Abstract
The HLLC (Harten–Lax–van Leer contact) approximate Riemann solver for computing solutions to hyperbolic systems by means of finite volume and discontinuous Galerkin methods is reviewed. HLLC was designed, as early as 1992, as an improvement to the classical HLL (Harten–Lax–van Leer) Riemann solver of Harten, Lax, and van Leer to solve systems with three or more characteristic fields, in order to avoid the excessive numerical dissipation of HLL for intermediate characteristic fields. Such numerical dissipation is particularly evident for slowly moving intermediate linear waves and for long evolution times. High-order accurate numerical methods can, to some extent, compensate for this shortcoming of HLL, but it is a costly remedy and for stationary or nearly stationary intermediate waves such compensation is very difficult to achieve in practice. It is therefore best to resolve the problem radically, at the first-order level, by choosing an appropriate numerical flux. The present paper is a review of the HLLC scheme, starting from some historical notes, going on to a description of the algorithm as applied to some typical hyperbolic systems, and ending with an overview of some of the most significant developments and applications in the last 25 years.
Similar content being viewed by others
References
Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983). https://doi.org/10.1137/1025002
Toro, E.F.: The weighted average flux method applied to the Euler equations. Philos. Trans. R. Soc. Lond. Ser. A Phys. Sci. Eng. A341, 499–530 (1992). https://doi.org/10.1098/rsta.1992.0113
Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL–Riemann solver. Technical Report CoA—9204, Department of Aerospace Science, College of Aeronautics, Cranfield Institute of Technology, UK (1992)
Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4, 25–34 (1994). https://doi.org/10.1007/BF01414629
Toro, E.F., Chakraborty, A.: Development of an approximate Riemann solver for the steady supersonic Euler equations. Aeronaut. J. 98, 325–339 (1994). https://doi.org/10.1017/S0001924000026890
Fraccarollo, L., Toro, E.F.: Experimental and numerical assessment of the shallow water model for two-dimensional dam-break type problems. J. Hydraul. Res. 33, 843–864 (1995). https://doi.org/10.1080/00221689509498555
Batten, P., Clarke, N., Lambert, C., Causon, D.: On the choice of wavespeeds for the HLLC Riemann solver. SIAM J. Sci. Stat. Comput. 18, 1553–1570 (1997). https://doi.org/10.1137/S1064827593260140
Batten, P., Leschziner, M.A., Goldberg, U.C.: Average-state Jacobians and implicit methods for compressible viscous and turbulent flows. J. Comput. Phys. 137, 38–78 (1997). https://doi.org/10.1006/jcph.1997.5793
Davis, S.F.: Simplified second-order Godunov-type methods. SIAM J. Sci. Stat. Comput. 9, 445–473 (1988). https://doi.org/10.1137/0909030
Einfeldt, B.: On Godunov-type methods for gas dynamics. SIAM J. Numer. Anal. 25(2), 294–318 (1988). https://doi.org/10.1137/0725021
Einfeldt, B., Munz, C.D., Roe, P.L., Sjögreen, B.: On Godunov-type methods near low densities. J. Comput. Phys. 92, 273–295 (1991). https://doi.org/10.1016/0021-9991(91)90211-3
Dumbser, M., Balsara, D.: A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems. J. Comput. Phys. 304, 275–319 (2016). https://doi.org/10.1016/j.jcp.2015.10.014
Toro, E.F.: Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley, Chichester (2001)
Toro, E.F.: Brain venous haemodynamics, neurological diseases and mathematical modelling. A review. Appl. Math. Comput. 272, 542–579 (2016). https://doi.org/10.1016/j.amc.2015.06.066
Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edn. Springer, Berlin (2009). https://doi.org/10.1007/b79761
Liou, M.S., Steffen, C.J.: A new flux splitting scheme. J. Comput. Phys. 107, 23–39 (1993). https://doi.org/10.1006/jcph.1993.1122
Toro, E.F., Vázquez-Cendón, M.E.: Flux splitting schemes for the Euler equations. Comput. Fluids 70, 1–12 (2012). https://doi.org/10.1016/j.compfluid.2012.08.023
Toro, E.F., Castro, C.E., Lee, B.J.: A novel numerical flux for the 3D Euler equations with general equation of state. J. Comput. Phys. 303, 80–94 (2015). https://doi.org/10.1016/j.jcp.2015.09.037
Tokareva, S.A., Toro, E.F.: A flux splitting method for the Baer–Nunziato equations of compressible two-phase flow. J. Comput. Phys. 323, 45–74 (2016). https://doi.org/10.1016/j.jcp.2016.07.019
Balsara, D.S., Montecinos, G.I., Toro, E.F.: Exploring various flux vector splittings for the magnetohydrodynamic system. J. Comput. Phys. 311, 1–21 (2016). https://doi.org/10.1016/j.jcp.2016.01.029
Steger, J.L., Warming, R.F.: Flux vector splitting of the inviscid gasdynamic equations with applications to finite-difference methods. J. Comput. Phys. 40, 263–293 (1981). https://doi.org/10.1016/0021-9991(81)90210-2
van Leer, B.: Flux-vector splitting for the Euler equations. Technical Report ICASE 82–30, NASA Langley Research Center, USA (1982)
Zha, G.-C., Bilgen, E.: Numerical solution of Euler equations by a new flux vector splitting scheme. Int. J. Numer. Methods Fluids 17, 115–144 (1993). https://doi.org/10.1002/fld.1650170203
Toro, E.F.: The Riemann problem: solvers and numerical fluxes. In: Abgrall, R., Shu, C.W. (eds.) Elsevier Handbook of Numerical Methods for Hyperbolic Problems. Chapter 2, vol. 17, pp. 19–54 (2016). https://doi.org/10.1016/bs.hna.2016.09.015
Godunov, S.K.: A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Sb. Math. 47, 357–393 (1959)
Toro, E.F.: Riemann problems and the WAF method for solving two-dimensional shallow water equations. Philos. Trans. R. Soc. Lond. Ser. A Phys. Sci. Eng. A338, 43–68 (1992). https://doi.org/10.1098/rsta.1992.0002
Formaggia, L., Quarteroni, A., Veneziani, A. (eds.): Cardiovascular Mathematics. Modeling and Simulation of the Circulatory System. Springer, Berlin (2009). https://doi.org/10.1007/978-88-470-1152-6
Müller, L.O., Toro, E.F.: A global multiscale model for the human circulation with emphasis on the venous system. Int. J. Numer. Methods Biomed. Eng. 30(7), 681–725 (2014). https://doi.org/10.1002/cnm.2622
Müller, L.O., Toro, E.F.: Enhanced global mathematical model for studying cerebral venous blood flow. J. Biomech. 47(13), 3361–3372 (2014). https://doi.org/10.1016/j.jbiomech.2014.08.005
Safranov, A.V.: Difference method for gasdynamical equations based on the jump conditions. Math. Modell. 20, 76–84 (2008). (in Russian)
Baer, M.R., Nunziato, J.W.: A Two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. J. Multiphase Flow 12, 861–889 (1986). https://doi.org/10.1016/0301-9322(86)90033-9
Tokareva, S.A., Toro, E.F.: HLLC-type Riemann solver for the Baer–Nunziato equations of compressible two-phase flow. J. Comput. Phys. 229, 3573–3604 (2010). https://doi.org/10.1016/j.jcp.2010.01.016
Gurski, K.F.: An HLLC-type approximate Riemann solver for ideal magnetohydrodynamics. SIAM J. Sci. Comput. 25(6), 2165–2187 (2004). https://doi.org/10.1137/S1064827502407962
Li, S.: An HLLC Riemann solver for magneto-hydrodynamics. J. Comput. Phys. 203(1), 344–357 (2005). https://doi.org/10.1016/j.jcp.2004.08.020
Mignone, A., Bodo, G.: An HLLC Riemann solver for relativistic flows - II. Magnetohydrodynamics. Mon. Not. R. Astron. Soc. 368(3), 1040–1054 (2006). https://doi.org/10.1111/j.1365-2966.2006.10162.x
Bouchut, F., Klingenberg, C., Waagan, K.: A multiwave approximate Riemann solver for ideal MHD based on relaxation, I: theoretical framework. Numer. Math. 108(1), 7–42 (2007). https://doi.org/10.1007/s00211-007-0108-8
Klingenberg, C., Schmidt, W., Waagan, K.: Numerical comparison of Riemann solvers for astrophysical hydrodynamics. J. Comput. Phys. 227(1), 12–35 (2007). https://doi.org/10.1016/j.jcp.2007.07.034
Honkkila, V., Janhunen, P.: HLLC solver for ideal relativistic MHD. J. Comput. Phys. 223(2), 643–656 (2007). https://doi.org/10.1016/j.jcp.2006.09.027
Fetcher, S., Munz, C.D., Rohde, C., Zeiler, C.: Approximate Riemann solver for compressible liquid vapor flow with phase transition and surface tension. Comput. Fluids 169, 169–185 (2018). https://doi.org/10.1016/j.compfluid.2017.03.026
Prebeg, M., FlĂĄtten, T., MĂĽller, B.: Large time step HLL and HLLC schemes. ESAIM: M2AN (2017). https://doi.org/10.1051/m2an/2017051
Pelanti, M.: Wave structure similarity of the HLLC and ROE Riemann solvers: application to low Mach number preconditioning. SIAM J. Sci. Comput. 40(3), A1836–A1859 (2018). https://doi.org/10.1137/17M1154965
Sweby, P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21, 995–1011 (1984). https://doi.org/10.1137/0721062
Harten, A., Osher, S.: Uniformly high-order accurate nonoscillatory schemes I. SIAM J. Numer. Anal. 24(2), 279–309 (1987). https://doi.org/10.1137/0724022
Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996). https://doi.org/10.1006/jcph.1996.0130
Balsara, D., Shu, C.W.: Monotonicity preserving weighted essentially nonoscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160, 405–452 (2000). https://doi.org/10.1006/jcph.2000.6443
Titarev, V.A., Toro, E.F.: Finite-volume WENO schemes for three-dimensional conservation laws. J. Comput. Phys. 201(1), 238–260 (2004). https://doi.org/10.1016/j.jcp.2004.05.015
Dumbser, M., Käser, M.: Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J. Comput. Phys. 221(2), 693–723 (2007). https://doi.org/10.1016/j.jcp.2006.06.043
Dumbser, M., Käser, M., Titarev, V.A., Toro, E.F.: Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J. Comput. Phys. 226(8), 204–243 (2007). https://doi.org/10.1016/j.jcp.2007.04.004
van der Vegt, J.J.W., van der Ven, H.: Space–time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows: I. General formulation. J. Comput. Phys. 182(2), 546–585 (2002). https://doi.org/10.1006/jcph.2002.7185
Pesch, L., van der Vegt, J.J.W.: A discontinuous Galerkin finite element discretization of the Euler equations for compressible and incompressible fluids. J. Comput. Phys. 227(11), 5426–5446 (2008). https://doi.org/10.1016/j.jcp.2008.01.046
Luo, H., Baum, J.D., Löhner, R.: A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids. J. Comput. Phys. 225(1), 686–713 (2007). https://doi.org/10.1016/j.jcp.2006.12.017
Wang, L., Mavriplis, D.J.: Implicit solution of the unsteady Euler equations for high-order accurate discontinuous Galerkin discretizations. J. Comput. Phys. 225(2), 1994–2015 (2007). https://doi.org/10.1016/j.jcp.2007.03.002
Qiu, J., Khoo, B.C., Shu, C.W.: A numerical study for the performance of the Runge–Kutta discontinuous Galerkin method based on different numerical fluxes. J. Comput. Phys. 212(2), 540–565 (2006). https://doi.org/10.1016/j.jcp.2005.07.011
Capdeville, G.: Towards a compact high-order method for non-linear hyperbolic systems, II. The Hermite-HLLC scheme. J. Comput. Phys. 227(22), 9428–9462 (2008). https://doi.org/10.1016/j.jcp.2008.06.024
Harris, R., Wang, Z.J., Liu, Y.: Efficient quadrature-free high-order spectral volume method on unstructured grids: Theory and 2D. J. Comput. Phys. 227(3), 1620–1642 (2008). https://doi.org/10.1016/j.jcp.2007.09.012
Li, L., Liu, X., Lou, J., Luo, H., Nishikawa, H., Ren, Y.: A discontinuous Galerkin method based on variational reconstruction for compressible flows on arbitrary grids. 2018 AIAA Aerospace Sciences Meeting, Kissimmee, Florida, AIAA Paper 2018-0831 (2018). https://doi.org/10.2514/6.2018-0831
Simon, S., Mandal, J.C.: A cure for numerical shock instability in HLLC Riemann solver using antidiffusion control. Comput. Fluids 174, 144–166 (2018). https://doi.org/10.1016/j.compfluid.2018.07.001
Balsara, D.S., Dumbser, M., Abgrall, R.: Multidimensional HLLC Riemann solver for unstructured meshes—With application to Euler and MHD flows. J. Comput. Phys. 261, 172–208 (2014). https://doi.org/10.1016/j.jcp.2013.12.029
Ambati, V.R., Bokhove, O.: Space–time discontinuous Galerkin discretization of rotating shallow water equations. J. Comput. Phys. 225(2), 1233–1261 (2007). https://doi.org/10.1016/j.jcp.2007.01.036
Castro DĂaz, M.J., Fernandez-Nieto, E.D., Morales de Luna, T., Narbona-Reina, G., ParĂ©s, C.: A HLLC scheme for non-conservative hyperbolic problems. Application to turbidity currents with sediment transport. ESAIM Math. Modell. Numer. Anal. 47(2), 1–32 (2013). https://doi.org/10.1051/m2an/2012017
Hosseinzadeh-Tabrizi, S.A., Ghaeini-Hessaroeyeh, M.: Application of bed load formulations for dam failure and overtopping. Civ. Eng. J. 3(10), 997–1007 (2017). https://doi.org/10.28991/cej-030932
Ziaeddini-Dashtkhaki, M., Ghaeini-Hessaroeyeh, M.: Numerical simulation of tidal wave over wavy bed. J. Coast. Mar. Eng. 1(1), 7–12 (2018)
Wells, B.V., Baines, M.J., Glaister, P.: Generation of arbitrary Lagrangian–Eulerian (ALE) velocities, based on monitor functions, for the solution of compressible fluid equations. Int. J. Numer. Methods Fluids 47, 1375–1381 (2005). https://doi.org/10.1002/fld.915
Nemec, A., Aftosmis, M.J.: Adjoint sensitivity computations for an embedded-boundary Cartesian mesh method. J. Comput. Phys. 227(4), 2724–2742 (2008). https://doi.org/10.1016/j.jcp.2007.11.018
Ball, G.J., East, R.A.: Shock and blast attenuation by aqueous foam barriers: influences of barrier geometry. Shock Waves 9(1), 37–47 (1999). https://doi.org/10.1007/s001930050137
Navarro-Martinez, S., Tutty, O.R.: Numerical simulation of Görtler vortices in hypersonic compression ramps. Comput. Fluids 34(2), 225–247 (2005). https://doi.org/10.1016/j.compfluid.2004.05.002
Berthon, C., Charrier, P., Dubroca, B.: An HLLC scheme to solve The \(M_1\) model of radiative transfer in two space dimensions. J. Sci. Comput. 31(3), 347–389 (2007). https://doi.org/10.1007/s10915-006-9108-6
Berthon, C., Coquel, F., Hérard, J.M., Uhlmann, M.: An approximate solution of the Riemann problem for a realisable second-moment turbulent closure. Shock Waves 11(4), 245–269 (2002). https://doi.org/10.1007/s001930100109
Gavrilyuk, S.L., Favrie, N., Saurel, R.: Modelling wave dynamics of compressible elastic materials. J. Comput. Phys. 227(5), 2941–2969 (2007). https://doi.org/10.1016/j.jcp.2007.11.030
Ohwada, T., Shibata, Y., Kato, T., Nakamura, T.: A simple, robust and efficient high-order accurate shock-capturing scheme for compressible flows: Towards minimalism. J. Comput. Phys. 362, 131–162 (2018). https://doi.org/10.1016/j.jcp.2018.02.019
White, J.A., Baurle, R.A., Passe, B.J., Spiegel, S.C., Nishikawa, H.: Geometrically flexible and efficient flow analysis of high speed vehicles via domain decomposition, Part 1: unstructured-grid solver for high speed flows. Conference: JANNAF 48th Combustion 36th Airbreathing Propulsion, pp. 1–22 (2017)
Pantano, C., Saurel, R., Schmitt, T.: An oscillation free shock-capturing method for compressible van der Waals supercritical fluid flows. J. Comput. Phys. 335, 780–811 (2017). https://doi.org/10.1016/j.jcp.2017.01.057
Daude, F., Tijsseling, A.S., Galon, P.: Numerical investigations of water-hammer with column-separation induced by vaporous cavitation using a one-dimensional finite-volume approach. J. Fluids Struct. 83, 91–118 (2018). https://doi.org/10.1016/j.jfluidstructs.2018.08.014
Sousa, J., Paniagua, G., Morata, E.C.: Thermodynamic analysis of a gas turbine engine with a rotating detonation combustor. Appl. Energy 195, 247–256 (2017). https://doi.org/10.1016/j.apenergy.2017.03.045
Knudsen, E., Doran, E.M., Mittal, V., Meng, J., Spurlock, W.: Compressible Eulerian needle-to-target large eddy simulations of a diesel fuel injector. Proc. Combust. Inst. 36(2), 2459–2466 (2017). https://doi.org/10.1016/j.proci.2016.08.016
Garrick, D.P., Owkes, M., Regele, J.D.: A finite-volume HLLC-based scheme for compressible interfacial flows with surface tension. J. Comput. Phys. 339, 46–67 (2017). https://doi.org/10.1016/j.jcp.2017.03.007
Fujisawa, K., Yamagata, T., Fujisawa, N.: Damping effect on impact pressure from liquid droplet impingement on wet wall. Ann. Nucl. Energy 121, 260–268 (2018). https://doi.org/10.1016/j.anucene.2018.07.008
Godunov, S.K., Klyuchinskiy, D.V., Safronov, A.V., Fortova, S.V., Shepelev, V.V.: Experimental study of numerical methods for the solution of gas dynamics problems with shock waves. J. Phys.: Conf. Ser. 946, 012048 (2018). https://doi.org/10.1088/1742-6596/946/1/012048
Godunov, S.K., Klyuchinskii, D.V., Fortova, S.V., Shepelev, V.V.: Experimental studies of difference gas dynamics models with shock waves. Comput. Math. Math. Phys. 58(8), 1201–1216 (2018). https://doi.org/10.1134/S0965542518080067
Varma, D., Chandrashekar, P.: A second-order well-balanced finite volume scheme for Euler equations with gravity. Comput. Fluids 181, 292–313 (2019). https://doi.org/10.1016/j.compfluid.2019.02.003
Wilkinson, S.D., Braithwaite, M., Nikiforakis, N., Michael, L.: A complete equation of state for non-ideal condensed phase explosives. J. Appl. Phys. 122(22), 225112 (2017). https://doi.org/10.1063/1.5006901
Goncalves, E., Hoarau, Y., Zeidan, D.: Simulation of shock-induced bubble collapse using a four-equation model. Shock Waves 29(1), 221–234 (2018). https://doi.org/10.1007/s00193-018-0809-1
Thornber, B., Griffond, J., Poujade, O., Attal, N., Varshochi, H., Bigdelou, P., Ramaprabhu, P., Olson, B., Greenough, J., Zhou, Y., et al.: Late-time growth rate, mixing, and anisotropy in the multimode narrowband Richtmyer–Meshkov instability: The \(\theta \)-group collaboration. Phys. Fluids 29(10), 105107 (2017). https://doi.org/10.1063/1.4993464
Joncquieres, V., Pechereau, F., Alvarez Laguna, A., Bourdon, A., Vermorel, O., Cuenot, B.: A 10-moment fluid numerical solver of plasma with sheaths in a Hall Effect Thruster. 2018 Joint Propulsion Conference, Cincinnati, Ohio, AIAA Paper 2018-4905 (2018). https://doi.org/10.2514/6.2018-4905
Wermelinger, F., Rasthofer, U., Hadjidoukas, P.E., Koumoutsakos, P.: Petascale simulations of compressible flows with interfaces. J. Comput. Sci. 26, 217–225 (2018). https://doi.org/10.1016/j.jocs.2018.01.008
Sangam, A.: An HLLC scheme for ten-moments approximation coupled with magnetic field. Int. J. Comput. Sci. Math. 2(1/2), 73–109 (2008). https://doi.org/10.1504/IJCSM.2008.019724
Singh, A.P.: A framework to improve turbulence models using full-field inversion and machine learning. PhD Thesis, Aerospace Engineering, The University of Michigan, USA (2018)
Ritos, K., Kokkinakis, I.W., Drikakis, D., Spottswood, S.M.: Implicit large eddy simulation of acoustic loading in supersonic turbulent boundary layers. Phys. Fluids 29(4), 046101 (2017). https://doi.org/10.1063/1.4979965
Walchli, B., Thornber, B.: Reynolds number effects on the single-mode Richtmyer–Meshkov instability. Phys. Rev. E 95(1), 013104 (2017). https://doi.org/10.1103/PhysRevE.95.013104
Islam, A., Thornber, B.: A high-order hybrid turbulence model with implicit large-eddy simulation. Comput. Fluids 167, 292–312 (2018). https://doi.org/10.1016/j.compfluid.2018.03.031
Park, M.A., Barral, N., Ibanez, ., Kamenetskiy, D.S., Krakos, J.A., Michal, T.R., Loseille, A.: Unstructured grid adaptation and solver technology for turbulent flows. 2018 AIAA Aerospace Sciences Meeting, Kissimmee, Florida, AIAA Paper 2018-1103 (2018). https://doi.org/10.2514/6.2018-1103
Pan, L., Padoan, P., Nordlund, Ă….: Detailed balance and exact results for density fluctuations in supersonic turbulence. Astrophys. J. Lett. 866, L17 (2018). https://doi.org/10.3847/2041-8213/aae57c
Hahn, M., Drikakis, D.: Large eddy simulation of compressible turbulence using high-resolution methods. Int. J. Numer. Methods Fluids 47, 971–977 (2005). https://doi.org/10.1002/fld.882
Kalveit, M., Drikakis, D.: Coupling strategies for hybrid molecular—continuum simulation methods. Proc. IMechE Part C J. Mech. Eng. Sci. 222, 797–806 (2008). https://doi.org/10.1243/09544062JMES716
Hahn, M., Drikakis, D.: Implicit large-eddy simulation of swept wing flow using high-resolution methods. AIAA J. 47, 618–629 (2009). https://doi.org/10.2514/1.37806
Pelanti, M.: Low Mach number preconditioning techniques for Roe-type and HLLC-type methods for a two-phase compressible flow model. Appl. Math. Comput. 310, 112–133 (2017). https://doi.org/10.1016/j.amc.2017.04.014
Saurel, R., Pantano, C.: Diffuse-interface capturing methods for compressible two-phase flows. Annu. Rev. Fluid Mech. 50, 105–130 (2018). https://doi.org/10.1146/annurev-fluid-122316-050109
Pan, S., Han, L., Hu, X., Adams, N.: A conservative sharp-interface method for compressible multi-material flows. J. Comput. Phys. 371, 870–895 (2018). https://doi.org/10.1016/j.jcp.2018.02.007
Daude, F., Galon, P.: A finite-volume approach for compressible single- and two-phase flows in flexible pipelines with fluid–structure interaction. J. Comput. Phys. 362, 375–408 (2018). https://doi.org/10.1016/j.jcp.2018.01.055
De Lorenzo, M.: Modelling and numerical simulation of metastable two-phase flows. PhD Thesis, Université Paris-Saclay (2018)
De Lorenzo, M., Pelanti, M., Lafon, P.: HLLC-type and path-conservative schemes for a single-velocity six-equation two-phase flow model: A comparative study. Appl. Math. Comput. 333, 95–117 (2018). https://doi.org/10.1016/j.amc.2018.03.092
Zheng, H.W., Shu, C., Chew, Y.T.: An object-oriented and quadrilateral-mesh based solution adaptive algorithm for compressible multi-fluid flows. J. Comput. Phys. 227(14), 6895–6921 (2008). https://doi.org/10.1016/j.jcp.2008.03.037
Tan, X.G., Przekwas, A.J., Gupta, R.K.: Computational modeling of blast wave interaction with a human body and assessment of traumatic brain injury. Shock Waves 27, 889–904 (2017). https://doi.org/10.1007/s00193-017-0740-x
Paxton, B., Schwab, J., Bauer, E.B., Bildsten, L., Blinnikov, S., Paul Duffell, R., Farmer, J.A., Goldberg, P.M., Sorokina, E., et al.: Modules for experiments in stellar astrophysics (MESA): Convective boundaries, element diffusion, and massive star explosions. Astrophys. J. Suppl. Ser. 234(2), 34 (2018). https://doi.org/10.3847/1538-4365/aaa5a8
Schneider, E.E., Robertson, B.E.: Hydrodynamical coupling of mass and momentum in multiphase galactic winds. Astrophys. J. 834(2), 144 (2017). https://doi.org/10.3847/1538-4357/834/2/144
Trebitsch, M., Blaizot, J., Rosdahl, J., Devriendt, J., Slyz, A.: Fluctuating feedback-regulated escape fraction of ionizing radiation in low-mass, high-redshift galaxies. Mon. Not. R. Astron. Soc. 470(1), 224–239 (2017). https://doi.org/10.1093/mnras/stx1060
Padnos, D., Mandelker, N., Birnboim, Y., Dekel, A., Krumholz, M.R., Steinberg, E.: Instability of supersonic cold streams feeding galaxies—II. Non-linear evolution of surface and body modes of Kelvin–Helmholtz instability. Mon. Not. R. Astron. Soc. 477(3), 2933–2968 (2018). https://doi.org/10.1093/mnras/sty789
Radice, D., Burrows, A., Vartanyan, D., Skinner, M.A., Dolence, J.C.: Electron-capture and low-mass iron-core-collapse supernovae: New neutrino-radiation–hydrodynamics simulations. Astrophys. J. 850(1), 43 (2017). https://doi.org/10.3847/1538-4357/aa92c5
Ohlmann, S.T., Röpke, F.K., Pakmor, R., Springel, V.: Constructing stable 3D hydrodynamical models of giant stars. Astron. Astrophys. 599, A5 (2017). https://doi.org/10.1051/0004-6361/201629692
Rosdahl, J., Katz, H., Blaizot, J., Kimm, T., Michel-Dansac, L., Garel, T., Haehnelt, M., Ocvirk, P., Teyssier, R.: The SPHINX cosmological simulations of the first billion years: the impact of binary stars on reionization. Mon. Not. R. Astron. Soc. 479(1), 994–1016 (2018). https://doi.org/10.1093/mnras/sty1655
Cielo, S., Bieri, R., Volonteri, M., Wagner, A.Y., Dubois, Y.: AGN feedback compared: jets versus radiation. Mon. Not. R. Astron. Soc. 477(1), 1336–1355 (2018). https://doi.org/10.1093/mnras/sty708
Bambic, C.J., Morsony, B.J., Reynolds, C.S.: Suppression of AGN-driven turbulence by magnetic fields in a magnetohydrodynamic model of the intracluster medium. Astrophys. J. 857(2), 84 (2018). https://doi.org/10.3847/1538-4357/aab558
Beckmann, R.S.: From seed to supermassive: simulating the origin, evolution and impact of massive black holes. PhD Thesis, University of Oxford (2017)
Miranda-Aranguren, S., Aloy, M.A., Rembiasz, T.: An HLLC Riemann solver for resistive relativistic magnetohydrodynamics. Mon. Not. R. Astron. Soc. 476(3), 3837–3860 (2018). https://doi.org/10.1093/mnras/sty419
Scannapieco, E., Safarzadeh, M.: Modeling star formation as a Markov process in a supersonic gravoturbulent medium. Astrophys. J. Lett. 865(2), L14 (2018). https://doi.org/10.3847/2041-8213/aae1f9
Leroy, M.H.J., Keppens, R.: On the influence of environmental parameters on mixing and reconnection caused by the Kelvin–Helmholtz instability at the magnetopause. Phys. Plasmas 24(1), 012906 (2017). https://doi.org/10.1063/1.4974758
Rasthofer, U., Wermelinger, F., Hadijdoukas, P., Koumoutsakos, P.: Large scale simulation of cloud cavitation collapse. Procedia Comput. Sci. 108, 1763–1772 (2017). https://doi.org/10.1016/j.procs.2017.05.158
Navarro, A., Lora-Clavijo, F.D., González, G.A.: Magnus: A new resistive MHD code with heat flow terms. Astrophys. J. 844(1), 57 (2017). https://doi.org/10.3847/1538-4357/aa7a13
Mignone, A.: MHD modeling: Aims, usage, scales assessed, caveats, codes. In: Torres, D. (ed.) Modelling Pulsar Wind Nebulae. Astrophysics and Space Science Library, vol. 446. Springer, Cham (2017)
Ryan, G.: Numerical simulations of black hole accretion. PhD Thesis, New York University (2017)
Suarez Noguez, T.: Understanding the distribution of gas in the Universe. PhD Thesis, UCL (University College London) (2018)
Harpole, A.: Multiscale modelling of neutron star oceans. PhD Thesis, University of Southampton (2018)
Takahiro, M., Kanya, K.: A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics. J. Comput. Phys. 208(1), 315–344 (2005). https://doi.org/10.1016/j.jcp.2005.02.017
Acknowledgements
The author gratefully acknowledges the significant contribution of M.S. Liou to the field of computational methods for fluid dynamics and applications.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Luo and C.-H. Chang.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Toro, E.F. The HLLC Riemann solver. Shock Waves 29, 1065–1082 (2019). https://doi.org/10.1007/s00193-019-00912-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00193-019-00912-4