Numerical symmetry-preserving techniques for low-dissipation shock-capturing schemes. (English) Zbl 1519.76207
Summary: Modern applications of computational fluid dynamics involve complex interactions across scales such as shock interactions with turbulent structures and multiphase interfaces. Such phenomena, which occur at very small physical viscosity, require high-resolution and low-dissipation compressible flow solvers. Many recent publications have focused on the design of high-order accurate numerical schemes and provide e.g. weighted essentially non-oscillatory (WENO) stencils up to 17th order for this purpose. As shown in detail by different authors, such schemes tremendously decrease adverse effects of numerical dissipation. However, such schemes are prone to numerically induced symmetry breaking which renders validation for the targeted problem range problematic.
In this paper, we show that symmetry-breaking relates to vanishing numerical viscosity and is driven systematically by algorithmic floating-point effects which are no longer hidden by numerical dissipation. We propose a systematic procedure to deal with such errors by numerical and algorithmic formulations which respect floating-point arithmetic. We show that by these procedures inherent symmetries are preserved for a broad range of test cases with high-order shock-capturing schemes in particular in the high-resolution limit for both 2D and 3D.
In this paper, we show that symmetry-breaking relates to vanishing numerical viscosity and is driven systematically by algorithmic floating-point effects which are no longer hidden by numerical dissipation. We propose a systematic procedure to deal with such errors by numerical and algorithmic formulations which respect floating-point arithmetic. We show that by these procedures inherent symmetries are preserved for a broad range of test cases with high-order shock-capturing schemes in particular in the high-resolution limit for both 2D and 3D.
MSC:
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 76L05 | Shock waves and blast waves in fluid mechanics |
Keywords:
symmetry breaking; floating-point arithmetic; low-dissipation schemes; high-resolution schemes; WENO; shock-capturingReferences:
| [1] | Hadjadj, A.; Kudryavtsev, A., Computation and flow visualization in high-speed aerodynamics, J Turbul, 6, N16 (2005) |
| [2] | Bermejo-Moreno, I.; Campo, L.; Larsson, J.; Bodart, J.; Helmer, D.; Eaton, J. K., Confinement effects in shock wave/turbulent boundary layer interactions through wall-modelled large-eddy simulations, J Fluid Mech, 758, 5-62 (2014) |
| [3] | Kannan, K.; Kedelty, D.; Herrmann, M., An in-cell reconstruction finite volume method for flows of compressible immiscible fluids, J Comput Phys, 373, 784-810 (2018) · Zbl 1416.76154 |
| [4] | Meng, J. C.; Colonius, T., Numerical simulation of the aerobreakup of a water droplet, J Fluid Mech, 835, 1108-1135 (2018) · Zbl 1419.76540 |
| [5] | Shi, J.; Zhang, Y.-T.; Shu, C.-W., Resolution of high order WENO schemes for complicated flow structures, J Comput Phys, 186, 2, 690-696 (2003) · Zbl 1047.76081 |
| [6] | Cheng, T.; Lee, K., Numerical simulations of underexpanded supersonic jet and free shear layer using WENO schemes, Int J Heat Fluid Flow, 26, 5, 755-770 (2005) |
| [7] | Zhang, Y.-T.; Shi, J.; Shu, C.-W.; Zhou, Y., Numerical viscosity and resolution of high-order weighted essentially nonoscillatory schemes for compressible flows with high reynolds numbers, Phys Rev E, 68, 4, 046709 (2003) |
| [8] | Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S. R., Uniformly high order accurate essentially non-oscillatory schemes, III, J Comput Phys, 71, 2, 231-303 (1987) · Zbl 0652.65067 |
| [9] | Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J Comput Phys, 115, 1, 200-212 (1994) · Zbl 0811.65076 |
| [10] | Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J Comput Phys, 126, 1, 202-228 (1996) · Zbl 0877.65065 |
| [11] | Zhang, P.; Wong, S. C.; Shu, C.-W., A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway, J Comput Phys, 212, 2, 739-756 (2006) · Zbl 1149.65319 |
| [12] | Carrillo, J. A.; Gamba, I. M.; Majorana, A.; Shu, C.-W., A WENO-solver for the transients of Boltzmann – poisson system for semiconductor devices: performance and comparisons with monte carlo methods, J Comput Phys, 184, 2, 498-525 (2003) · Zbl 1034.82063 |
| [13] | Filbet, F.; Shu, C.-W., Approximation of hyperbolic models for chemosensitive movement, SIAM J Sci Comput, 27, 3, 850-872 (2005) · Zbl 1141.35396 |
| [14] | Aràndiga, F.; Belda, A. M., Weighted ENO interpolation and applications, Commun Nonlinear Sci Numer Simulat, 9, 2, 187-195 (2004) · Zbl 1037.41001 |
| [15] | Shu, C.-W., High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Review, 51, 1, 82-126 (2009) · Zbl 1160.65330 |
| [16] | Henrick, A. K.; Aslam, T. D.; Powers, J. M., Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points, J Comput Phys, 207, 2, 542-567 (2005) · Zbl 1072.65114 |
| [17] | Borges, R.; Carmona, M.; Costa, B.; Don, W. S., An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J Comput Phys, 227, 6, 3191-3211 (2008) · Zbl 1136.65076 |
| [18] | Balsara, D. S.; Shu, C.-W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J Comput Phys, 160, 2, 405-452 (2000) · Zbl 0961.65078 |
| [19] | Gerolymos, G.; Sénéchal, D.; Vallet, I., Very-high-order WENO schemes, J Comput Phys, 228, 23, 8481-8524 (2009) · Zbl 1176.65088 |
| [20] | Balsara, D. S.; Garain, S.; Shu, C.-W., An efficient class of WENO schemes with adaptive order, J Comput Phys, 326, 780-804 (2016) · Zbl 1422.65146 |
| [21] | Fu, L.; Hu, X. Y.; Adams, N. A., A family of high-order targeted ENO schemes for compressible-fluid simulations, J Comput Phys, 305, 333-359 (2016) · Zbl 1349.76462 |
| [22] | Fu, L.; Hu, X. Y.; Adams, N. A., Targeted ENO schemes with tailored resolution property for hyperbolic conservation laws, J Comput Phys, 349, 97-121 (2017) · Zbl 1380.65154 |
| [23] | Rossinelli D., Hejazialhosseini B., Hadjidoukas P., Bekas C., Curioni A., Bertsch A., et al. 11 PFLOP/S simulations of cloud cavitation collapse. Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis2013;:1-13.; Rossinelli D., Hejazialhosseini B., Hadjidoukas P., Bekas C., Curioni A., Bertsch A., et al. 11 PFLOP/S simulations of cloud cavitation collapse. Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis2013;:1-13. |
| [24] | Remacle, J.-F.; Flaherty, J. E.; Shephard, M. S., An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems, SIAM Rev., 45, 1, 53-72 (2003) · Zbl 1127.65323 |
| [25] | Xu, Z.; Shu, C.-W., Anti-diffusive flux corrections for high order finite difference WENO schemes, J Comput Phys, 205, 2, 458-485 (2005) · Zbl 1087.76080 |
| [26] | Sun, Z.-S.; Luo, L.; Ren, Y.-X.; Zhang, S.-Y., A sixth order hybrid finite difference scheme based on the minimized dispersion and controllable dissipation technique, J Comput Phys, 270, 238-254 (2014) · Zbl 1349.76537 |
| [27] | Delin, C.; Zhongguo, S.; Zhu, H.; Guang, X., Improvement of the weighted essentially nonoscillatory scheme based on the interaction of smoothness indicators, Int J Numer Methods Fluids, 85, 12, 693-711 (2017) |
| [28] | 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) · Zbl 1349.76426 |
| [29] | Sutherland, R. S., A new computational fluid dynamics code i: Fyris alpha, Astrophys Space Sci, 327, 2, 173-206 (2010) · Zbl 1195.85004 |
| [30] | Toro, E. F., Riemann solvers and numerical methods for fluid dynamics: a practical introduction (2013), Springer Science & Business Media |
| [31] | LeVeque, R. J., Finite volume methods for hyperbolic problems, 31 (2002), Cambridge University Press · Zbl 1010.65040 |
| [32] | Gottlieb, S.; Shu, C.-W.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev, 43, 1, 89-112 (2001) · Zbl 0967.65098 |
| [33] | Roe, P. L., Approximate riemann solvers, parameter vectors, and difference schemes, J Comput Phys, 43, 2, 357-372 (1981) · Zbl 0474.65066 |
| [34] | Harten, A.; Lax, P. D.; Leer, B.v., On upstream differencing and godunov-type schemes for hyperbolic conservation laws, SIAM Rev, 25, 1, 35-61 (1983) · Zbl 0565.65051 |
| [35] | Toro, E. F.; Spruce, M.; Speares, W., Restoration of the contact surface in the HLL-Riemann solver, Shock Waves, 4, 1, 25-34 (1994) · Zbl 0811.76053 |
| [36] | Batten, P.; Clarke, N.; Lambert, C.; Causon, D. M., On the choice of wavespeeds for the HLLC riemann solver, SIAM J Sci Comput, 18, 6, 1553-1570 (1997) · Zbl 0992.65088 |
| [37] | Hu, X. Y.; Adams, N. A.; Shu, C.-W., Positivity-preserving method for high-order conservative schemes solving compressible euler equations, J Comput Phys, 242, 169-180 (2013) · Zbl 1311.76088 |
| [38] | Johnsen, E.; Colonius, T., Implementation of WENO schemes in compressible multicomponent flow problems, J Comput Phys, 219, 2, 715-732 (2006) · Zbl 1189.76351 |
| [39] | Coralic, V.; Colonius, T., Finite-volume WENO scheme for viscous compressible multicomponent flows, J Comput Phys, 274, 95-121 (2014) · Zbl 1351.76100 |
| [40] | Einfeldt, B.; Munz, C.-D.; Roe, P. L.; Sjögreen, B., On godunov-type methods near low densities, J Comput Phys, 92, 2, 273-295 (1991) · Zbl 0709.76102 |
| [41] | Qiu, J.; Shu, C.-W., On the construction, comparison, and local characteristic decomposition for high-order central WENO schemes, J Comput Phys, 183, 1, 187-209 (2002) · Zbl 1018.65106 |
| [42] | Liska, R.; Wendroff, B., Comparison of several difference schemes on 1D and 2D test problems for the euler equations, SIAM J Sci Comput, 25, 3, 995-1017 (2003) · Zbl 1096.65089 |
| [43] | Hu, X.; Wang, Q.; Adams, N. A., An adaptive central-upwind weighted essentially non-oscillatory scheme, J Comput Phys, 229, 23, 8952-8965 (2010) · Zbl 1204.65103 |
| [44] | Schneider, E. E.; Robertson, B. E., CHOLLA: A new massively parallel hydrodynamics code for astrophysical simulation, Astrophys J Supplement Ser, 217, 2, 24 (2015) |
| [45] | Schulz-Rinne, C. W.; Collins, J. P.; Glaz, H. M., Numerical solution of the Riemann problem for two-dimensional gas dynamics, SIAM J Sci Comput, 14, 6, 1394-1414 (1993) · Zbl 0785.76050 |
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