A new class of moving-least-squares WENO-SPH schemes. (English) Zbl 1349.76661
Summary: We present a new class of meshless Lagrangian particle methods based on the SPH formulation of Vila and Ben Moussa, combined with a new weighted essentially non-oscillatory (WENO) reconstruction technique on moving point clouds in multiple space dimensions. The key idea is to produce for each particle first a set of high order accurate Moving-Least-Squares (MLS) reconstructions on a set of different reconstruction stencils. Then, these reconstructions are combined with each other using a non-linear WENO technique in order to capture at the same time discontinuities and to maintain accuracy and low numerical dissipation in smooth regions. The numerical fluxes between interacting particles are subsequently evaluated using this MLS-WENO reconstruction at the midpoint between two particles, in combination with a Riemann solver that provides the necessary stabilization of the scheme based on the underlying physics of the governing equations. We propose the use of two different Riemann solvers: the Rusanov flux and an Osher-type flux. The use of monotone fluxes together with a WENO reconstruction ensures accuracy, stability, robustness and an essentially non-oscillatory solution without the artificial viscosity term usually employed in conventional SPH schemes. To our knowledge, this is the first time that the WENO method, which has originally been developed for mesh-based schemes in the Eulerian framework on fixed grids, is extended to meshfree Lagrangian particle methods like SPH in multiple space dimensions. We test the new algorithm on two dimensional blast wave problems and on the classical one-dimensional Sod shock tube problem for the Euler equations of compressible gas dynamics. We obtain a good agreement with the exact or numerical reference solution in all cases and an improved accuracy and robustness compared to existing standard SPH schemes.
MSC:
| 76M28 | Particle methods and lattice-gas methods |
| 65M75 | Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs |
| 76Nxx | Compressible fluids and gas dynamics |
| 35L65 | Hyperbolic conservation laws |
Keywords:
smooth particle hydrodynamics (SPH); meshless Lagrangian particle methods; moving-least-squares (MLS); high order meshless WENO reconstruction; Riemann solvers; hyperbolic conservation lawsSoftware:
HE-E1GODFReferences:
| [1] | Liu, M. B.; Liu, G. R., Smoothed particle hydrodynamics (SPH): an overview and recent developments, Arch. Comput. Methods Eng., 17, 25-76 (2010) · Zbl 1348.76117 |
| [2] | Lucy, L. B., A numerical approach to the testing of the fission hypothesis, Astron. J., 82, 1013-1024 (1977) |
| [3] | Gingold, R. A.; Monaghan, J. J., Smoothed particle hydrodynamics - Theory and application to non-spherical stars, Mon. Not. R. Astron. Soc., 181, 375-389 (November 1977) · Zbl 0421.76032 |
| [4] | Libersky, L. D.; Petschek, A. G., Smooth particle hydrodynamics with strength of materials, (Trease, H. E.; Fritts, M. F.; Crowley, W. P., Advances in the Free-Lagrange Method Including Contributions on Adaptive Gridding and the Smooth Particle Hydrodynamics Method. Advances in the Free-Lagrange Method Including Contributions on Adaptive Gridding and the Smooth Particle Hydrodynamics Method, Lecture Notes in Physics, vol. 395 (1991), Springer: Springer Berlin, Heidelberg), 248-257 |
| [5] | Libersky, L. D.; Petschek, A. G.; Carney, T. C.; Hipp, J. R.; Allahdadi, F. A., High strain lagrangian hydrodynamics: a three-dimensional SPH code for dynamic material response, J. Comput. Phys., 109, 1, 67-75 (1993) · Zbl 0791.76065 |
| [6] | Johnson, G. R.; Beissel, S. R., Normalized smoothing functions for SPH impact computations, Int. J. Numer. Methods Eng., 39, 16, 2725-2741 (1996) · Zbl 0880.73076 |
| [7] | Monaghan, J. J., Simulating free surface flows with SPH, J. Comput. Phys., 110, 399-406 (1994) · Zbl 0794.76073 |
| [8] | Monaghan, J. J., Smoothed particle hydrodynamics, Rep. Prog. Phys., 68, 8, 1703 (2005) · Zbl 1160.76399 |
| [9] | Ferrari, A.; Dumbser, M.; Toro, E. F.; Armanini, A., A new 3d parallel SPH scheme for free surface flows, Comput. Fluids, 38, 6, 1203-1217 (2009) · Zbl 1242.76270 |
| [10] | Hu, X. Y.; Adams, N. A., A multi-phase SPH method for macroscopic and mesoscopic flows, J. Comput. Phys., 213, 2, 844-861 (2006) · Zbl 1136.76419 |
| [11] | Deng, L.; Liu, Y.; Wang, W.; Ge, W.; Li, J., A two-fluid smoothed particle hydrodynamics (tf-SPH) method for gas-solid fluidization, Chem. Eng. Sci., 99, 89-101 (2013) |
| [12] | Adami, S.; Hu, X. Y.; Adams, N. A., A new surface-tension formulation for multi-phase {SPH} using a reproducing divergence approximation, J. Comput. Phys., 229, 13, 5011-5021 (2010) · Zbl 1346.76161 |
| [13] | Cleary, P. W.; Monaghan, J. J., Conduction modelling using smoothed particle hydrodynamics, J. Comput. Phys., 148, 1, 227-264 (1999) · Zbl 0930.76069 |
| [14] | Español, P.; Revenga, M., Smoothed dissipative particle dynamics, Phys. Rev. E, 67, 026705 (Feb 2003) |
| [15] | Herrera, P. A.; Massabó, M.; Beckie, R. D., A meshless method to simulate solute transport in heterogeneous porous media, Adv. Water Resour., 32, 3, 413-429 (2009) |
| [16] | Herrera, P. A.; Valocchi, A. J.; Beckie, R. D., A multidimensional streamline-based method to simulate reactive solute transport in heterogeneous porous media, Adv. Water Resour., 33, 7, 711-727 (2010) |
| [17] | Boso, F.; Bellin, A.; Dumbser, M., Numerical simulations of solute transport in highly heterogeneous formations: a comparison of alternative numerical schemes, Adv. Water Resour., 52, 178-189 (2013) |
| [18] | Vignjevic, R.; Campbell, J., Review of development of the smooth particle hydrodynamics (SPH) method, (Hiermaier, S., Predictive Modeling of Dynamic Processes (2009), Springer), 367-396 |
| [19] | Monaghan, J. J., Smoothed particle hydrodynamics and its diverse applications, Annu. Rev. Fluid Mech., 44, 1, 323-346 (2012) · Zbl 1361.76019 |
| [20] | Fang, J.; Parriaux, A.; Rentschler, M.; Ancey, C., Improved {SPH} methods for simulating free surface flows of viscous fluids, Appl. Numer. Math., 59, 2, 251-271 (2009) · Zbl 1194.76202 |
| [21] | Liu, M. B.; Liu, G. R.; Lam, K. Y.; Zong, Z., Smoothed particle hydrodynamics for numerical simulation of underwater explosion, Comput. Mech., 30, 2, 106-118 (2003) · Zbl 1128.76352 |
| [22] | Liu, M. B.; Liu, G. R.; Lam, K. Y., A one-dimensional meshfree particle formulation for simulating shock waves, Shock Waves, 13, 201-211 (2003) · Zbl 1063.76079 |
| [23] | Adami, S.; Hu, X. Y.; Adams, N. A., A conservative SPH method for surfactant dynamics, J. Comput. Phys., 229, 5, 1909-1926 (2010) · Zbl 1329.76281 |
| [24] | Colagrossi, A.; Landrini, M., Numerical simulation of interfacial flows by smoothed particle hydrodynamics, J. Comput. Phys., 191, 2, 448-475 (2003) · Zbl 1028.76039 |
| [25] | Ellero, M., (Proceedings of the IUTAM Symposium on Advances in Micro- and Nanofluidics. Proceedings of the IUTAM Symposium on Advances in Micro- and Nanofluidics, Dresden, Germany, 6-8 September, 2007. Proceedings of the IUTAM Symposium on Advances in Micro- and Nanofluidics. Proceedings of the IUTAM Symposium on Advances in Micro- and Nanofluidics, Dresden, Germany, 6-8 September, 2007, IUTAM (2009), Springer) |
| [26] | Gholami, B.; Comerford, A.; Ellero, M., A multiscale SPH particle model of the near-wall dynamics of leukocytes in flow, Int. J. Numer. Methods Biomed. Eng., 30, 83-102 (2014) |
| [27] | Vázquez-Quesada, A.; Ellero, M.; Español, P., A SPH-based particle model for computational microrheology, Microfluid. Nanofluid., 13, 2, 249-260 (2012) |
| [28] | Belytschko, T.; Krongauz, Y.; Dolbow, J.; Gerlach, C., On the completeness of meshfree particle methods, Int. J. Numer. Methods Eng., 43, 5, 785-819 (1998) · Zbl 0939.74076 |
| [29] | Swegle, J. W.; Hicks, D. L.; Attaway, S. W., Smoothed particle hydrodynamics stability analysis, J. Comput. Phys., 116, 1, 123-134 (1995) · Zbl 0818.76071 |
| [30] | Liu, W. K.; Jun, S.; Zhang, Y. F., Reproducing kernel particle methods, Int. J. Numer. Methods Fluids, 20, 8-9, 1081-1106 (1995) · Zbl 0881.76072 |
| [31] | Dilts, G. A., Moving-least-squares-particle hydrodynamics—I. Consistency and stability, Int. J. Numer. Methods Eng., 44, 8, 1115-1155 (1999) · Zbl 0951.76074 |
| [32] | Zhang, G. M.; Batra, R. C., Modified smoothed particle hydrodynamics method and its application to transient problems, Comput. Mech., 34, 2, 137-146 (2004) · Zbl 1138.74422 |
| [33] | Liu, M. B.; Liu, G. R., Restoring particle consistency in smoothed particle hydrodynamics, Appl. Numer. Math., 56, 1, 19-36 (2006) · Zbl 1329.76285 |
| [34] | Monaghan, J. J., Sph without a tensile instability, J. Comput. Phys., 159, 2, 290-311 (2000) · Zbl 0980.76065 |
| [35] | Balsara, D. S., Von Neumann stability analysis of smoothed particle hydrodynamics—suggestions for optimal algorithms, J. Comput. Phys., 121, 2, 357-372 (1995) · Zbl 0835.76070 |
| [36] | Hu, X. Y.; Adams, N. A., A constant-density approach for incompressible multi-phase SPH, J. Comput. Phys., 228, 6, 2082-2091 (2009) · Zbl 1280.76053 |
| [37] | Xu, R.; Stansby, P.; Laurence, D., Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach, J. Comput. Phys., 228, 18, 6703-6725 (2009) · Zbl 1261.76047 |
| [38] | Ellero, M.; Serrano, M.; Español, P., Incompressible smoothed particle hydrodynamics, J. Comput. Phys., 226, 2, 1731-1752 (October 2007) · Zbl 1121.76050 |
| [39] | Adami, S.; Hu, X. Y.; Adams, N. A., A transport-velocity formulation for smoothed particle hydrodynamics, J. Comput. Phys., 241, 292-307 (2013) · Zbl 1349.76659 |
| [40] | Vila, J. P., On particle weighted methods and smooth particle hydrodynamics, Math. Models Methods Appl. Sci., 09, 02, 161-209 (1999) · Zbl 0938.76090 |
| [41] | Ben Moussa, B., On the convergence of SPH method for scalar conservation laws with boundary conditions, Methods Appl. Anal., 13, 29-62 (2006) · Zbl 1202.65121 |
| [42] | Inutsuka, S., Reformulation of smoothed particle hydrodynamics with Riemann solver, J. Comput. Phys., 179, 1, 238-267 (2002) · Zbl 1060.76094 |
| [43] | Puri, K.; Ramachandran, P., A comparison of SPH schemes for the compressible Euler equations, J. Comput. Phys., 256, 308-333 (2014) · Zbl 1349.76728 |
| [44] | Abel, T., rpSPH: A novel smoothed particle hydrodynamics algorithm, Mon. Not. R. Astron. Soc., 413, 271-285 (2011) |
| [45] | Renaut, G. A.; Marongiu, J. C.; Leduc, J.; Leboeuf, F., On the high-order reconstruction for Meshfree Particle Methods in Numerical Flow Simulation, (2nd ECCOMAS Young Investigators Conference (YIC 2013) (2013), Bordeaux: Bordeaux France) |
| [46] | Xueying, Z.; Haiyan, T.; Leihsin, K.; Wen, C., A contact SPH method with high-order limiters for simulation of inviscid compressible flows, Commun. Comput. Phys., 14, 2 (2013) · Zbl 1373.76289 |
| [47] | Cheng, J.; Shu, C. W., A high order ENO conservative Lagrangian type scheme for the compressible Euler equations, J. Comput. Phys., 227, 1567-1596 (2007) · Zbl 1126.76035 |
| [48] | Liu, W.; Cheng, J.; Shu, C. W., High order conservative Lagrangian schemes with Lax-Wendroff type time discretization for the compressible Euler equations, J. Comput. Phys., 228, 8872-8891 (2009) · Zbl 1287.76181 |
| [49] | Boscheri, W.; Dumbser, M., Arbitrary-Lagrangian-Eulerian one-step WENO finite volume schemes on unstructured triangular meshes, Commun. Comput. Phys., 14, 1174-1206 (2013) · Zbl 1388.65075 |
| [50] | Dumbser, M.; Boscheri, W., High-order unstructured Lagrangian one-step WENO finite volume schemes for non-conservative hyperbolic systems: applications to compressible multi-phase flows, Comput. Fluids, 86, 405-432 (2013) · Zbl 1290.76081 |
| [51] | Boscheri, W.; Balsara, D. S.; Dumbser, M., Lagrangian ADER-WENO finite volume schemes on unstructured triangular meshes based on genuinely multidimensional HLL Riemann solvers, J. Comput. Phys., 267, 112-135 (2014) · Zbl 1349.76309 |
| [52] | Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (1997), Springer · Zbl 0888.76001 |
| [53] | Dumbser, M.; Toro, E. F., On universal Osher-type schemes for general nonlinear hyperbolic conservation laws, Commun. Comput. Phys., 10, 635-671 (2011) · Zbl 1373.76125 |
| [54] | Dumbser, M.; Toro, E. F., A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems, J. Sci. Comput., 48, 1-3, 70-88 (2011) · Zbl 1220.65110 |
| [55] | Boscheri, W.; Dumbser, M., Arbitrary-Lagrangian-Eulerian one-step WENO finite volume schemes on unstructured triangular meshes, Commun. Comput. Phys., 14, 1174-1206 (2013) · Zbl 1388.65075 |
| [56] | Friedrich, O., Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids, J. Comput. Phys., 144, 1, 194-212 (1998) · Zbl 1392.76048 |
| [57] | Käser, M.; Iske, A., ADER schemes on adaptive triangular meshes for scalar conservation laws, J. Comput. Phys., 205, 486-508 (2005) · Zbl 1072.65116 |
| [58] | 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, 1, 204-243 (2007) · Zbl 1124.65074 |
| [59] | Jiang, G. S.; Shu, C. W., Efficient implementation of weighted eno schemes, J. Comput. Phys., 126, 1, 202-228 (1996) · Zbl 0877.65065 |
| [60] | Hu, C.; Shu, C. W., Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150, 1, 97-127 (1999) · Zbl 0926.65090 |
| [61] | 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) · Zbl 1110.65077 |
| [62] | Hockney, R. W.; Eastwood, J. W., Computer Simulation Using Particles (1981), McGraw-Hill: McGraw-Hill New York · Zbl 0662.76002 |
| [63] | Monaghan, J. J., Smoothed particle hydrodynamics, Annu. Rev. Astron. Astrophys., 30, 543-574 (1992) |
| [65] | Agossler, A., Moving least-squares: a numerical differentiation method for irregularly spaced calculation points (2001), Sandia National Labs., Technical report |
| [66] | Dumbser, M.; Balsara, D. S.; Toro, E. F.; Munz, C. D., A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes, J. Comput. Phys., 227, 18, 8209-8253 (2008) · Zbl 1147.65075 |
| [67] | Dumbser, M.; Enaux, C.; Toro, E. F., Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, J. Comput. Phys., 227, 8, 3971-4001 (2008) · Zbl 1142.65070 |
| [68] | 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 |
| [69] | Titarev, V. A.; Tsoutsanis, P.; Drikakis, D., WENO schemes for mixed-element unstructured meshes, Commun. Comput. Phys., 8, 585-609 (2010) · Zbl 1364.76121 |
| [70] | Tsoutsanis, P.; Titarev, V. A.; Drikakis, D., WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions, J. Comput. Phys., 230, 1585-1601 (2011) · Zbl 1210.65160 |
| [71] | Titarev, V. A.; Toro, E. F., Finite-volume {WENO} schemes for three-dimensional conservation laws, J. Comput. Phys., 201, 1, 238-260 (2004) · Zbl 1059.65078 |
| [72] | Gottlieb, S.; Shu, C. W., Total variation diminishing Runge-Kutta schemes, Math. Comput., 67, 73-85 (1998) · Zbl 0897.65058 |
| [73] | Xu, F.; Zhao, Y.; Yan, R.; Furukawa, T., Multidimensional discontinuous SPH method and its application to metal penetration analysis, Int. J. Numer. Methods Eng., 93, 11, 1125-1146 (2013) · Zbl 1352.74121 |
| [74] | Sod, G. A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 27, 1, 1-31 (1978) · Zbl 0387.76063 |
| [75] | Titarev, V. A.; Toro, E. F., ADER schemes for three-dimensional nonlinear hyperbolic systems, J. Comput. Phys., 204, 715-736 (2005) · Zbl 1060.65641 |
| [76] | Aboiyar, T.; Georgoulis, E. H.; Iske, A., Adaptive ADER methods using kernel-based polyharmonic spline WENO reconstruction, SIAM J. Sci. Comput., 32, 6, 3251-3277 (Nov 2010) · Zbl 1221.65236 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
