for hyperbolic-elliptic phase transition problems. SIAM J. Sci. Comput. 34 (2012A) 1753–A1776. | DOI | MR | Zbl

C. Chalons, P. Engel and C. Rohde, A conservative and convergent scheme for undercompressive shock waves. SIAM J. Numer. Anal. 52 (2014) 554–579. | DOI | MR | Zbl

C. Chalons and P.G. Lefloch, Computing undercompressive waves with the random choice scheme. Nonclassical shock waves. Interfaces Free Bound. 5 (2003) 129–158. | DOI | MR | Zbl

C. Chen and H. Hattori, Exact Riemann solvers for conservation laws with phase change. Appl. Numer. Math., 94 (2015) 222–240. | DOI | MR | Zbl

A. Chertock, S. Karni and A. Kurganov, Interface tracking method for compressible multifluids. ESAIM: M2AN 42 (2008) 991–1019. | DOI | Numdam | MR | Zbl

R.M. Colombo and F.S. Priuli, Characterization of Riemann solvers for the two phase $p$-system. Commun. Partial Differ. Equ. 28 (2003) 1371–1389. | DOI | MR | Zbl

R.M. Colombo and M.D. Rosini, Pedestrian flows and non-classical shocks. Math. Methods Appl. Sci. 28 (2005) 1553–1567. | DOI | MR | Zbl

Y. Di, R. Li, T. Tang and P. Zhang, Moving mesh finite element methods for the incompressible Navier-Stokes equations. SIAM J. Sci. Comput. 26 (2005) 1036–1056. | DOI | MR | Zbl

A. Dressel and C. Rohde, A finite-volume approach to liquid-vapour fluids with phase transition. In Finite volumes for complex Applications V. Iste, London (2008) 53–68. | MR

S. Fechter, C.-D. Munz, C. Rohde and C. Zeiler, A sharp interface method for compressible liquidvapor flow with phase transition and surface tension. J. Comput. Phys. 336 (2017) 347–374. | DOI | MR | Zbl

R.P. Fedkiw, The ghost fluid method for numerical treatment of discontinuities and interfaces. Godunov Methods. Edited by E. Toro. Springer US (2001) 309–317. | MR | Zbl

M. Hantke, W. Dreyer and G. Warnecke, Exact solutions to the Riemann problem for compressible isothermal Euler equations for two-phase flows with and without phase transition. Quart. Appl. Math. 71 (2013) 509–540. | DOI | MR | Zbl

A. Harten and J. M. Hyman, Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comput. Phys. 50 (1983) 235–269. | DOI | MR | Zbl

B.T. Hayes and P.G. Lefloch, Nonclassical shocks and kinetic relations: finite difference schemes. SIAM J. Numer. Anal. 35 (1998) 2169–2194. | DOI | MR | Zbl

T.Y. Hou, P. Rosakis and P. Lefloch, A level-set approach to the computation of twinning and phase-transition dynamics. J. Comput. Phys. 150 (1999) 302–331. | DOI | MR | Zbl

F. Kissling and C. Rohde, The computation of nonclassical shock waves in porous media with a heterogeneous multiscale method: The multidimensional case. Multiscale Model. Simul. 13 (2015) 1507–1541. | DOI | MR | Zbl

A. Kluwick, E. Cox and S. Scheichl, Non-classical kinematic shocks in suspensions of particles in fluids. Acta Mech. 144 (2000) 197–210. | DOI | Zbl

D. Kröner, Numerical schemes for conservation laws. Wiley-Teubner Series Advances in Numerical Mathematics. John Wiley and Sons Ltd., Chichester (1997). | MR | Zbl

P.G. Lefloch, Propagating phase boundaries: formulation of the problem and existence via the Glimm method. Arch. Rational Mech. Anal. 123 (1993) 153–197. | DOI | MR | Zbl

P.G. LeFloch, Hyperbolic systems of conservation laws. The theory of classical and nonclassical shock waves. Lecture in Mathematics. ETH Zürich, Birkhäuser, Basel (2002). | MR | Zbl

P.G. Lefloch and C. Rohde, High-order schemes, entropy inequalities, and nonclassical shocks. SIAM J. Numer. Anal. 37 (2000) 2023–2060. | DOI | MR | Zbl

P.G. Lefloch and M.D. Thanh, Non-classical Riemann solvers and kinetic relations. II. An hyperbolic-elliptic model of phase-transition dynamics. Proc. R. Soc. Edinburgh Sect. A 132 (2002) 181–219. | DOI | MR | Zbl

R.J. LeVeque, Finite volume methods for hyperbolic problems, vol. 31. Cambridge university press (2002). | MR | Zbl

R. Li and T. Tang, Moving mesh discontinuous Galerkin method for hyperbolic conservation laws. J. Sci. Comput. 27 (2006) 347–363. | DOI | MR | Zbl

C. Merkle and C. Rohde, Computation of dynamical phase transitions in solids. Appl. Numer. Math. 56 (2006) 1450–1463. | DOI | MR | Zbl

C. Merkle and C. Rohde, The sharp-interface approach for fluids with phase change: Riemann problems and ghost fluid techniques. ESAIM: M2AN 41 (2007) 1089–1123. | DOI | Numdam | MR | Zbl

J. Ning, X. Yuan, T. Ma and C. Wang, Positivity-preserving moving mesh scheme for two-step reaction model in two dimensions. Comput. Fluids 123 (2015) 72–86. | DOI | MR | Zbl

C. Rohde and C. Zeiler, A relaxation Riemann solver for compressible two-phase flow with phase transition and surface tension. Appl. Numer. Math. 95 (2015) 267–279. | DOI | MR | Zbl

R. Sanders, The moving grid method for nonlinear hyperbolic conservation laws. SIAM J. Numer. Anal. 22 (1985) 713–728. | DOI | MR | Zbl

J. M. Stockie, J. A. Mackenzie and R.D. Russell, A moving mesh method for one-dimensional hyperbolic conservation laws. SIAM J. Sci. Comput. 22 (2000) 1791–1813. | DOI | MR | Zbl

F. Svensson, Moving meshes and higher order finite volume reconstructions. Int. J. Finite Vol. 3 (2006) 27. | MR | Zbl

H. Tang and T. Tang. Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws. SIAM J. Numer. Anal. 41 (2003) 487–515. | DOI | MR | Zbl

The CGAL Project, CGAL User and Reference Manual. CGAL Editorial Board, 4.7 edition (2016).

L. Truskinovsky, Kinks versus shocks. In Shock Induced Transitions and Phase Structures in General Media, edited by J. Dunn. Vol. 52 of The IMA Volumes in Mathematics and its Applications. Springer (1993) 185–229. | MR | Zbl

C.J. Van Duijn, L.A. Peletier and I.S. Pop, A new class of entropy solutions of the Buckley-Leverett equation. SIAM J. Math. Anal. 39 (2007) 507–536. | DOI | MR | Zbl

M. Yvinec, 2D triangulation. In CGAL User and Reference Manual. CGAL Editorial Board, 4.7 edition (2015).

C. Zeiler, Liquid Vapor Phase Transitions: Modeling, Riemann Solvers and Computation. Ph.D. thesis. Universität Stuttgart (2015).

X. Zhong, T.Y. Hou and P.G. Lefloch, Computational methods for propagating phase boundaries. J. Comput. Phys. 124 (1996) 192–216. | DOI | MR | Zbl