Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-30T21:13:38.156Z Has data issue: false hasContentIssue false
  • English
  • Français

Essentially non-oscillatory and weighted essentially non-oscillatory schemes

Published online by Cambridge University Press:  30 November 2020

Chi-Wang Shu
Chi-Wang Shu*
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI02912, USA E-mail: chi-wang_shu@brown.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes were designed for solving hyperbolic and convection–diffusion equations with possibly discontinuous solutions or solutions with sharp gradient regions. The main idea of ENO and WENO schemes is actually an approximation procedure, aimed at achieving arbitrarily high-order accuracy in smooth regions and resolving shocks or other discontinuities sharply and in an essentially non-oscillatory fashion. Both finite volume and finite difference schemes have been designed using the ENO or WENO procedure, and these schemes are very popular in applications, most noticeably in computational fluid dynamics but also in other areas of computational physics and engineering. Since the main idea of the ENO and WENO schemes is an approximation procedure not directly related to partial differential equations (PDEs), ENO and WENO schemes also have non-PDE applications. In this paper we will survey the basic ideas behind ENO and WENO schemes, discuss their properties, and present examples of their applications to different types of PDEs as well as to non-PDE problems.

Type
Research Article
Information
Acta Numerica , Volume 29 , May 2020 , pp. 701 - 762
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

Research supported by NSF grant DMS-1719410.

References

Abbett, W. P. (2007), ‘The magnetic connection between the convection zone and corona in the quiet sun’, Astrophys. J. 665, 1469–1488.CrossRefGoogle Scholar
Abgrall, R. (1994), ‘On essentially non-oscillatory schemes on unstructured meshes: Analysis and implementation’, J. Comput. Phys. 114, 45–58.CrossRefGoogle Scholar
Abgrall, R. (1996), ‘Numerical discretization of the first-order Hamilton–Jacobi equation on triangular meshes’, Comm. Pure Appl. Math. 49, 1339–1373.3.0.CO;2-B>CrossRefGoogle Scholar
Adams, N. A. and Shariff, K. (1996), ‘A high-resolution hybrid compact-ENO scheme for shock–turbulence interaction problems’, J. Comput. Phys. 127, 27–51.CrossRefGoogle Scholar
Aihara, S., Takaki, T. and Takada, N. (2019), ‘Multi-phase-field modeling using a conservative Allen–Cahn equation for multiphase flow’, Comput. Fluids 178, 141–151.CrossRefGoogle Scholar
Balakrishnan, K. and Bellan, J. (2019), ‘A multi-species modeling framework for describing supersonic jet-induced cratering in a granular bed: Cratering on Titan case study’, Internat. J. Multiphase Flow 118, 205–241.CrossRefGoogle Scholar
Balsara, D. S. and Shu, C.-W. (2000), ‘Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy’, J. Comput. Phys. 160, 405–452.CrossRefGoogle Scholar
Banks, J. W., Odu, A. G., Berger, R., Chapman, T., Arrighi, W. and Brunner, S. (2019), ‘High-order accurate conservative finite difference methods for Vlasov equations in 2D+2V’, SIAM J. Sci. Comput. 41, B953–B982.CrossRefGoogle Scholar
Banoo, K., Rhew, J.-H., Lundstrom, M., Shu, C.-W. and Jerome, J. W. (2001), ‘Simulating quasi-ballistic transport in Si nanotransistors’, VLSI Design 13, 5–13.CrossRefGoogle Scholar
Belkacem, K., Kupka, F., Samadi, R. and Grimm-Strele, H. (2019), ‘Solar $p$ -mode damping rates: Insight from a 3D hydrodynamical simulation’, Astronom. Astrophys. 625, A20.CrossRefGoogle Scholar
Ben-Artzi, M., Li, J. and Warnecke, G. (2006), ‘A direct Eulerian GRP scheme for compressible fluid flows’, J. Comput. Phys. 218, 19–43.CrossRefGoogle Scholar
Berger, L., Kleinheinz, K., Attili, A. and Pitsch, H. (2019), ‘Characteristic patterns of thermodiffusively unstable premixed lean hydrogen flames’, Proc. Combustion Institute 37, 1879–1886.CrossRefGoogle Scholar
Berger, M. J., Helzel, C. and LeVeque, R. J. (2003), ‘ $h$ -box methods for the approximation of hyperbolic conservation laws on irregular grids’, SIAM J. Numer. Anal. 41, 893–918.CrossRefGoogle Scholar
Bihs, H., Kamath, A., Aggarwal, A. and Pakozdi, C. (2019a), ‘Efficient wave modeling using nonhydrostatic pressure distribution and free surface tracking on fixed grids’, J . Offshore Mech. Arctic Engrg Trans. ASME 141, 041805.CrossRefGoogle Scholar
Bihs, H., Kamath, A., Chella, M. A. and Arntsen, O. A. (2019b), ‘Extreme wave generation, breaking, and impact simulations using wave packets in REEF3D’, J . Offshore Mech. Arctic Engrg Trans. ASME 141, 041802.CrossRefGoogle Scholar
Borges, R., Carmona, M., Costa, B. and Don, W. S. (2008), ‘An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws’, J. Comput. Phys. 227, 3191–3211.CrossRefGoogle Scholar
Burger, R., Chowell, G., Gavilan, E., Mulet, P. and Villada, L. M. (2019), ‘Numerical solution of a spatio-temporal predator–prey model with infected prey’, Math. Biosci. Engrg 16, 438–473.CrossRefGoogle Scholar
Capdeville, G. (2008), ‘A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes’, J. Comput. Phys. 227, 2977–3014.CrossRefGoogle Scholar
Carlini, E., Ferretti, R. and Russo, G. (2005), ‘A weighted essentially nonoscillatory, large time-step scheme for Hamilton–Jacobi equations’, SIAM J. Sci. Comput. 27, 1071–1091.CrossRefGoogle Scholar
Carrillo, J. A., Gamba, I. M., Majorana, A. and Shu, C.-W. (2003), ‘A WENO-solver for the transients of Boltzmann–Poisson system for semiconductor devices: Performance and comparisons with Monte Carlo methods’, J. Comput. Phys. 184, 498–525.CrossRefGoogle Scholar
Carrillo, J. A., Gamba, I. M., Majorana, A. and Shu, C.-W. (2006), ‘2D semiconductor device simulations by WENO–Boltzmann schemes: Efficiency, boundary conditions and comparison to Monte Carlo methods’, J. Comput. Phys. 214, 55–80.CrossRefGoogle Scholar
Casper, J., Shu, C.-W. and Atkins, H. (1994), ‘Comparison of two formulations for high-order accurate essentially nonoscillatory schemes’, AIAA J. 32, 1970–1977.CrossRefGoogle Scholar
Castro, M. J. and Semplice, M. (2019), ‘Third- and fourth-order well-balanced schemes for the shallow water equations based on the CWENO reconstruction’, Internat. J. Numer. Methods Fluids 89, 304–325.CrossRefGoogle Scholar
Castro, M., Gallardo, J. M. and Parés, C. (2006), ‘High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products: Application to shallow-water systems’, Math. Comput. 75, 1103–1134.CrossRefGoogle Scholar
Castro, M., Costa, B. and Don, W. S. (2011), ‘High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws’, J. Comput. Phys. 230, 1766–1792.CrossRefGoogle Scholar
Cercignani, C., Gamba, I. M., Jerome, J. W. and Shu, C.-W. (1998), ‘Applicability of the high field model: A preliminary numerical study’, VLSI Design 8, 275–282.CrossRefGoogle Scholar
Ceresiat, L., Grosshans, H. and Papalexandris, M. V. (2019), ‘Powder electrification during pneumatic transport: The role of the particle properties and flow rates’, J. Loss Prevent. Process Indust. 58, 60–69.CrossRefGoogle Scholar
Chella, M. A., Bihs, H. and Myrhaug, D. (2019a), ‘Wave impact pressure and kinematics due to breaking wave impingement on a monopile’, J. Fluids Structures 86, 94–123.CrossRefGoogle Scholar
Chella, M. A., Bihs, H., Myrhaug, D. and Arntsen, O. A. (2019b), ‘Numerical modeling of breaking wave kinematics and wave impact pressures on a vertical slender cylinder’, J. Offshore Mech. Arctic Engrg Trans. ASME 141, 051802.CrossRefGoogle Scholar
Chen, D. P., Eisenberg, R. S., Jerome, J. W. and Shu, C.-W. (1995), ‘Hydrodynamic model of temperature change in open ionic channels’, Biophys. J. 69, 2304–2322.CrossRefGoogle Scholar
Chen, G.-Q., Jerome, J. W. and Shu, C.-W. (1998), ‘Analysis and simulation of extended hydrodynamic models: The multi-valley Gunn oscillator and MESFET symmetries’, VLSI Design 6, 277–282.CrossRefGoogle Scholar
Chen, S., Sun, Q., Klioutchnikov, I. and Olivier, H. (2019a), ‘Numerical study of chemically reacting flow in a shock tube using a high-order point-implicit scheme’, Comput. Fluids 184, 107–118.CrossRefGoogle Scholar
Chen, X., Dou, H.-S., Liu, Q., Zhu, Z. and Zhang, W. (2019b), ‘Comparative study of Reynolds stress budgets of thermally and calorically perfect gases for high-temperature supersonic turbulent channel flow’, Proc. Inst. Mech. Engrs G J. Aerosp. Engrg 233, 4222–4234.CrossRefGoogle Scholar
Chen, X., Li, X. and Zhu, Z. (2019c), ‘Effects of dimensional wall temperature on velocity-temperature correlations in supersonic turbulent channel flow of thermally perfect gas’, Sci. China Phys. Mech. Astron. 62, 064711.CrossRefGoogle Scholar
Chen, Y. and Lowengrub, J. S. (2019), ‘Tumor growth and calcification in evolving microenvironmental geometries’, J. Theoret. Biol. 463, 138–154.CrossRefGoogle Scholar
Cheng, J. and Shu, C.-W. (2007), ‘A high order ENO conservative Lagrangian type scheme for the compressible Euler equations’, J. Comput. Phys. 227, 1567–1596.CrossRefGoogle Scholar
Cheng, J. and Shu, C.-W. (2008a), ‘A high order accurate conservative remapping method on staggered meshes’, Appl. Numer. Math. 58, 1042–1060.CrossRefGoogle Scholar
Cheng, J. and Shu, C.-W. (2008b), ‘A third order conservative Lagrangian type scheme on curvilinear meshes for the compressible Euler equations’, Commun. Comput. Phys. 4, 1008–1024.Google Scholar
Cheng, J. and Shu, C.-W. (2014), ‘Positivity-preserving Lagrangian scheme for multi-material compressible flow’, J. Comput. Phys. 257, 143–168.CrossRefGoogle Scholar
Cheng, J., Shu, C.-W. and Zeng, Q. (2012), ‘A conservative Lagrangian scheme for solving compressible fluid flows with multiple internal energy equations’, J. Comput. Phys. 12, 1307–1328.Google Scholar
Cheng, Z., Liu, Y., Zhang, M. and Wang, J. (2019), ‘IB-WENO method for incompressible flow with elastic boundaries’, J. Comput. Appl. Math. 362, 498–509.CrossRefGoogle Scholar
Choi, J.-I., Oberoi, R. C., Edwards, J. R. and Rosati, J. A. (2007), ‘An immersed boundary method for complex incompressible flows’, J. Comput. Phys. 224, 757–784.CrossRefGoogle Scholar
Chou, C.-S. and Shu, C.-W. (2006), ‘High order residual distribution conservative finite difference WENO schemes for steady state problems on non-smooth meshes’, J. Comput. Phys. 214, 698–724.CrossRefGoogle Scholar
Chou, C.-S. and Shu, C.-W. (2007), ‘High order residual distribution conservative finite difference WENO schemes for convection–diffusion steady state problems on non-smooth meshes’, J. Comput. Phys. 224, 992–1020.CrossRefGoogle Scholar
Christofi, S. N. (1996), The study of building blocks for essentially non-oscillatory (ENO) schemes. PhD thesis, Brown University.Google Scholar
Cockburn, B. and Shu, C.-W. (1989), ‘TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws, II: General framework’, Math. Comput. 52, 411–435.Google Scholar
Cockburn, B. and Shu, C.-W. (1998), ‘The Runge–Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems’, J. Comput. Phys. 141, 199–224.CrossRefGoogle Scholar
Cockburn, B. and Shu, C.-W. (2001), ‘Runge–Kutta discontinuous Galerkin methods for convection-dominated problems’, J. Sci. Comput. 16, 173–261.CrossRefGoogle Scholar
Crandall, M. G. and Lions, P. L. (1984), ‘Two approximations of solutions of Hamilton–Jacobi equations’, Math. Comput. 43, 1–19.CrossRefGoogle Scholar
Crandall, M. G. and Majda, A. (1980), ‘Monotone difference approximations for scalar conservation laws’, Math. Comput. 34, 1–21.CrossRefGoogle Scholar
Deng, F., Han, G., Liu, M., Ding, J., Weng, P. and Jiang, Z. (2019), ‘Numerical simulation of the interaction of two shear layers in double backward-facing steps’, Phys. Fluids 31, 056106.Google Scholar
Deng, X. and Maekawa, H. (1997), ‘Compact high-order accurate nonlinear schemes’, J. Comput. Phys. 130, 77–91.CrossRefGoogle Scholar
Deng, X. and Zhang, H. (2000), ‘Developing high-order weighted compact nonlinear schemes’, J. Comput. Phys. 165, 22–44.CrossRefGoogle Scholar
Dong, H., Fu, L., Zhang, F., Liu, Y. and Liu, J. (2019), ‘Detonation simulations with a fifth-order TENO scheme’, Commun. Comput. Phys. 25, 1357–1393.CrossRefGoogle Scholar
Donnert, J. M. F., Jang, H., Mendygral, P., Brunetti, G., Ryu, D. and Jones, T. W. (2019), ‘WENO-WOMBAT: Scalable fifth-order constrained-transport magnetohydrodynamics for astrophysical applications’, Astrophys. J. Suppl. Ser. 241, 23.CrossRefGoogle Scholar
Dudson, B. D., Umansky, M. V., Xu, X. Q., Snyder, P. B. and Wilson, H. R. (2009), ‘BOUT++: A framework for parallel plasma fluid simulations’, Comput. Phys. Commun. 180, 1467–1480.CrossRefGoogle Scholar
Dumbser, M. and Käser, M. (2007), ‘Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems’, J. Comput. Phys. 221, 693–723.CrossRefGoogle Scholar
Dumbser, M., Käser, M., Titarev, V. A. and Toro, E. F. (2007), ‘Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems’, J. Comput. Phys. 226, 204–243.CrossRefGoogle Scholar
Erlebacher, G., Hussaini, M. Y. and Shu, C.-W. (1997), ‘Interaction of a shock with a longitudinal vortex’, J. Fluid. Mech. 337, 129–153.CrossRefGoogle Scholar
Fatemi, E., Jerome, J. W. and Osher, S. (1991), ‘Solution of the hydrodynamic device model using high-order nonoscillatory shock capturing algorithms’, IEEE Trans. Computer-Aided Design Integrated Circuits Systems 10, 232–244.CrossRefGoogle Scholar
Fedkiw, R., Aslam, T., Merriman, B. and Osher, S. (1999), ‘A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method)’, J. Comput. Phys. 152, 457–492.CrossRefGoogle Scholar
Feng, L.-L., Shu, C.-W. and Zhang, M. (2004), ‘A hybrid cosmological hydrodynamic/ $N$ -body code based on a weighted essentially nonoscillatory scheme’, Astrophys. J. 612, 1–13.CrossRefGoogle Scholar
Fievet, R., Raman, V., Voelkel, S. and Varghese, P. L. (2019), ‘Numerical investigation of the coupling of vibrational nonequilibrium and turbulent mixing using state-specific description’, Phys. Rev. Fluids 4, 013401.CrossRefGoogle Scholar
Filbet, F. and Shu, C.-W. (2005), ‘Approximation of hyperbolic models for chemosensitive movement’, SIAM J. Sci. Comput. 27, 850–872.CrossRefGoogle Scholar
Fjordholm, U. S. and Ray, D. (2016), ‘A sign preserving WENO reconstruction method’, J. Sci. Comput. 68, 1111–1140.CrossRefGoogle Scholar
Fjordholm, U. S., Mishra, S. and Tadmor, E. (2013a), ‘Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws’, SIAM J. Numer. Anal. 50, 544–573.CrossRefGoogle Scholar
Fjordholm, U. S., Mishra, S. and Tadmor, E. (2013b), ‘ENO reconstruction and ENO interpolation are stable’, Found. Comput. Math. 13, 139–159.CrossRefGoogle Scholar
Foucarta, F., Duez, M. D., Kidder, L. E., Nissanke, S. M., Pfeiffer, H. P. and Scheel, M. A. (2019), ‘Numerical simulations of neutron star–black hole binaries in the near-equal-mass regime’, Phys. Rev. D 99, 103025.CrossRefGoogle Scholar
Freret, L., Ivan, L., Sterck, H. D. and Groth, C. P. T. (2019), ‘High-order finite-volume method with block-based AMR for magnetohydrodynamics flows’, J. Sci. Comput. 79, 176–208.CrossRefGoogle Scholar
Friedrichs, O. (1998), ‘Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids’, J. Comput. Phys. 144, 194–212.CrossRefGoogle Scholar
Gadiou, A. and Tenaud, C. (2004), ‘Implicit WENO shock capturing scheme for unsteady flows: Application to one-dimensional Euler equations’, Internat. J. Numer. Methods Fluids 45, 197–229.Google Scholar
Ghosh, D., Chapman, T. D., Berger, R. L., Dimits, A. and Banks, J. W. (2019), ‘A multispecies, multifluid model for laser-induced counterstreaming plasma simulations’, Comput. Fluids 186, 38–57.CrossRefGoogle Scholar
Gibou, F., Fedkiw, R., Caflisch, R. and Osher, S. (2003), ‘A level set approach for the numerical simulation of dendritic growth’, J. Sci. Comput. 19, 183–199.CrossRefGoogle Scholar
Goldberg, M. (1977), ‘On a boundary extrapolation theorem by Kreiss’, Math. Comput. 31, 469–477.CrossRefGoogle Scholar
Goldberg, M. and Tadmor, E. (1978), ‘Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems, I’, Math. Comput. 32, 1097–1107.CrossRefGoogle Scholar
Goldberg, M. and Tadmor, E. (1981), ‘Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems, II’, Math. Comput. 36, 603–626.CrossRefGoogle Scholar
Gottlieb, S., Ketcheson, D. and Shu, C.-W. (2009), ‘High order strong stability preserving time discretizations’, J. Sci. Comput. 38, 251–289.CrossRefGoogle Scholar
Gottlieb, S., Ketcheson, D. and Shu, C.-W. (2011), Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations, World Scientific.CrossRefGoogle Scholar
Gottlieb, S., Mullen, J. S. and Ruuth, S. J. (2006), ‘A fifth order flux implicit WENO method’, J. Sci. Comput. 27, 271–287.CrossRefGoogle Scholar
Gottlieb, S., Shu, C.-W. and Tadmor, E. (2001), ‘Strong stability preserving high order time discretization methods’, SIAM Rev. 43, 89–112.CrossRefGoogle Scholar
Guo, Y., Yang, W., Zhang, H., Wang, J. and Song, S. (2019), ‘A splitting method for the Degasperis–Procesi equation using an optimized WENO scheme and the Fourier pseudospectral method’, Adv. Appl. Math. Mech. 11, 53–71.CrossRefGoogle Scholar
Ha, Y., Gardner, C. L., Gelb, A. and Shu, C.-W. (2005), ‘Numerical simulation of high Mach number astrophysical jets with radiative cooling’, J. Sci. Comput. 24, 597–612.CrossRefGoogle Scholar
Han, W., Ma, W., Qian, C., Wen, J. and Wang, C. (2019a), ‘Bifurcation of pulsation instability in one-dimensional $\text{H}_{\mathsf{2}}$ – $\text{O}_{\mathsf{2}}$ detonation with detailed reaction mechanism’, Phys. Rev. Fluids 4, 103202.CrossRefGoogle Scholar
Han, W., Wang, C. and Law, C. K. (2019b), ‘Pulsation in one-dimensional $\text{H}_{\mathsf{2}}$ – $\text{O}_{\mathsf{2}}$ detonation with detailed reaction mechanism’, Combust. Flame 200, 242–261.CrossRefGoogle Scholar
Han, W., Wang, C. and Law, C. K. (2019c), ‘Role of transversal concentration gradient in detonation propagation’, J. Fluid. Mech. 865, 602–649.CrossRefGoogle Scholar
Han, W., Wang, C. and Law, C. K. (2019d), ‘Three-dimensional simulation of oblique detonation waves attached to cone’, Phys. Rev. Fluids 4, 053201.CrossRefGoogle Scholar
Hao, J., Xiong, S. and Yang, Y. (2019), ‘Tracking vortex surfaces frozen in the virtual velocity in non-ideal flows’, J. Fluid. Mech. 863, 513–544.CrossRefGoogle Scholar
Harten, A. (1983), ‘High resolution schemes for hyperbolic conservation laws’, J. Comput. Phys. 49, 357–393.CrossRefGoogle Scholar
Harten, A., Engquist, B., Osher, S. and Chakravarthy, S. (1987), ‘Uniformly high order essentially non-oscillatory schemes, III’, J. Comput. Phys. 71, 231–303.CrossRefGoogle Scholar
Harten, A., Engquist, B., Osher, S. and Chakravarthy, S. (1997), ‘Uniformly high order essentially non-oscillatory schemes, III’, J. Comput. Phys. 131, 3–47.CrossRefGoogle Scholar
Harten, A., Osher, S., Engquist, B. and Chakravarthy, S. (1986), ‘Some results on uniformly high order accurate essentially non-oscillatory schemes’, Appl. Numer. Math. 2, 347–377.CrossRefGoogle Scholar
Hejranfar, K. and Rahmani, S. (2019), ‘Numerical simulation of shock–disturbances interaction in high-speed compressible inviscid flow over a blunt nose using weighted essentially non-oscillatory scheme’, Wave Motion 88, 167–195.CrossRefGoogle Scholar
Henrick, A. K., Aslam, T. D. and Powers, J. M. (2005), ‘Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points’, J. Comput. Phys. 207, 542–567.CrossRefGoogle Scholar
Hill, D. J. and Pullin, D. I. (2004), ‘Hybrid tuned center-difference-WENO method for large eddy simulations in the presence of strong shocks’, J. Comput. Phys. 194, 435–450.CrossRefGoogle Scholar
Hu, C. and Shu, C.-W. (1999), ‘Weighted essentially non-oscillatory schemes on triangular meshes’, J. Comput. Phys. 150, 97–127.CrossRefGoogle Scholar
Huang, J., Bretzke, J.-V. and Duan, L. (2019), ‘Assessment of turbulence models in a hypersonic cold-wall turbulent boundary layer’, Fluids 4, 37.CrossRefGoogle Scholar
Huang, L., Shu, C.-W. and Zhang, M. (2008), ‘Numerical boundary conditions for the fast sweeping high order WENO methods for solving the Eikonal equation’, J. Comput. Math. 26, 336–346.Google Scholar
Huang, L., Wong, S. C., Zhang, M., Shu, C.-W. and Lam, W. H. K. (2009), ‘Revisiting Hughes’ dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm’, Transport. Res. B 43, 127–141.CrossRefGoogle Scholar
Huang, Z. and Wang, H. (2019), ‘Linear interaction of two-dimensional free-stream disturbances with an oblique shock wave’, J. Fluid. Mech. 873, 1179–1205.CrossRefGoogle Scholar
Jackson, H. and Nikiforakis, N. (2019), ‘A numerical scheme for non-Newtonian fluids and plastic solids under the GPR model’, J. Comput. Phys. 387, 410–429.CrossRefGoogle Scholar
Jain, A. and Kim, S. H. (2019), ‘On the non-equilibrium models for subfilter scalar variance in large eddy simulation of turbulent mixing and combustion’, Phys. Fluids 31, 025112.CrossRefGoogle Scholar
Jerome, J. W. and Shu, C.-W. (1995), ‘Transport effects and characteristic modes in the modeling and simulation of submicron devices’, IEEE Trans. Computer-Aided Design Integrated Circuits Systems 14, 917–923.CrossRefGoogle Scholar
Jiang, G. and Peng, D.-P. (2000), ‘Weighted ENO schemes for Hamilton–Jacobi equations’, SIAM J. Sci. Comput. 21, 2126–2143.CrossRefGoogle Scholar
Jiang, G. and Shu, C.-W. (1996), ‘Efficient implementation of weighted ENO schemes’, J. Comput. Phys. 126, 202–228.CrossRefGoogle Scholar
Jiang, G. and Wu, C. C. (1999), ‘A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics’, J. Comput. Phys. 150, 561–594.CrossRefGoogle Scholar
Jiang, L., Shan, H. and Liu, C. Q. (2001), ‘Weighted compact scheme’, Internat. J. Comput. Fluid Dynamics 15, 147–155.CrossRefGoogle Scholar
Jiang, Y., Shu, C.-W. and Zhang, M. (2013), ‘An alternative formulation of finite difference weighted ENO schemes with Lax–Wendroff time discretization for conservation laws’, SIAM J. Sci. Comput. 35, A1137–A1160.CrossRefGoogle Scholar
Jiang, Y., Shu, C.-W. and Zhang, M. (2015), ‘High order finite difference WENO schemes with positivity-preserving limiter for correlated random walk with density-dependent turning rates’, Math. Models Methods Appl. Sci. 25, 1553–1588.CrossRefGoogle Scholar
Johnsen, E. and Colonius, T. (2006), ‘Implementation of WENO schemes in compressible multicomponent flow problems’, J. Comput. Phys. 219, 715–732.CrossRefGoogle Scholar
Kamath, A., Fleit, G. and Bihs, H. (2019), ‘Investigation of free surface turbulence damping in RANS simulations for complex free surface flows’, Water 11, 456.CrossRefGoogle Scholar
Khosronejad, A., Kang, S. and Flora, K. (2019), ‘Fully coupled free-surface flow and sediment transport modelling of flash floods in a desert stream in the Mojave Desert, California’, Hydrol. Process. 33, 2772–2791.Google Scholar
Klingenberg, C., Puppo, G. and Semplice, M. (2019), ‘Arbitrary order finite volume well-balanced schemes for the Euler equations with gravity’, SIAM J. Sci. Comput. 41, A695–A721.CrossRefGoogle Scholar
Kremeyer, K., Sebastian, K. and Shu, C.-W. (2006), ‘Computational study of shock mitigation and drag reduction by pulsed energy lines’, AIAA J. 44, 1720–1731.CrossRefGoogle Scholar
Kumar, S. and Singh, P. (2019), ‘High order WENO finite volume approximation for population density neuron model’, Appl. Math. Comput. 356, 173–189.Google Scholar
Lehmann, M., Schmidt, J. and Salo, H. (2019), ‘Density waves and the viscous overstability in Saturn’s rings’, Astronom. Astrophys. 623, A121.CrossRefGoogle Scholar
Lele, S. K. (1992), ‘Compact finite difference schemes with spectral-like resolution’, J. Comput. Phys. 103, 16–42.CrossRefGoogle Scholar
LeVeque, R. J. (1990), Numerical Methods for Conservation Laws, Birkhäuser.CrossRefGoogle Scholar
Levy, D., Puppo, G. and Russo, G. (1999), ‘Central WENO schemes for hyperbolic systems of conservation laws’, Math. Model. Numer. Anal. 33, 547–571.CrossRefGoogle Scholar
Levy, D., Puppo, G. and Russo, G. (2000), ‘Compact central WENO schemes for multidimensional conservation laws’, SIAM J. Sci. Comput. 22, 656–672.CrossRefGoogle Scholar
Li, C. and Chen, X. (2019), ‘Simulating nonhydrostatic atmospheres on planets (SNAP): Formulation, validation, and application to the Jovian atmosphere’, Astrophys. J. Suppl. Ser. 240, 37.CrossRefGoogle Scholar
Li, J. and Du, Z. (2016), ‘A two-stage fourth order time-accurate discretization for Lax–Wendroff type flow solvers, I: Hyperbolic conservation laws’, SIAM J. Sci. Comput. 38, A3046–A3069.CrossRefGoogle Scholar
Li, P., Li, C., Wang, H., Sun, M., Liu, C., Wang, Z. and Huang, Y. (2019a), ‘Distribution characteristics and mixing mechanism of a liquid jet injected into a cavity-based supersonic combustor’, Aerosp. Sci. Technol. 94, 105401.CrossRefGoogle Scholar
Li, T., Shu, C.-W. and Zhang, M. (2016), ‘Stability analysis of the inverse Lax–Wendroff boundary treatment for high order upwind-biased finite difference schemes’, J. Comput. Appl. Math. 299, 140–158.CrossRefGoogle Scholar
Li, T., Shu, C.-W. and Zhang, M. (2017), ‘Stability analysis of the inverse Lax–Wendroff boundary treatment for high order central difference schemes for diffusion equations’, J. Sci. Comput. 70, 576–607.CrossRefGoogle Scholar
Li, W., Fang, Y., Modesti, D. and Cheng, C. (2019b), ‘Decomposition of the mean skin-friction drag in compressible turbulent channel flows’, J. Fluid. Mech. 875, 101–123.CrossRefGoogle Scholar
Li, Y. L. and Yu, C. H. (2019), ‘Research on dam-break flow induced front wave impacting a vertical wall based on the CLSVOF and level set methods’, Ocean Engrg 178, 442–462.CrossRefGoogle Scholar
Lipanov, A. M., Rusyak, I. G., Korolev, S. A. and Karskanov, S. A. (2019), ‘Numerical solution of the problem of flow past projected bodies for determining their aerodynamic coefficients’, J. Engrg Phys. Thermophys. 92, 477–485.CrossRefGoogle Scholar
Liu, H., Gao, Z., Jiang, C. and Lee, C. (2019), ‘Numerical study of combustion effects on the development of supersonic turbulent mixing layer flows with WENO schemes’, Comput. Fluids 189, 82–93.CrossRefGoogle Scholar
Liu, W., Cheng, J. and Shu, C.-W. (2009a), ‘High order conservative Lagrangian schemes with Lax–Wendroff type time discretization for the compressible Euler equations’, J. Comput. Phys. 228, 8872–8891.CrossRefGoogle Scholar
Liu, W., Yuan, L. and Shu, C.-W. (2011a), ‘A conservative modification to the ghost fluid method for compressible multiphase flows’, Commun. Comput. Phys. 10, 785–806.CrossRefGoogle Scholar
Liu, X.-D., Osher, S. and Chan, T. (1994), ‘Weighted essentially non-oscillatory schemes’, J. Comput. Phys. 115, 200–212.CrossRefGoogle Scholar
Liu, Y. and Zhang, Y.-T. (2013), ‘A robust reconstruction for unstructured WENO schemes’, J. Sci. Comput. 54, 603–621.CrossRefGoogle Scholar
Liu, Y.-Y., Shu, C.-W. and Zhang, M.-P. (2009b), ‘On the positivity of linear weights in WENO approximations’, Acta Math. Appl. Sinica Eng. Ser. 25, 503–538.CrossRefGoogle Scholar
Liu, Y.-Y., Shu, C.-W. and Zhang, M.-P. (2011b), ‘High order finite difference WENO schemes for nonlinear degenerate parabolic equations’, SIAM J. Sci. Comput. 33, 939–965.CrossRefGoogle Scholar
Louati, M. and Ghidaoui, M. S. (2019), ‘The need for high order numerical schemes to model dispersive high frequency acoustic waves in water-filled pipes’, J. Hydraul. Research 57, 405–425.CrossRefGoogle Scholar
Lu, J., Fang, J., Tan, S., Shu, C.-W. and Zhang, M. (2016), ‘Inverse Lax–Wendroff procedure for numerical boundary conditions of convection–diffusion equations’, J. Comput. Phys. 317, 276–300.CrossRefGoogle Scholar
Macklin, P. and Lowengrub, J. (2007), ‘Nonlinear simulation of the effect of microenvironment on tumor growth’, J. Theoret. Biol. 245, 677–704.CrossRefGoogle Scholar
Mahesh, K., Lele, S. K. and Moin, P. (1997), ‘The influence of entropy fluctuations on the interaction of turbulence with a shock wave’, J. Fluid. Mech. 334, 353–379.CrossRefGoogle Scholar
Mansell, E. R., Ziegler, C. L. and Bruning, E. C. (2010), ‘Simulated electrification of a small thunderstorm with two-moment bulk microphysics’, J. Atmos. Sci. 67, 171–194.CrossRefGoogle Scholar
Martin, M. P., Taylor, E. M., Wu, M. and Weirs, V. G. (2006), ‘A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence’, J. Comput. Phys. 220, 270–289.CrossRefGoogle Scholar
Mehmani, Y. and Tchelepi, H. A. (2019), ‘Multiscale formulation of two-phase flow at the pore scale’, J. Comput. Phys. 389, 164–188.CrossRefGoogle Scholar
Mignone, A., Bodo, G., Massaglia, S., Matsakos, T., Tesileanu, O., Zanni, C. and Ferrari, A. (2007), ‘PLUTO: A numerical code for computational astrophysics’, Astrophys. J. Suppl. Ser. 170, 228–242.CrossRefGoogle Scholar
Mignone, A., Zanni, C., Tzeferacos, P., van Straalen, B., Colella, P. and Bodo, G. (2012), ‘The Pluto code for adaptive mesh computations in astrophysical fluid dynamics’, Astrophys. J. Suppl. Ser. 198, 7.CrossRefGoogle Scholar
Mo, H., Lien, F.-S., Zhang, F. and Cronin, D. S. (2019), ‘A mesoscale study on explosively dispersed granular material using direct simulation’, J. Appl. Phys. 125, 214302.CrossRefGoogle Scholar
Mouronval, A.-S., Tie, B., Hadjadj, A. and Moebs, G. (2019), ‘Investigation of shock/ elastic obstacles interactions by means of a coupling technique’, J. Fluids Structures 84, 345–367.CrossRefGoogle Scholar
Murillo, J., Navas-Montilla, A. and Garcia-Navarro, P. (2019), ‘Formulation of exactly balanced solvers for blood flow in elastic vessels and their application to collapsed states’, Comput. Fluids 186, 74–98.CrossRefGoogle Scholar
Nishiguchi, K., Bale, R., Okazawa, S. and Tsubokura, M. (2019), ‘Full Eulerian deformable solid–fluid interaction scheme based on building-cube method for large-scale parallel computing’, Internat. J. Numer. Methods Engrg 117, 221–248.CrossRefGoogle Scholar
Noelle, S., Pankratz, N., Puppo, G. and Natvig, J. R. (2006), ‘Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows’, J. Comput. Phys. 213, 474–499.CrossRefGoogle Scholar
Noelle, S., Xing, Y. and Shu, C.-W. (2007), ‘High-order well-balanced finite volume WENO schemes for shallow water equation with moving water’, J. Comput. Phys. 226, 29–58.CrossRefGoogle Scholar
Novoselov, A. G., Law, C. K. and Mueller, M. E. (2019a), ‘Direct numerical simulation of turbulent nonpremixed “cool” flames: Applicability of flamelet models’, Proc. Combustion Institute 37, 2143–2150.CrossRefGoogle Scholar
Novoselov, A. G., Reuter, C. B., Yehia, O. R., Won, S. H., Fu, M. K., Kokmanian, K., Hultmark, M., Ju, Y. and Mueller, M. E. (2019b), ‘Turbulent nonpremixed cool flames: Experimental measurements, direct numerical simulation, and manifold-based combustion modeling’, Combust. Flame 209, 144–154.CrossRefGoogle Scholar
Ochs, B. A., Ranjan, R., Ranjan, D. and Menon, S. (2019), ‘Turbulent premixed flame kernels in supersonic flows: Topology and flame speeds of turbulent premixed flame kernels in supersonic flows’, Combust. Flame 210, 83–99.CrossRefGoogle Scholar
Osher, S. and Chakravarthy, S. (1984), ‘High resolution schemes and the entropy condition’, SIAM J . Numer. Anal. 21, 955–984.CrossRefGoogle Scholar
Osher, S. and Chakravarthy, S. (1996), Very high order accurate TVD schemes. In Oscillation Theory, Computation, and Methods of Compensated Compactness (Dafermos, C. et al., eds), Vol. 2 of IMA Volumes in Mathematics and its Applications, Springer, pp. 229–274.CrossRefGoogle Scholar
Osher, S. and Sethian, J. (1988), ‘Fronts propagating with curvature dependent speed: Algorithms based on Hamilton–Jacobi formulations’, J. Comput. Phys. 79, 12–49.CrossRefGoogle Scholar
Osher, S. and Shu, C.-W. (1991), ‘High-order essentially nonoscillatory schemes for Hamilton–Jacobi equations’, SIAM J. Numer. Anal. 28, 907–922.CrossRefGoogle Scholar
Ou, J. and Chen, J. (2019), ‘DSMC data-improved numerical simulation of hypersonic flow past a flat plate in near-continuum regime’, Comput. Fluids 194, 104308.CrossRefGoogle Scholar
Ou, J. and Zhai, Z. (2019), ‘Effects of aspect ratio on shock–cylinder interaction’, Acta Mech. Sinica 35, 61–69.CrossRefGoogle Scholar
Paula, T., Adami, S. and Adams, N. A. (2019), ‘Analysis of the early stages of liquid-water-drop explosion by numerical simulation’, Phys. Rev. Fluids 4, 044003.CrossRefGoogle Scholar
Peshkova, I., Boscheri, W., Loubere, R., Romenski, E. and Dumbser, M. (2019), ‘Theoretical and numerical comparison of hyperelastic and hypoelastic formulations for Eulerian non-linear elastoplasticity’, J. Comput. Phys. 387, 481–521.CrossRefGoogle Scholar
Petermann, J., Jung, Y. S., Baeder, J. and Rauleder, J. (2019), ‘Validation of higher-order interactional aerodynamics simulations on full helicopter configurations’, J. American Helicopter Soc. 64, 042002.Google Scholar
Pietri, R. D., Drago, A., Feo, A., Pagliara, G., Pasquali, M., Traversi, S. and Wiktorowicz, G. (2019), ‘Merger of compact stars in the two-families scenario’, Astrophys. J. 881, 122.CrossRefGoogle Scholar
Pillai, A. L. and Kurose, R. (2019), ‘Combustion noise analysis of a turbulent spray flame using a hybrid DNS/APE-RF approach’, Combust . Flame 200, 168–191.CrossRefGoogle Scholar
Pirozzoli, S. (2002), ‘Conservative hybrid compact-WENO schemes for shock–turbulence interaction’, J. Comput. Phys. 178, 81–117.CrossRefGoogle Scholar
Pirozzoli, S., Grasso, F. and Gatski, T. B. (2004), ‘Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at $M=2.25$ ’, Phys. Fluids 16, 530–545.CrossRefGoogle Scholar
Puppo, G. and Russo, G. (2006), ‘Staggered finite difference schemes for conservation laws’, J. Sci. Comput. 27, 403–418.CrossRefGoogle Scholar
Qian, C., Wang, C., Liu, J., Brandenburg, A., Haugen, N. E. L. and Liberman, M. A. (2020), ‘Convergence properties of detonation simulations’, Geophys. Astrophys. Fluid Dynamics 114, 58–76.CrossRefGoogle Scholar
Qiu, J.-M. and Shu, C.-W. (2011a), ‘Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow’, J. Comput. Phys. 230, 863–889.CrossRefGoogle Scholar
Qiu, J.-M. and Shu, C.-W. (2011b), ‘Conservative semi-Lagrangian finite difference WENO formulations with applications to the Vlasov equation’, Commun. Comput. Phys. 10, 979–1000.CrossRefGoogle Scholar
Qiu, J.-M., Feng, L.-L., Shu, C.-W. and Fang, L.-Z. (2007), ‘A WENO algorithm of the temperature and ionization profiles around a point source’, New Astronomy 12, 398–409.CrossRefGoogle Scholar
Qiu, J.-M., Shu, C.-W., Feng, L.-L. and Fang, L.-Z. (2006), ‘A WENO algorithm for the radiative transfer and ionized sphere at reionization’, New Astronomy 12, 1–10.CrossRefGoogle Scholar
Qiu, J.-M., Shu, C.-W., Liu, J.-R. and Fang, L.-Z. (2008), ‘A WENO algorithm for the growth of ionized regions at the reionization epoch’, New Astronomy 13, 1–11.CrossRefGoogle Scholar
Qiu, J.-X. and Shu, C.-W. (2002), ‘On the construction, comparison, and local characteristic decomposition for high order central WENO schemes’, J. Comput. Phys. 183, 187–209.CrossRefGoogle Scholar
Qiu, J.-X. and Shu, C.-W. (2003a), ‘Finite difference WENO schemes with Lax–Wendroff type time discretization’, SIAM J. Sci. Comput. 24, 2185–2198.CrossRefGoogle Scholar
Qiu, J.-X. and Shu, C.-W. (2003b), ‘Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method: One-dimensional case’, J. Comput. Phys. 193, 115–135.CrossRefGoogle Scholar
Qiu, J.-X. and Shu, C.-W. (2005a), ‘A comparison of troubled-cell indicators for Runge–Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters’, SIAM J. Sci. Comput. 27, 995–1013.CrossRefGoogle Scholar
Qiu, J.-X. and Shu, C.-W. (2005b), ‘Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method, II: Two dimensional case’, Comput. Fluids 34, 642–663.CrossRefGoogle Scholar
Qiu, J.-X. and Shu, C.-W. (2005c), ‘Hermite WENO schemes for Hamilton–Jacobi equations’, J. Comput. Phys. 204, 82–99.CrossRefGoogle Scholar
Qiu, J.-X. and Shu, C.-W. (2005d), ‘Runge–Kutta discontinuous Galerkin method using WENO limiters’, SIAM J. Sci. Comput. 26, 907–929.CrossRefGoogle Scholar
Rahman, S. M. and San, O. (2019), ‘A relaxation filtering approach for two-dimensional Rayleigh–Taylor instability-induced flows’, Fluids 4, 78.CrossRefGoogle Scholar
Rasthofer, U., Wermelinger, F., Karnakov, P., Sukys, J. and Koumoutsakos, P. (2019), ‘Computational study of the collapse of a cloud with $12500$ gas bubbles in a liquid’, Phys. Rev. Fluids 4, 063602.CrossRefGoogle Scholar
Ren, H., Li, Q., Li, C., Liu, T. and Wang, Z. (2019), ‘Applicability analysis of the prevailing algorithms for reliable solution of the bipolar charge transport model’, Physica Scripta 94, 075802.CrossRefGoogle Scholar
Rodriguez, M. and Johnsen, E. (2019), ‘A high-order accurate five-equations compressible multiphase approach for viscoelastic fluids and solids with relaxation and elasticity’, J. Comput. Phys. 379, 70–90.CrossRefGoogle Scholar
Roe, P. (1978), ‘Approximate Riemann solvers, parameter vectors and difference schemes’, J. Comput. Phys. 27, 1–31.Google Scholar
Rogerson, A. and Meiburg, E. (1990), ‘A numerical study of the convergence properties of ENO schemes’, J. Sci. Comput. 5, 151–167.CrossRefGoogle Scholar
Roh, S., Ryu, D., Kang, H., Hai, S. and Jung, H. (2019), ‘Turbulence dynamo in the stratified medium of Galaxy clusters’, Astrophys. J. 883, 138.CrossRefGoogle Scholar
Roy, I., Qiu, J.-M., Shu, C.-W. and Fang, L.-Z. (2009a), ‘A WENO algorithm for radiative transfer with resonant scattering and the Wouthuysen-field coupling’, New Astronomy 14, 513–520.CrossRefGoogle Scholar
Roy, I., Shu, C.-W. and Fang, L.-Z. (2010), ‘Resonant scattering and Ly $\alpha$ radiation emergent from neutral hydrogen halos’, Astrophys. J. 716, 604–614.CrossRefGoogle Scholar
Roy, I., Xu, W., Qiu, J.-M., Shu, C.-W. and Fang, L.-Z. (2009b), ‘Time evolution of Wouthuysen-field coupling’, Astrophys. J. 694, 1121–1130.CrossRefGoogle Scholar
Sabelnikov, V. A., Yu, R. and Lipatnikov, A. N. (2019), ‘Thin reaction zones in constant-density turbulent flows at low Damköhler numbers: Theory and simulations’, Phys . Fluids 31, 055104.CrossRefGoogle Scholar
Sakov, P., Counillon, F., Bertino, L., Lisaeter, K. A., Oke, P. R. and Korablev, A. (2012), ‘TOPAZ4: An ocean–sea ice data assimilation system for the North Atlantic and Arctic’, Ocean Science 8, 633–656.CrossRefGoogle Scholar
Sasikumar, A., Kamath, A., Musch, O., Bihs, H. and Arntsen, O. A. (2019), ‘Numerical modeling of berm breakwater optimization with varying berm geometry using REEF3D’, J . Offshore Mech. Arctic Engrg Trans. ASME 141, 011801.CrossRefGoogle Scholar
Schneider, T., Kaul, C. M. and Pressel, K. G. (2019), ‘Possible climate transitions from breakup of stratocumulus decks under greenhouse warming’, Nature Geosci. 12, 163–167.CrossRefGoogle Scholar
Schuller, T., Ducruix, S., Durox, D. and Candel, S. (2003a), ‘Modeling tools for the prediction of premixed flame transfer functions’, Proc . Combustion Institute 29, 107–113.CrossRefGoogle Scholar
Schuller, T., Durox, D. and Candel, S. (2003b), ‘A unified model for the prediction of laminar flame transfer functions: Comparisons between conical and V-flame dynamics’, Combust . Flame 134, 21–34.CrossRefGoogle Scholar
Seal, D. C., Guclu, Y. and Christlieb, A. J. (2014), ‘High-order multiderivative time integrators for hyperbolic conservation laws’, J. Sci. Comput. 60, 101–140.CrossRefGoogle Scholar
Sebastian, K. and Shu, C.-W. (2003), ‘Multidomain WENO finite difference method with interpolation at subdomain interfaces’, J. Sci. Comput. 19, 405–438.CrossRefGoogle Scholar
Shahmardi, A., Zade, S., Ardekani, M. N., Poole, R. J., Lundell, F., Rosti, M. E. and Brandt, L. (2019), ‘Turbulent duct flow with polymers’, J. Fluid. Mech. 859, 1057–1083.CrossRefGoogle Scholar
Sharma, S., Shadloo, M. S. and Hadjadj, A. (2019), ‘Turbulent flow topology in supersonic boundary layer with wall heat transfer’, Internat. J. Heat Fluid Flow 78, 108430.CrossRefGoogle Scholar
Shen, J., Shu, C.-W. and Zhang, M. (2007), ‘A high order WENO scheme for a hierarchical size-structured population model’, J. Sci. Comput. 33, 279–291.CrossRefGoogle Scholar
Shi, J., Hu, C. and Shu, C.-W. (2002), ‘A technique of treating negative weights in WENO schemes’, J. Comput. Phys. 175, 108–127.CrossRefGoogle Scholar
Shi, J., Zhang, Y.-T. and Shu, C.-W. (2003), ‘Resolution of high order WENO schemes for complicated flow structures’, J. Comput. Phys. 186, 690–696.CrossRefGoogle Scholar
Shi, M., Xu, L., Wang, Z. and Lv, H. (2019), ‘Effect of a roughness element on the hypersonic boundary layer receptivity due to different types of free-stream disturbance with a single frequency’, Entropy 21, 255.CrossRefGoogle Scholar
Shu, C.-W. (1987), ‘TVB uniformly high-order schemes for conservation laws’, Math. Comput. 49, 105–121.CrossRefGoogle Scholar
Shu, C.-W. (1988), ‘Total-variation-diminishing time discretizations’, SIAM J. Sci. Statist. Comput. 9, 1073–1084.CrossRefGoogle Scholar
Shu, C.-W. (1990), ‘Numerical experiments on the accuracy of ENO and modified ENO schemes’, J. Sci. Comput. 5, 127–149.CrossRefGoogle Scholar
Shu, C.-W. (1997), ‘Preface to the republication of “Uniform high order essentially non-oscillatory schemes, III”’, J. Comput. Phys. 131, 1–2.CrossRefGoogle Scholar
Shu, C.-W. (1998), Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In Advanced Numerical Approximation of Nonlinear Hyperbolic Equations (Cockburn, B. et al., eds), Vol. 1697 of Lecture Notes in Mathematics, Springer, pp. 325–432.CrossRefGoogle Scholar
Shu, C.-W. (2007), High order numerical methods for time dependent Hamilton–Jacobi equations. In Mathematics and Computation in Imaging Science and Information Processing (Goh, S. et al., eds), Vol. 11 of Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, World Scientific, pp. 47–91.CrossRefGoogle Scholar
Shu, C.-W. (2009), ‘High order weighted essentially nonoscillatory schemes for convection dominated problems’, SIAM Rev. 51, 82–126.CrossRefGoogle Scholar
Shu, C.-W. and Osher, S. (1988), ‘Efficient implementation of essentially non-oscillatory shock-capturing schemes’, J. Comput. Phys. 77, 439–471.CrossRefGoogle Scholar
Shu, C.-W. and Osher, S. (1989), ‘Efficient implementation of essentially non-oscillatory shock-capturing schemes, II’, J. Comput. Phys. 83, 32–78.CrossRefGoogle Scholar
Shu, C.-W. and Zeng, Y. (1997), ‘High-order essentially non-oscillatory scheme for viscoelasticity with fading memory’, Quart. Appl. Math. 55, 459–484.CrossRefGoogle Scholar
Siddiqi, K., Kimia, B. B. and Shu, C.-W. (1997), ‘Geometric shock-capturing ENO schemes for subpixel interpolation, computation and curve evolution’, Graphical Models and Image Processing (CVGIP:GMIP) 59, 278–301.CrossRefGoogle Scholar
Singh, P., Kumar, S. and Koksal, M. E. (2019), ‘High-order finite volume approximation for population density model based on quadratic integrate-and-fire neuron’, Engrg Comput. 36, 84–102.CrossRefGoogle Scholar
Smoller, J. (1983), Shock Waves and Reaction-Diffusion Equations, Springer.CrossRefGoogle Scholar
Tan, S. and Shu, C.-W. (2010), ‘Inverse Lax–Wendroff procedure for numerical boundary conditions of conservation laws’, J. Comput. Phys. 229, 8144–8166.CrossRefGoogle Scholar
Tan, S. and Shu, C.-W. (2011), ‘A high order moving boundary treatment for compressible inviscid flows’, J. Comput. Phys. 230, 6023–6036.CrossRefGoogle Scholar
Tan, S., Wang, C., Shu, C.-W. and Ning, J. (2012), ‘Efficient implementation of high order inverse Lax–Wendroff boundary treatment for conservation laws’, J. Comput. Phys. 231, 2510–2527.CrossRefGoogle Scholar
Tanguy, S. and Berlemont, A. (2005), ‘Application of a level set method for simulation of droplet collisions’, Internat. J. Multiphase Flow 31, 1015–1035.CrossRefGoogle Scholar
Tanguy, S., Menard, T. and Berlemont, A. (2007), ‘A level set method for vaporizing two-phase flows’, J. Comput. Phys. 221, 837–853.CrossRefGoogle Scholar
Titarev, V. A. and Toro, E. F. (2005), ‘ADER schemes for three-dimensional nonlinear hyperbolic systems’, J. Comput. Phys. 204, 715–736.CrossRefGoogle Scholar
Toro, E. (1997), Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer.CrossRefGoogle Scholar
van Eerten, H., Zhang, W. and MacFadyen, A. (2010), ‘Off-axis gamma-ray burst afterflow modeling based on a two-dimensional axisymmetric hydrodynamics simulation’, Astrophys. J. 722, 235–247.CrossRefGoogle Scholar
Verma, P. S., Teissier, J.-M., Henze, O. and Muller, W.-C. (2019), ‘Fourth-order accurate finite-volume CWENO scheme for astrophysical MHD problems’, Monthly Notices Roy. Astron. Soc. 482, 416–437.CrossRefGoogle Scholar
Vevek, U. S., Zang, B. and New, T. H. (2019), ‘On alternative setups of the double Mach reflection problem’, J. Sci. Comput. 78, 1291–1303.CrossRefGoogle Scholar
Vigano, D., Martinez-Gomez, D., Pons, J. A., Palenzuela, C., Carrasco, F., Minano, B., Arbona, A., Bona, C. and Masso, J. (2019), ‘A Simflowny-based high-performance 3D code for the generalized induction equation’, Comput. Phys. Commun. 237, 168–183.CrossRefGoogle Scholar
Wang, C., Shu, C.-W., Han, W. and Ning, J. (2013), ‘High resolution WENO simulation of 3D detonation waves’, Combust. Flame 160, 447–462.CrossRefGoogle Scholar
Wang, D. and Byambaakhuu, T. (2019), ‘High-order Lax–Friedrichs WENO fast sweeping methods for the ${\mathsf{S}}_{\mathsf{N}}$ neutron transport equation’, Nuclear Sci. Engrg 193, 982–990.CrossRefGoogle Scholar
Wang, J.-H., Pan, S., Hu, X. Y. and Adams, N. A. (2019a), ‘Partial characteristic decomposition for multi-species Euler equations’, Comput. Fluids 181, 364–382.CrossRefGoogle Scholar
Wang, J.-X., Huang, J., Duan, L. and Xiao, H. (2019b), ‘Prediction of Reynolds stresses in high-Mach-number turbulent boundary layers using physics-informed machine learning’, Theoret. Comput. Fluid Dynamics 33, 1–19.CrossRefGoogle Scholar
Wang, W., Shu, C.-W., Yee, H. C. and Sjögreen, B. (2009), ‘High-order well-balanced schemes and applications to non-equilibrium flow’, J. Comput. Phys. 228, 6682–6702.CrossRefGoogle Scholar
Wang, W., Shu, C.-W., Yee, H. C., Kotov, D. V. and Sjögreen, B. (2015), ‘High order finite difference methods with subcell resolution for stiff multispecies detonation capturing’, Commun. Comput. Phys. 17, 317–336.CrossRefGoogle Scholar
Wang, X., Li, G., Qian, S., Li, J. and Wang, Z. (2019c), ‘High order well-balanced finite difference WENO schemes for shallow water flows along channels with irregular geometry’, Appl. Math. Comput. 363, 124587.Google Scholar
Wise, S. M., Lowengrub, J. S., Frieboes, H. B. and Cristini, V. (2008), ‘Three-dimensional multispecies nonlinear tumor growth, I: Model and numerical method’, J. Theoret. Biol. 253, 524–543.CrossRefGoogle Scholar
Wu, H., Ma, P. C., Jaravel, T. and Ihme, M. (2019a), ‘Pareto-efficient combustion modeling for improved CO-emission prediction in LES of a piloted turbulent dimethyl ether jet flame’, Proc. Combustion Institute 37, 2267–2276.CrossRefGoogle Scholar
Wu, J., Dong, G. and Li, Y. (2019b), ‘Parallel chemistry acceleration algorithm with ISAT table-size control in the application of gaseous detonation’, Shock Waves 29, 523–535.CrossRefGoogle Scholar
Wu, J., Dong, G. and Li, Y. (2019c), ‘Performance of parallel chemistry acceleration algorithm in simulations of gaseous detonation: Effects of fuel type and numerical scheme resolution’, Combust. Sci. Technol. 191, 2185–2207.CrossRefGoogle Scholar
Xing, Y. and Shu, C.-W. (2005), ‘High order finite difference WENO schemes with the exact conservation property for the shallow water equations’, J. Comput. Phys. 208, 206–227.CrossRefGoogle Scholar
Xing, Y. and Shu, C.-W. (2006a), ‘High-order well-balanced finite difference WENO schemes for a class of hyperbolic systems with source terms’, J. Sci. Comput. 27, 477–494.CrossRefGoogle Scholar
Xing, Y. and Shu, C.-W. (2006b), ‘High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms’, J. Comput. Phys. 214, 567–598.CrossRefGoogle Scholar
Xing, Y. and Shu, C.-W. (2006c), ‘A new approach of high order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms’, Commun. Comput. Phys. 1, 101–135.Google Scholar
Xing, Y. and Shu, C.-W. (2011), ‘High-order finite volume WENO schemes for the shallow water equations with dry states’, Adv. Water Resources 34, 1026–1038.CrossRefGoogle Scholar
Xing, Y. and Shu, C.-W. (2013), ‘High order well-balanced WENO scheme for the gas dynamics equations under gravitational fields’, J. Sci. Comput. 54, 645–662.CrossRefGoogle Scholar
Xing, Z., Zhang, C., Cui, H., Hai, Y., Wu, Q. and Min, D. (2019), ‘Space charge accumulation and decay in dielectric materials with dual discrete traps’, Appl. Sci. 9, 4253.CrossRefGoogle Scholar
Xiong, T., Shu, C.-W. and Zhang, M. (2012), ‘WENO scheme with subcell resolution for computing nonconservative Euler equations with applications to one-dimensional compressible two-medium flows’, J. Sci. Comput. 53, 222–247.CrossRefGoogle Scholar
Xiong, T., Zhang, M., Shu, C.-W., Wong, S. C. and Zhang, P. (2011), ‘High-order computational scheme for a dynamic continuum model for bi-directional pedestrian flow’, Computer Aided Civil Infrastructure Engrg 26, 298–310.CrossRefGoogle Scholar
Xu, J. and Zhao, H. (2003), ‘An Eulerian formulation for solving partial differential equations along a moving interface’, J. Sci. Comput. 19, 573–594.CrossRefGoogle Scholar
Xu, J., Li, Z., Lowengrub, J. and Zhao, H. (2006), ‘A level-set method for interfacial flows with surfactant’, J. Comput. Phys. 212, 590–616.CrossRefGoogle Scholar
Yang, L., Li, T., Wong, S. C., Shu, C.-W. and Zhang, M. (2019a), ‘Modeling and simulation of urban air pollution from the dispersion of vehicle exhaust: A continuum modeling approach’, Internat. J. Sustainable Transport. 13, 722–740.CrossRefGoogle Scholar
Yang, Y., Roy, I., Shu, C.-W. and Fang, L.-Z. (2011), ‘Effect of dust on Ly $\alpha$ photon transfer in optically thick halo’, Astrophys. J. 739, 91.CrossRefGoogle Scholar
Yang, Y., Roy, I., Shu, C.-W. and Fang, L.-Z. (2013), ‘Angular distribution of Ly $\alpha$ resonant photons emerging from optically thick medium’, Astrophys. J. 772, 3.CrossRefGoogle Scholar
Yang, Y., Wang, H., Sun, M., Wang, Z. and Wang, Y. (2019b), ‘Numerical investigation of transverse jet in supersonic crossflow using a high-order nonlinear filter scheme’, Acta Astronautica 154, 74–81.CrossRefGoogle Scholar
Yu, J.-L. and Lu, X.-Y. (2019), ‘Topological evolution near the turbulent/non-turbulent interface in turbulent mixing layer’, J. Turbul. 20, 300–321.CrossRefGoogle Scholar
Yu, P., Watanabe, H., Zhang, W., Kurose, R. and Kitagawa, T. (2019a), ‘Flamelet model for a three-feed non-premixed combustion system with a diluent stream: Analysis and validation of quasi-two-dimensional flamelet (Q2DF) models’, Energy Fuels 33, 4640–4650.CrossRefGoogle Scholar
Yu, R. and Lipatnikov, A. N. (2019), ‘Surface-averaged quantities in turbulent reacting flows and relevant evolution equations’, Phys. Rev. E 100, 013107.CrossRefGoogle Scholar
Yu, R., Nillson, T., Bai, X.-S. and Lipatnikov, A. N. (2019b), ‘Evolution of averaged local premixed flame thickness in a turbulent flow’, Combust. Flame 207, 232–249.CrossRefGoogle Scholar
Zanna, L. D., Zanotti, O., Bucciantini, N. and Londrillo, P. (2007), ‘ECHO: A Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics’, Astronom. Astrophys. 473, 11–30.CrossRefGoogle Scholar
Zhang, D., Tan, J. and Yao, X. (2019a), ‘Direct numerical simulation of spatially developing highly compressible mixing layer: Structural evolution and turbulent statistics’, Phys. Fluids 31, 036102.CrossRefGoogle Scholar
Zhang, J., Liu, H. and Ba, Y. (2019b), ‘Numerical study of droplet dynamics on a solid surface with insoluble surfactants’, Langmuir 24, 7858–7870.CrossRefGoogle Scholar
Zhang, M., Shu, C.-W., Wong, G. C. K. and Wong, S. C. (2003a), ‘A weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill–Whitham–Richards traffic flow model’, J. Comput. Phys. 191, 639–659.CrossRefGoogle Scholar
Zhang, P., Wong, S. C. and Shu, C.-W. (2006a), ‘A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway’, J. Comput. Phys. 212, 739–756.CrossRefGoogle Scholar
Zhang, S., Jiang, S. and Shu, C.-W. (2008), ‘Development of nonlinear weighted compact schemes with increasingly higher order accuracy’, J. Comput. Phys. 227, 7294–7321.CrossRefGoogle Scholar
Zhang, S., Jiang, S., Zhang, Y.-T. and Shu, C.-W. (2009a), ‘The mechanism of sound generation in the interaction between a shock wave and two counter-rotating vortices’, Phys. Fluids 21, 076101.CrossRefGoogle Scholar
Zhang, S., Li, H., Liu, X., Zhang, H. and Shu, C.-W. (2013), ‘Classification and sound generation of two-dimensional interaction of two Taylor vortices’, Phys. Fluids 25, 056103.CrossRefGoogle Scholar
Zhang, S., Zhang, H. and Shu, C.-W. (2009b), ‘Topological structure of shock induced vortex breakdown’, J. Fluid. Mech. 639, 343–372.CrossRefGoogle Scholar
Zhang, S., Zhang, Y.-T. and Shu, C.-W. (2005), ‘Multistage interaction of a shock wave and a strong vortex’, Phys. Fluids 17, 116101.CrossRefGoogle Scholar
Zhang, S., Zhang, Y.-T. and Shu, C.-W. (2006b), ‘Interaction of an oblique shock wave with a pair of parallel vortices: Shock dynamics and mechanism of sound generation’, Phys. Fluids 18, 126101.CrossRefGoogle Scholar
Zhang, X. and Shu, C.-W. (2010a), ‘On maximum-principle-satisfying high order schemes for scalar conservation laws’, J. Comput. Phys. 229, 3091–3120.CrossRefGoogle Scholar
Zhang, X. and Shu, C.-W. (2010b), ‘On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes’, J. Comput. Phys. 229, 8918–8934.CrossRefGoogle Scholar
Zhang, X. and Shu, C.-W. (2011), ‘Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: Survey and new developments’, Proc. Roy. Soc. A 467, 2752–2776.CrossRefGoogle Scholar
Zhang, X. and Shu, C.-W. (2012), ‘Positivity-preserving high order finite difference WENO schemes for compressible Euler equations’, J. Comput. Phys. 231, 2245–2258.CrossRefGoogle Scholar
Zhang, X., Xia, Y. and Shu, C.-W. (2012), ‘Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes’, J. Sci. Comput. 50, 29–62.CrossRefGoogle Scholar
Zhang, Y.-T. and Shu, C.-W. (2003), ‘High order WENO schemes for Hamilton–Jacobi equations on triangular meshes’, SIAM J . Sci. Comput. 24, 1005–1030.Google Scholar
Zhang, Y.-T. and Shu, C.-W. (2009), ‘Third order WENO scheme on three dimensional tetrahedral meshes’, Commun. Comput. Phys. 5, 836–848.Google Scholar
Zhang, Y.-T., Shi, J., Shu, C.-W. and Zhou, Y. (2003b), ‘Numerical viscosity and resolution of high-order weighted essentially nonoscillatory schemes for compressible flows with high Reynolds numbers’, Phys. Rev. E 68, 046709.CrossRefGoogle Scholar
Zhang, Y.-T., Shu, C.-W. and Zhou, Y. (2006c), ‘Effects of shock waves on Rayleigh–Taylor instability’, Phys. Plasmas 13, 062705.CrossRefGoogle Scholar
Zhao, G., Sun, M., Wu, J., Cui, X. and Wang, H. (2019), ‘Investigation of flame flashback phenomenon in a supersonic crossflow with ethylene injection upstream of cavity flameholder’, Aerosp. Sci. Technol. 87, 190–206.CrossRefGoogle Scholar
Zheng, F., Shu, C.-W. and Qiu, J. (2017), ‘Finite difference Hermite WENO schemes for the Hamilton–Jacobi equations’, J. Comput. Phys. 337, 27–41.CrossRefGoogle Scholar
Zheng, F., Shu, C.-W. and Qiu, J. (2019), ‘High order finite difference Hermite WENO schemes for the Hamilton–Jacobi equations on unstructured meshes’, Comput. Fluids 183, 53–65.CrossRefGoogle Scholar
Zhong, X. and Shu, C.-W. (2013), ‘A simple weighted essentially nonoscillatory limiter for Runge–Kutta discontinuous Galerkin methods’, J. Comput. Phys. 232, 397–415.CrossRefGoogle Scholar
Zhou, H., You, J., Xiong, S., Yang, Y., Thevenin, D. and Chen, S. (2019), ‘Interactions between the premixed flame front and the three-dimensional Taylor–Green vortex’, Proc. Combustion Institute 37, 2461–2468.CrossRefGoogle Scholar
Zhu, J. and Qiu, J. (2016), ‘A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws’, J. Comput. Phys. 318, 110–121.CrossRefGoogle Scholar
Zhu, J. and Qiu, J. (2017), ‘A new type of finite volume WENO schemes for hyperbolic conservation laws’, J. Sci. Comput. 73, 1338–1359.CrossRefGoogle Scholar
Zhu, J. and Shu, C.-W. (2018), ‘A new type of multi-resolution WENO schemes with increasingly higher order of accuracy’, J. Comput. Phys. 375, 659–683.CrossRefGoogle Scholar
Zhu, J. and Shu, C.-W. (2019), ‘A new type of multi-resolution WENO schemes with increasingly higher order of accuracy on triangular meshes’, J. Comput. Phys. 392, 19–33.CrossRefGoogle Scholar
Zhu, J. and Shu, C.-W. (2020), ‘A new type of third-order finite volume multi-resolution WENO schemes on tetrahedral meshes’, J. Comput. Phys. 406, 109212.CrossRefGoogle Scholar
Zhu, J., Qiu, J. and Shu, C.-W. (2020), ‘High-order Runge–Kutta discontinuous Galerkin methods with a new type of multi-resolution WENO limiters’, J. Comput. Phys. 404, 109105.CrossRefGoogle Scholar
Zhu, J., Zhong, X., Shu, C.-W. and Qiu, J. (2017), ‘Runge–Kutta discontinuous Galerkin method with a simple and compact Hermite WENO limiter on unstructured meshes’, Commun. Comput. Phys. 21, 623–649.CrossRefGoogle Scholar
Zhu, W., Feng, L.-L., Xia, Y., Shu, C.-W., Gu, Q. and Fang, L.-Z. (2013), ‘Turbulence in the intergalactic medium: Solenoidal and dilatational motions and the impact of numerical viscosity’, Astrophys. J. 777, 48.CrossRefGoogle Scholar
Zhu, Y., Gao, L., Luo, K. H., Pan, J., Pan, Z. and Zhang, P. (2019a), ‘Flame evolution in shock-accelerated flow under different reactive gas mixture gradients’, Phys. Rev. E 100, 013111.CrossRefGoogle Scholar
Zhu, Y., Yang, Z., Luo, K. H., Pan, J. and Pan, Z. (2019b), ‘Numerical investigation of planar shock wave impinging on spherical gas bubble with different densities’, Phys. Fluids 31, 056101.Google Scholar