×

zbMATH — the first resource for mathematics

High-order unstructured Lagrangian one-step WENO finite volume schemes for non-conservative hyperbolic systems: applications to compressible multi-phase flows. (English) Zbl 1290.76081
Summary: In this article we present the first better than second order accurate unstructured Lagrangian-type one-step WENO finite volume scheme for the solution of hyperbolic partial differential equations with non-conservative products. The method achieves high order of accuracy in space together with essentially non-oscillatory behavior using a non-linear WENO reconstruction operator on unstructured triangular meshes. High order accuracy in time is obtained via a local Lagrangian space-time Galerkin predictor method that evolves the spatial reconstruction polynomials in time within each element. The final one-step finite volume scheme is derived by integration over a moving space-time control volume, where the non-conservative products are treated by a path-conservative approach that defines the jump terms on the element boundaries. The entire method is formulated as an Arbitrary-Lagrangian-Eulerian (ALE) method, where the mesh velocity can be chosen independently of the fluid velocity.The new scheme is applied to the full seven-equation Baer-Nunziato model of compressible multi-phase flows with relaxation source terms in two space dimensions. The use of a Lagrangian approach allows an excellent resolution of the solid contact and the resolution of jumps in the volume fraction. The high order of accuracy of the scheme in space and time is confirmed via a numerical convergence study. Finally, the proposed method is also applied to a reduced version of the compressible Baer-Nunziato model for the simulation of free surface water waves in moving domains. In particular, the phenomenon of sloshing is studied in a moving water tank and comparisons with experimental data are provided.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76N15 Gas dynamics (general theory)
76T99 Multiphase and multicomponent flows
Software:
ReALE; RIEMANN; CHIC
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Abgrall, R., On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, Journal of Computational Physics, 144, 45-58, (1994) · Zbl 0822.65062
[2] Abgrall, R.; Karni, S., Computations of compressible multifluids, Journal of Computational Physics, 169, 594-623, (2001) · Zbl 1033.76029
[3] Abgrall, R.; Karni, S., A comment on the computation of non-conservative products, Journal of Computational Physics, 229, 2759-2763, (2010) · Zbl 1188.65134
[4] Abgrall, R.; Nkonga, B.; Saurel, R., Efficient numerical approximation of compressible multi-material flow for unstructured meshes, Computers and Fluids, 32, 571-605, (2003) · Zbl 1084.76543
[5] Abgrall, R.; Saurel, R., Discrete equations for physical and numerical compressible multiphase mixtures, Journal of Computational Physics, 186, 361-396, (2003) · Zbl 1072.76594
[6] Aboiyar, T.; Georgoulis, E. H.; Iske, A., Adaptive ADER methods using kernel-based polyharmonic spline WENO reconstruction, SIAM Journal on Scientific Computing, 32, 3251-3277, (2010) · Zbl 1221.65236
[7] Akyildiz, H.; Unal, E., Experimental investigation of pressure distribution on a rectangular tank due to the liquid sloshing, Ocean Engineering, 32, 1503-1516, (2005)
[8] Andrianov, N.; Saurel, R.; Warnecke, G., A simple method for compressible multiphase mixtures and interfaces, International Journal for Numerical Methods in Fluids, 41, 109-131, (2003) · Zbl 1025.76025
[9] Andrianov, N.; Warnecke, G., The Riemann problem for the Baer-Nunziato two-phase flow model, Journal of Computational Physics, 212, 434-464, (2004) · Zbl 1115.76414
[10] Armenio, V.; La Rocca, M., On the analysis of sloshing of water in rectangular containers: numerical and experimental investigation, Ocean Engineering, 23, 705-739, (1996)
[11] Baer, M. R.; Nunziato, J. W., A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials, Journal of Multiphase Flow, 12, 861-889, (1986) · Zbl 0609.76114
[12] Balsara, D., Second-order accurate schemes for magnetohydrodynamics with divergence-free reconstruction, The Astrophysical Journal Supplement Series, 151, 149-184, (2004)
[13] Balsara, D.; Shu, C. W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, Journal of Computational Physics, 160, 405-452, (2000) · Zbl 0961.65078
[14] Balsara, D. S., Multidimensional HLLE Riemann solver: application to Euler and magnetohydrodynamic flows, Journal of Computational Physics, 229, 1970-1993, (2010) · Zbl 1303.76140
[15] Balsara, D. S., A two-dimensional HLLC Riemann solver for conservation laws: application to Euler and magnetohydrodynamic flows, Journal of Computational Physics, 231, 7476-7503, (2012) · Zbl 1284.76261
[16] Barth TJ, Frederickson PO. Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA paper no. 90-0013, 28th aerospace sciences meeting; January 1990.
[17] Batchelor, G. K., An introduction to fluid mechanics, (1974), Cambridge University Press
[18] Benson, D. J., Computational methods in Lagrangian and Eulerian hydrocodes, Computer Methods in Applied Mechanics and Engineering, 99, 235-394, (1992) · Zbl 0763.73052
[19] Berndt, M.; Breil, J.; Galera, S.; Kucharik, M.; Maire, P. H.; Shashkov, M., Two-step hybrid conservative remapping for multimaterial arbitrary Lagrangian-Eulerian methods, Journal of Computational Physics, 230, 6664-6687, (2011) · Zbl 1408.65077
[20] Boscheri, W.; Dumbser, M., Arbitrary-Lagrangian-Eulerian one-step WENO finite volume schemes on unstructured triangular meshes, Commun Comput Phys, 14, 5, 1174-1206, (2013) · Zbl 1388.65075
[21] Boscheri, W.; Dumbser, M.; Righetti, M., A semi-implicit scheme for 3d free surface flows with high order velocity reconstruction on unstructured Voronoi meshes, International Journal for Numerical Methods in Fluids, 72, 607-631, (2013)
[22] Breil, J.; Galera, S.; Maire, P. H., Multi-material ALE computation in inertial confinement fusion code CHIC, Computers and Fluids, 46, 161-167, (2011) · Zbl 1433.76190
[23] Breil, J.; Harribey, T.; Maire, P. H.; Shashkov, M., A multi-material reale method with MOF interface reconstruction, Computers and Fluids, 83, 115-125, (2013) · Zbl 1290.76094
[24] Caramana, E. J.; Burton, D. E.; Shashkov, M. J.; Whalen, P. P., The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, Journal of Computational Physics, 146, 227-262, (1998) · Zbl 0931.76080
[25] Carré, G.; Del Pino, S.; Després, B.; Labourasse, E., A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension, Journal of Computational Physics, 228, 5160-5183, (2009) · Zbl 1168.76029
[26] Castro, M. J.; Gallardo, J. M.; López, J. A.; Parés, C., Well-balanced high order extensions of godunov’s method for semilinear balance laws, SIAM Journal of Numerical Analysis, 46, 1012-1039, (2008) · Zbl 1159.74045
[27] Castro, M. J.; Gallardo, J. M.; Parés, C., High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products: applications to shallow-water systems, Mathematics of Computation, 75, 1103-1134, (2006) · Zbl 1096.65082
[28] Castro, M. J.; LeFloch, P. G.; Muñoz-Ruiz, M. L.; Parés, C., Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes, Journal of Computational Physics, 227, 8107-8129, (2008) · Zbl 1176.76084
[29] Casulli, V., Semi-implicit finite difference methods for the two-dimensional shallow water equations, Journal of Computational Physics, 86, 56-74, (1990) · Zbl 0681.76022
[30] Casulli, V.; Cheng, R. T., Semi-implicit finite difference methods for three-dimensional shallow water flow, International Journal of Numerical Methods in Fluids, 15, 629-648, (1992) · Zbl 0762.76068
[31] Cesenek, J.; Feistauer, M.; Horacek, J.; Kucera, V.; Prokopova, J., Simulation of compressible viscous flow in time-dependent domains, Applied Mathematics and Computation, 219, 7139-7150, (2013) · Zbl 1426.76233
[32] Chen, B. F., Viscous fluid in a tank under coupled surge, heave and pitch motions, Journal of Waterway, Port, Coastal, and Ocean Engineering - ASCE, 131, 239-256, (2005)
[33] Chen, B. F.; Chiang, H. W., Complete 2d and fully nonlinear analysis of ideal fluid in tanks, Journal of Engineering Mechanics - ASCE, 125, 70-78, (1999)
[34] Chen, B. F.; Nokes, R., Time-independent finite difference analysis of 2d and nonlinear viscous liquid sloshing in a rectangular tank, Journal of Computational Physics, 209, 47-81, (2005) · Zbl 1329.76224
[35] Chen, W.; Haroun, M. A.; Liu, F., Large amplitude liquid sloshing in seismically excited tanks, Earthquake Engineering & Structural Dynamics, 25, 653-669, (1996)
[36] Cheng, J.; Shu, C. W., A high order ENO conservative Lagrangian type scheme for the compressible Euler equations, Journal of Computational Physics, 227, 1567-1596, (2007) · Zbl 1126.76035
[37] Cheng, J.; Shu, C. W., A cell-centered Lagrangian scheme with the preservation of symmetry and conservation properties for compressible fluid flows in two-dimensional cylindrical geometry, Journal of Computational Physics, 229, 7191-7206, (2010) · Zbl 1425.35142
[38] Cheng, J.; Shu, C. W., Improvement on spherical symmetry in two-dimensional cylindrical coordinates for a class of control volume Lagrangian schemes, Communications in Computational Physics, 11, 1144-1168, (2012) · Zbl 1373.76158
[39] Clain, S.; Diot, S.; Loubère, R., A high-order finite volume method for systems of conservation laws - multi-dimensional optimal order detection (MOOD), Journal of Computational Physics, 230, 4028-4050, (2011) · Zbl 1218.65091
[40] Claisse, A.; Després, B.; Labourasse, E.; Ledoux, F., A new exceptional points method with application to cell-centered Lagrangian schemes and curved meshes, Journal of Computational Physics, 231, 4324-4354, (2012) · Zbl 1426.76350
[41] Cockburn, B.; Karniadakis, G. E.; Shu, C. W., Discontinuous Galerkin Methods, Lecture Notes in Computational Science and Engineering, (2000), Springer
[42] Courant, R.; Isaacson, E.; Rees, M., On the solution of nonlinear hyperbolic differential equations by finite differences, Communications on Pure and Applied Mathematics, 5, 243-255, (1952) · Zbl 0047.11704
[43] Deledicque, V.; Papalexandris, M. V., An exact Riemann solver for compressible two-phase flow models containing non-conservative products, Journal of Computational Physics, 222, 217-245, (2007) · Zbl 1216.76044
[44] Després, B.; Mazeran, C., Symmetrization of Lagrangian gas dynamic in dimension two and multimdimensional solvers, Comptes Rendus Mecanique, 331, 475-480, (2003) · Zbl 1293.76089
[45] Després, B.; Mazeran, C., Lagrangian gas dynamics in two-dimensions and Lagrangian systems, Archive for Rational Mechanics and Analysis, 178, 327-372, (2005) · Zbl 1096.76046
[46] Dubcova, L.; Feistauer, M.; Horacek, J.; Svacek, P., Numerical simulation of interaction between turbulent flow and a vibrating airfoil, Computing and Visualization in Science, 12, 207-225, (2009) · Zbl 1426.74127
[47] Dubiner, M., Spectral methods on triangles and other domains, Journal of Scientific Computing, 6, 345-390, (1991) · Zbl 0742.76059
[48] Dumbser, M., Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations, Computers & Fluids, 39, 60-76, (2010) · Zbl 1242.76161
[49] Dumbser, M., A simple two-phase method for the simulation of complex free surface flows, Computer Methods in Applied Mechanics and Engineering, 200, 1204-1219, (2011) · Zbl 1225.76210
[50] Dumbser, M., A diffuse interface method for complex three-dimensional free surface flows, Computer Methods in Applied Mechanics and Engineering, 257, 47-64, (2013) · Zbl 1286.76099
[51] Dumbser, M.; Balsara, D.; Toro, E. F.; Munz, C. D., A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes, Journal of Computational Physics, 227, 8209-8253, (2008) · Zbl 1147.65075
[52] Dumbser, M.; Balsara, D. S., High-order unstructured one-step PNPM schemes for the viscous and resistive MHD equations, CMES - Computer Modeling in Engineering & Sciences, 54, 301-333, (2009) · Zbl 1231.76345
[53] Dumbser, M.; Castro, M.; Parés, C.; Toro, E. F., ADER schemes on unstructured meshes for non-conservative hyperbolic systems: applications to geophysical flows, Computers and Fluids, 38, 1731-1748, (2009) · Zbl 1177.76222
[54] Dumbser, M.; Enaux, C.; Toro, E. F., Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, Journal of Computational Physics, 227, 3971-4001, (2008) · Zbl 1142.65070
[55] Dumbser, M.; Hidalgo, A.; Castro, M.; Parés, C.; Toro, E. F., FORCE schemes on unstructured meshes II: non-conservative hyperbolic systems, Computer Methods in Applied Mechanics and Engineering, 199, 625-647, (2010) · Zbl 1227.76043
[56] Dumbser, M.; Käser, M., Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, Journal of Computational Physics, 221, 693-723, (2007) · Zbl 1110.65077
[57] Dumbser, M.; Käser, M.; Titarev, V. A.; Toro, E. F., Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, Journal of Computational Physics, 226, 204-243, (2007) · Zbl 1124.65074
[58] Dumbser, M.; Toro, E. F., On universal osher-type schemes for general nonlinear hyperbolic conservation laws, Communications in Computational Physics, 10, 635-671, (2011) · Zbl 1373.76125
[59] Dumbser, M.; Toro, E. F., A simple extension of the osher Riemann solver to non-conservative hyperbolic systems, Journal of Scientific Computing, 48, 70-88, (2011) · Zbl 1220.65110
[60] Dumbser, M.; Uuriintsetseg, A.; Zanotti, O., On arbitrary-Lagrangian-Eulerian one-step WENO schemes for stiff hyperbolic balance laws, Communications in Computational Physics, 14, 301-327, (2013) · Zbl 1373.76126
[61] Dumbser, M.; Uuriintsetseg, A.; Zanotti, O., On arbitrary-Lagrangian-Eulerian one-step WENO schemes for stiff hyperbolic balance laws, Communications in Computational Physics, 14, 301-327, (2013) · Zbl 1373.76126
[62] Dumbser, M.; Zanotti, O., Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations, Journal of Computational Physics, 228, 6991-7006, (2009) · Zbl 1261.76028
[63] Faltinsen, O. M., A numerical nonlinear method of sloshing in tanks with two-dimensional flow, Journal of Ship Research, 22, 193-202, (1978)
[64] Faltinsen, O. M.; Rognebakke, O. F.; Lukovsky, I. A.; Timokha, A. N., Adaptive multimodal approach to nonlinear sloshing in a rectangular tank, Journal of Fluid Mechanics, 407, 201-234, (2000) · Zbl 0990.76006
[65] Faltinsen, O. M.; Timokha, A. N., Adaptive multimodal approach to nonlinear sloshing in a rectangular tank, Journal of Fluid Mechanics, 432, 167-200, (2001) · Zbl 0991.76009
[66] Farhat, C.; Roux, F. X., A method of finite element tearing and interconnecting and its parallel solution algorithm, International Journal for Numerical Methods in Engineering, 32, 1205-1227, (1991) · Zbl 0758.65075
[67] Fedkiw, R.; Aslam, T.; Merriman, B.; Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), Journal of Computational Physics, 152, 457-492, (1999) · Zbl 0957.76052
[68] Fedkiw, R. P.; Aslam, T.; Xu, S., The ghost fluid method for deflagration and detonation discontinuities, Journal of Computational Physics, 154, 393-427, (1999) · Zbl 0955.76071
[69] Feistauer, M.; Horacek, J.; Ruzicka, M.; Svacek, P., Numerical analysis of flow-induced nonlinear vibrations of an airfoil with three degrees of freedom, Computers and Fluids, 49, 110-127, (2011) · Zbl 1271.76165
[70] Feistauer, M.; Kucera, V.; Prokopova, J.; Horacek, J., The ALE discontinuous Galerkin method for the simulatio of air flow through pulsating human vocal folds, AIP Conference Proceedings, 1281, 83-86, (2010)
[71] Ferrari, A.; Dumbser, M.; Toro, E. F.; Armanini, A., A new stable version of the SPH method in Lagrangian coordinates, Communications in Computational Physics, 4, 378-404, (2008) · Zbl 1364.76175
[72] Ferrari, A.; Dumbser, M.; Toro, E. F.; Armanini, A., A new 3D parallel SPH scheme for free surface flows, Computers & Fluids, 38, 1203-1217, (2009) · Zbl 1242.76270
[73] Ferrari, A.; Fraccarollo, L.; Dumbser, M.; Toro, E. F.; Armanini, A., Three-dimensional flow evolution after a dambreak, Journal of Fluid Mechanics, 663, 456-477, (2010) · Zbl 1205.76228
[74] Ferrari, A.; Munz, C. D.; Weigand, B., A high order sharp interface method with local timestepping for compressible multiphase flows, Communications in Computational Physics, 9, 205-230, (2011) · Zbl 1284.76265
[75] Friedrich, O., Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids, Journal of Computational Physics, 144, 194-212, (1998) · Zbl 1392.76048
[76] Gassner, G.; Dumbser, M.; Hindenlang, F.; Munz, C. D., Explicit one-step time discretizations for discontinuous Galerkin and finite volume schemes based on local predictors, Journal of Computational Physics, 230, 4232-4247, (2011) · Zbl 1220.65122
[77] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes, III, Journal of Computational Physics, 71, 231-303, (1987) · Zbl 0652.65067
[78] Healy, R. W.; Russel, T. F., Solution of the advection-dispersion equation in two dimensions by a finite-volume eulerian-Lagrangian localized adjoint method, Advances in Water Resources, 21, 11-26, (1998)
[79] Hidalgo, A.; Dumbser, M., ADER schemes for nonlinear systems of stiff advection diffusion reaction equations, Journal of Scientific Computing, 48, 173-189, (2011) · Zbl 1221.65231
[80] Hill, D. F., Transient and steady-state amplitudes of forced waves in rectangular basins, Physics of Fluids, 15, 1576-1587, (2003) · Zbl 1186.76228
[81] Hirt, C.; Amsden, A.; Cook, J., An arbitrary Lagrangian-Eulerian computing method for all flow speeds, Journal of Computational Physics, 14, 227-253, (1974) · Zbl 0292.76018
[82] Hirt, C. W.; Nichols, B. D., Volume of fluid (VOF) method for dynamics of free boundaries, Journal of Computational Physics, 39, 201-225, (1981) · Zbl 0462.76020
[83] Hu, C.; Shu, C. W., Weighted essentially non-oscillatory schemes on triangular meshes, Journal of Computational Physics, 150, 97-127, (1999) · Zbl 0926.65090
[84] Huang, C. S.; Arbogast, T.; Qiu, J., An eulerian-Lagrangian weno finite volume scheme for advection problems, Journal of Computational Physics, 231, 4028-4052, (2012) · Zbl 1260.65083
[85] Idelsohn, S. R.; Oñate, E.; Del Pin, F., The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves, International Journal for Numerical Methods in Engineering, 61, 964-984, (2004) · Zbl 1075.76576
[86] Idelsohn, S. R.; Mier-Torrecilla, M.; Oñate, E., Multi-fluid flows with the particle finite element method, Computer Methods in Applied Mechanics and Engineering, 198, 2750-2767, (2009) · Zbl 1228.76086
[87] Jiang, G. S.; Shu, C. W., Efficient implementation of weighted ENO schemes, Journal of Computational Physics, 126, 202-228, (1996) · Zbl 0877.65065
[88] Kapila, A. K.; Menikoff, R.; Bdzil, J. B.; Son, S. F.; Stewart, D. S., Two-phase modelling of DDT in granular materials: reduced equations, Physics of Fluids, 13, 3002-3024, (2001) · Zbl 1184.76268
[89] Karniadakis, G. E.; Sherwin, S. J., Spectral/hp element methods in CFD, (1999), Oxford University Press · Zbl 0954.76001
[90] Käser, M.; Iske, A., ADER schemes on adaptive triangular meshes for scalar conservation laws, Journal of Computational Physics, 205, 486-508, (2005) · Zbl 1072.65116
[91] Kurganov, A.; Tadmor, E., Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Numerical Methods for Partial Differential Equations, 18, 584-608, (2002) · Zbl 1058.76046
[92] Larese, A.; Rossi, R.; Oñate, E.; Idelsohn, S. R., Validation of the particle finite element method (PFEM) for simulation of the free-surface flows, Engineering Computations, 25, 385-425, (2008) · Zbl 1257.76091
[93] Lax, P. D.; Wendroff, B., Systems of conservation laws, Communications in Pure and Applied Mathematics, 13, 217-237, (1960) · Zbl 0152.44802
[94] Lentine, M.; Grétarsson, Jón Tómas; Fedkiw, R., An unconditionally stable fully conservative semi-Lagrangian method, Journal of Computational Physics, 230, 2857-2879, (2011) · Zbl 1316.76076
[95] Liu, W.; Cheng, J.; Shu, C. W., High order conservative Lagrangian schemes with laxwendroff type time discretization for the compressible Euler equations, Journal of Computational Physics, 228, 8872-8891, (2009) · Zbl 1287.76181
[96] Löhner, R.; Yang, C.; Onate, E., On the simulation of flows with violent free surface motion, Computer Methods in Applied Mechanics and Engineering, 195, 5597-5620, (2006) · Zbl 1122.76070
[97] Loubère, R.; Maire, P. H.; Shashkov, M., Reale: a reconnection arbitrary-Lagrangian-Eulerian method in cylindrical geometry, Computers and Fluids, 46, 59-69, (2011) · Zbl 1305.76066
[98] Loubère, R.; Maire, P. H.; Váchal, P., A second-order compatible staggered Lagrangian hydrodynamics scheme using a cell-centered multidimensional approximate Riemann solver, Procedia Computer Science, 1, 1931-1939, (2010) · Zbl 1432.76206
[99] Luo, H.; Luo, L.; Nourgaliev, R.; Mousseau, V. A.; Dinh, N., A reconstructed discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids, Journal of Computational Physics, 229, 6961-6978, (2010) · Zbl 1425.35138
[100] Luo, H.; Xia, Y.; Spiegel, S.; Nourgaliev, R.; Jiang, Z., A reconstructed discontinuous Galerkin method based on a hierarchical WENO reconstruction for compressible flows on tetrahedral grids, Journal of Computational Physics, 236, 477-492, (2013) · Zbl 1286.65125
[101] Maire, P.-H., A high-order one-step sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids, Computers and Fluids, 46, 1, 341-347, (2011) · Zbl 1433.76137
[102] Maire, P.-H., A unified sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids, International Journal for Numerical Methods in Fluids, 65, 1281-1294, (2011) · Zbl 1429.76089
[103] Maire, P. H., A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry, Journal of Computational Physics, 228, 6882-6915, (2009) · Zbl 1261.76021
[104] Maire, P. H., A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes, Journal of Computational Physics, 228, 2391-2425, (2009) · Zbl 1156.76434
[105] Maire, P. H.; Abgrall, R.; Breil, J.; Ovadia, J., A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, SIAM Journal on Scientific Computing, 29, 1781-1824, (2007) · Zbl 1251.76028
[106] Maire, P. H.; Nkonga, B., Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics, Journal of Computational Physics, 228, 799-821, (2009) · Zbl 1156.76039
[107] Dal Maso, G.; LeFloch, P. G.; Murat, F., Definition and weak stability of nonconservative products, Journal de Mathématiques Pures et Appliquées, 74, 483-548, (1995) · Zbl 0853.35068
[108] Le Métayer, O.; Massoni, J.; Saurel, R., Modelling evaporation fronts with reactive Riemann solvers, Journal of Computational Physics, 205, 567-610, (2005) · Zbl 1088.76051
[109] Monaghan, J. J., Simulating free surface flows with SPH, Journal of Computational Physics, 110, 399-406, (1994) · Zbl 0794.76073
[110] Muñoz, M. L.; Parés, C., Godunov method for nonconservative hyperbolic systems, Mathematical Modelling and Numerical Analysis, 41, 169-185, (2007) · Zbl 1124.65077
[111] Mulder, W.; Osher, S.; Sethian, J. A., Computing interface motion in compressible gas dynamics, Journal of Computational Physics, 100, 209-228, (1992) · Zbl 0758.76044
[112] Munz, C. D., On Godunov-type schemes for Lagrangian gas dynamics, SIAM Journal on Numerical Analysis, 31, 17-42, (1994) · Zbl 0796.76057
[113] Murrone, A.; Guillard, H., A five equation reduced model for compressible two phase flow problems, Journal of Computational Physics, 202, 664-698, (2005) · Zbl 1061.76083
[114] Nakayama, T.; Washizu, K., Nonlinear analysis of liquid motion in a container subjected to forced pitching oscillation, International Journal for Numerical Methods in Engineering, 15, 1207-1220, (1980) · Zbl 0438.76012
[115] Le Métayer, R. SaurelO.; Massoni, J., Modelling evaporation fronts with reactive Riemann solvers, Journal of Computational Physics, 205, 567-610, (2005) · Zbl 1088.76051
[116] Okamoto, T.; Kawahara, M., Two-dimensional sloshing analysis by Lagrangian finite element method, International Journal for Numerical Methods in Fluids, 11, 453-477, (1990) · Zbl 0711.76008
[117] Okamoto, T.; Kawahara, M., 3D sloshing analysis by an arbitrary Lagrangian-Eulerian finite element method, International Journal on Computational Fluid Dynamics, 8, 129-146, (1997) · Zbl 0894.76039
[118] Oñate, E.; Celigueta, M.; Idelsohn, S.; Salazar, F.; Suarez, B., Possibilities of the particle finite element method for fluid-soil-structure interaction problems, Journal of Computational Mechanics, 48, 307-318, (2011) · Zbl 1398.76120
[119] Oñate, E.; Idelsohn, S. R.; Celigueta, M. A.; Rossi, R., Advances in the particle finite element method for the analysis of fluid-multibody interaction and bed erosion in free-surface flows, Computer Methods in Applied Mechanics and Engineering, 197, 1777-1800, (2008) · Zbl 1194.74460
[120] López Ortega, A.; Scovazzi, G., A geometrically-conservative, synchronized, flux-corrected remap for arbitrary Lagrangian-Eulerian computations with nodal finite elements, Journal of Computational Physics, 230, 6709-6741, (2011) · Zbl 1284.76255
[121] Osher, S.; Sethian, J. A., Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79, 12-49, (1988) · Zbl 0659.65132
[122] Osher, S.; Solomon, F., Upwind difference schemes for hyperbolic conservation laws, Mathematics and Computation, 38, 339-374, (1982) · Zbl 0483.65055
[123] Parés, C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM Journal on Numerical Analysis, 44, 300-321, (2006) · Zbl 1130.65089
[124] Parés, C.; Castro, M. J., On the well-balance property of roe’s method for nonconservative hyperbolic systems: applications to shallow-water systems, Mathematical Modelling and Numerical Analysis, 38, 821-852, (2004) · Zbl 1130.76325
[125] Peery, J. S.; Carroll, D. E., Multi-material ale methods in unstructured grids, Computer Methods in Applied Mechanics and Engineering, 187, 591-619, (2000) · Zbl 0980.74068
[126] Petitpas, F.; Massoni, J.; Saurel, R.; Lapebie, E.; Munier, L., Diffuse interface model for high speed cavitating underwater systems, International Journal of Multiphase Flow, 35, 747-759, (2009)
[127] Del Pin, F.; Idelsohn, S. R.; Oñate, E.; Aubry, R., The ALE/Lagrangian particle finite element method: a new approach to computation of free-surface flows and fluid-object interactions, Computers and Fluids, 36, 27-38, (2007) · Zbl 1181.76095
[128] Qiu, Jing-Mei; Shu, Chi-Wang, Conservative high order semi-Lagrangian finite difference weno methods for advection in incompressible flow, Journal of Computational Physics, 230, 863-889, (2011) · Zbl 1391.76489
[129] Rhebergen, S.; Bokhove, O.; van der Vegt, J. J.W., Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations, Journal of Computational Physics, 227, 1887-1922, (2008) · Zbl 1153.65097
[130] Rieber, M.; Frohn, A., A numerical study on the mechanism of splashing, International Journal of Heat and Fluid Flow, 20, 455-461, (1999)
[131] Riemslagh, K.; Vierendeels, J.; Dick, E., An arbitrary Lagrangian-Eulerian finite-volume method for the simulation of rotary displaecment pump flow, Applied Numerical Mathematics, 32, 419-433, (2000) · Zbl 0965.76055
[132] Saurel, R.; Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows, Journal of Computational Physics, 150, 425-467, (1999) · Zbl 0937.76053
[133] Saurel, R.; Gavrilyuk, S.; Renaud, F., A multiphase model with internal degrees of freedom: application to shock-bubble interaction, Journal of Fluid Mechanics, 495, 283-321, (2003) · Zbl 1080.76062
[134] Saurel, R.; Larini, M.; Loraud, J. C., Exact and approximate Riemann solvers for real gases, Journal of Computational Physics, 112, 126-137, (1994) · Zbl 0799.76058
[135] Saurel, R.; Massoni, J.; Renaud, F., A numerical method for one-dimensional compressible multiphase flows on moving meshes, International Journal for Numerical Methods in Fluids, 54, 1425-1450, (2007) · Zbl 1205.76202
[136] Saurel, R.; Petitpas, F.; Abgrall, R., Modelling phase transition in metastable liquids: application to cavitating and flashing flows, Journal of Fluid Mechanics, 607, 313-350, (2008) · Zbl 1147.76060
[137] Saurel, R.; Petitpas, F.; Berry, R. A., Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures, Journal of Computational Physics, 228, 1678-1712, (2009) · Zbl 1409.76105
[138] Schwendeman, D. W.; Wahle, C. W.; Kapila, A. K., The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow, Journal of Computational Physics, 212, 490-526, (2006) · Zbl 1161.76531
[139] Scovazzi, G., Lagrangian shock hydrodynamics on tetrahedral meshes: a stable and accurate variational multiscale approach, Journal of Computational Physics, 231, 8029-8069, (2012)
[140] Shao, J. R.; Li, H. Q.; Liu, G. R.; Liu, M. B., An improved sph method for modeling liquid sloshing dynamics, Computers and Structures, 101, 18-26, (2012)
[141] Smagorinsky, S., General circulation experiments with the primitive equations, Monthly Weather Review, 91, 99-164, (1963)
[142] Smith, R. W., AUSM(ALE): a geometrically conservative arbitrary Lagrangian-Eulerian flux splitting scheme, Journal of Computational Physics, 150, 268-286, (1999) · Zbl 0936.76046
[143] Sonar, T., On the construction of essentially non-oscillatory finite volume approximations to hyperbolic conservation laws on general triangulations: polynomial recovery, accuracy and stencil selection, Computer Methods in Applied Mechanics and Engineering, 140, 157-181, (1997) · Zbl 0898.76086
[144] Stroud, A. H., Approximate calculation of multiple integrals, (1971), Prentice-Hall Inc. Englewood Cliffs, New Jersey · Zbl 0379.65013
[145] Tian, B.; Toro, E. F.; Castro, C. E., A path-conservative method for a five-equation model of two-phase flow with an hllc-type Riemann solver, Computers and Fluids, 46, 122-132, (2011) · Zbl 1433.76166
[146] Titarev, V. A.; Tsoutsanis, P.; Drikakis, D., WENO schemes for mixed-element unstructured meshes, Communications in Computational Physics, 8, 585-609, (2010) · Zbl 1364.76121
[147] Tokareva, S. A.; Toro, E. F., Hllc-type Riemann solver for the baernunziato equations of compressible two-phase flow, Journal of Computational Physics, 229, 3573-3604, (2010) · Zbl 1391.76440
[148] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer · Zbl 0923.76004
[149] Toro, E. F.; Hidalgo, A.; Dumbser, M., FORCE schemes on unstructured meshes I: conservative hyperbolic systems, Journal of Computational Physics, 228, 3368-3389, (2009) · Zbl 1168.65377
[150] Toumi, I., A weak formulation of roe’s approximate Riemann solver, Journal of Computational Physics, 102, 360-373, (1992) · Zbl 0783.65068
[151] Tsoutsanis, P.; Titarev, V. A.; Drikakis, D., WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions, Journal of Computational Physics, 230, 1585-1601, (2011) · Zbl 1210.65160
[152] van Leer, B., Towards the ultimate conservative difference scheme V: a second order sequel to godunov’s method, Journal of Computational Physics, 32, 101-136, (1979) · Zbl 1364.65223
[153] Verhagen, H. G.; Wijingaarden, L., Nonlinear oscillation of fluid in a container, Journal of Fluid Mechanics, 22, 737-751, (1965) · Zbl 0128.43101
[154] von Neumann, J.; Richtmyer, R. D., A method for the calculation of hydrodynamics shocks, Journal of Applied Physics, 21, 232-237, (1950) · Zbl 0037.12002
[155] Wu, G. X.; Ma, Q. A.; Taylor, R. E., Numerical simulation of sloshing waves in a 3d tank based on a finite element method, Applied Ocean Research, 20, 337-355, (1998)
[156] Zein, A.; Hantke, M.; Warnecke, G., Modeling phase transition for compressible two-phase flows applied to metastable liquids, Journal of Computational Physics, 229, 2964-2998, (2010) · Zbl 1307.76079
[157] Zhang, T.; Zheng, Y., Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics systems, SIAM Journal on Mathematical Analysis, 21, 593-630, (1990) · Zbl 0726.35081
[158] Zhang, Y. T.; Shu, C. W., Third order WENO scheme on three dimensional tetrahedral meshes, Communications in Computational Physics, 5, 836-848, (2009) · Zbl 1364.65177
[159] Zhao, W.; Yang, J.; Hu, Z.; Xiao, L., Experimental investigation of effects of inner-tank sloshing on hydrodynamics of an FLNG system, Journal of Hydrodynamics, 24, 107-115, (2012)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.