Abstract
In this work, we review the family of direct Arbitrary-Lagrangian–Eulerian (ALE) finite vlume (FV) and discontinuous Galerkin (DG) schemes on moving meshes that at each time step are rearranged by explicitly allowing topology changes, in order to guarantee a robust mesh evolution even for high shear flow and very long evolution times. Two different techniques are presented: a local nonconforming approach for dealing with sliding lines, and a global regeneration of Voronoi tessellations for treating general unpredicted movements. Corresponding elements at consecutive times are connected in space-time to construct closed space-time control volumes, whose bottom and top faces may be polygons with a different number of nodes, with different neighbors, and even degenerate space-time sliver elements. Our final ALE FV-DG scheme is obtained by integrating, over these arbitrary shaped space-time control volumes, the space-time conservation formulation of the governing hyperbolic PDE system: so, we directly evolve the solution in time avoiding any remapping stage, being conservative and satisfying the GCL by construction. Arbitrary high order of accuracy in space and time is achieved through a fully discrete one-step predictor–corrector ADER approach, also integrated with well balancing techniques to further improve the accuracy and to maintain exactly even at discrete level many physical invariants of the studied system. A large set of different numerical tests has been carried out in order to check the accuracy and the robustness of our methods for both smooth and discontinuous problems, in particular in the case of vortical flows.
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References
Balsara D (2004) Second-order accurate schemes for magnetohydrodynamics with divergence-free reconstruction. Astrophys J Suppl Ser 151:149–184
Balsara D, Dumbser M (2015) Divergence-free MHD on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers. J Comput Phys 299:687–715
Barlow A, Maire P, Rider W, Rieben R, Shashkov M (2016) Arbitrary Lagrangian–Eulerian methods for modeling high-speed compressible multimaterial flows. J Comput Phys 322:603–665
Barth T, Frederickson P (1990) Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA paper 90–0013
Barth T, Jespersen D (1989) The design and application of upwind schemes on unstructured meshes. AIAA Pap 89–0366:1–12
Benson DJ (1992) Computational methods in Lagrangian and Eulerian hydrocodes. Comput Methods Appl Mech Eng 99(2):235–394. https://doi.org/10.1016/0045-7825(92)90042-I
Berndt M, Breil J, Galera S, Kucharik M, Maire P, Shashkov M (2011) Two-step hybrid conservative remapping for multimaterial arbitrary Lagrangian–Eulerian methods. J Comput Phys 230:6664–6687
Bertoluzza S, Pino SD, Labourasse E (2016) A conservative slide line method for cell-centered semi-Lagrangian and ALE schemes in 2D. ESAIM Math Model Numer Anal 50(2016):187–214
Bochev P, Ridzal D, Shashkov M (2013) Fast optimization-based conservative remap of scalar fields through aggregate mass transfer. J Comput Phys 246:37–57
Bonazzoli, Gaburro E, Dolean V, Rapetti F (2014) High order edge finite element approximations for the time-harmonic Maxwell’s equations. In: 2014 IEEE conference on antenna measurements & applications (CAMA), Antibes Juan-les-Pins, pp 1–4
Boscheri W (2017) An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics. Int J Numer Methods Fluids 84:76–106
Boscheri W (2017) An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics. Int J Numer Methods Fluids 84(2):76–106
Boscheri W (2017) High order direct Arbitrary-Lagrangian–Eulerian (ALE) finite volume schemes for hyperbolic systems on unstructured meshes. Arch Comput Methods Eng 24(4):751–801
Boscheri W, Balsara D, Dumbser M (2014) Lagrangian ADER-WENO finite volume schemes on unstructured triangular meshes based on genuinely multidimensional HLL Riemann solvers. J Comput Phys 267:112–138
Boscheri W, Dumbser M (2013) Arbitrary-Lagrangian–Eulerian one-step WENO finite volume schemes on unstructured triangular meshes. Commun Comput Phys 14:1174–1206
Boscheri W, Dumbser M (2014) A direct Arbitrary-Lagrangian–Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3D. J Comput Phys 275:484–523
Boscheri W, Dumbser M (2016) High order accurate direct Arbitrary-Lagrangian–Eulerian ADER-WENO finite volume schemes on moving curvilinear unstructured meshes. Comput Fluids 136:48–66
Boscheri W, Dumbser M (2017) Arbitrary-Lagrangian–Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes. J Comput Phys 346:449–479
Boscheri W, Dumbser M, Balsara D (2014) High-order ADER-WENO ale schemes on unstructured triangular meshes application of several node solvers to hydrodynamics and magnetohydrodynamics. Int J Numer Methods Fluids 76(10):737–778
Boscheri W, Dumbser M, Balsara D (2014) High order Lagrangian ADER-WENO schemes on unstructured meshes—application of several node solvers to hydrodynamics and magnetohydrodynamics. Int J Numer Methods Fluids 76:737–778
Boscheri W, Dumbser M, Righetti M (2013) A semi-implicit scheme for 3D free surface flows with high-order velocity reconstruction on unstructured voronoi meshes. Int J Numer Methods Fluids 72(6):607–631
Boscheri W, Dumbser M, Zanotti O (2015) High order cell-centered Lagrangian-type finite volume schemes with time-accurate local time stepping on unstructured triangular meshes. J Comput Phys 291:120–150
Boscheri W, Loubère R (2017) High order accurate direct Arbitrary-Lagrangian–Eulerian ADER-MOOD finite volume schemes for non-conservative hyperbolic systems with stiff source terms. Commun Comput Phys 21:271–312
Boscheri W, Loubère R, Dumbser M (2015) Direct Arbitrary-Lagrangian–Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws. J Comput Phys 292:56–87
Boscheri W, Pisaturo GR, Righetti M (2019) High-order divergence-free velocity reconstruction for free surface flows on unstructured voronoi meshes. Int J Numer Methods Fluids 90(6):296–321
Boscheri W, Semplice M, Dumbser M (2019) Central WENO subcell finite volume limiters for ADER discontinuous Galerkin schemes on fixed and moving unstructured meshes. Commun Comput Phys 25:311–346
Busto S, Ferrín J, Toro EF, Vázquez-Cendón ME (2018) A projection hybrid high order finite volume/finite element method for incompressible turbulent flows. J Comput Phys 353:169–192
Busto S, Chiocchetti S, Dumbser M, Gaburro E, Peshkov I (2020) High order ADER schemes for continuum mechanics. Front Phys. https://doi.org/10.3389/fphy.2020.00032
Caramana E (2009) The implementation of slide lines as a combined force and velocity boundary condition. J Comput Phys 228:3911–3916
Caramana E, Burton D, Shashkov M, Whalen P (1998) The construction of compatible hydrodynamics algorithms utilizing conservation of total energy. J Comput Phys 146:227–262
Carré G, Pino SD, Després B, Labourasse E (2009) A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension. J Comput Phys 228:5160–5183
Castro M, Gallardo J, López J, Parés C (2008) Well-balanced high order extensions of Godunov’s method for semilinear balance laws. SIAM J Numer Anal 46:1012–1039
Castro M, Gallardo J, Marquina A (2016) Approximate Osher–Solomon schemes for hyperbolic systems. Appl Math Comput 272:347–368
Castro M, Gallardo J, Parés C (2006) High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Math Comput 75:1103–1134
Castro M, Pardo A, Parés C (2007) Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique. Math Models Methods Appl Sci 17(12):2055–2113
Castro M, Pardo A, Parés C, Toro E (2010) On some fast well-balanced first order solvers for nonconservative systems. Math Comput 79(271):1427–1472
Castro MJ, Fernández E, Ferriero A, García JA, Parés C (2009) High order extensions of Roe schemes for two dimensional nonconservative hyperbolic systems. J Sci Comput 39:67–114
Castro Díaz MJ, Fernández-Nieto ED (2012) A class of computationally fast first order finite volume solvers: PVM methods. SIAM J Sci Comput 34(4):A2173–A2196
Cheng J, Shu C (2007) A high order ENO conservative Lagrangian type scheme for the compressible Euler equations. J Comput Phys 227:1567–1596
Cheng J, Shu C (2010) A cell-centered Lagrangian scheme with the preservation of symmetry and conservation properties for compressible fluid flows in two-dimensional cylindrical geometry. J Comput Phys 229:7191–7206
Clain S, Diot S, Loubère R (2011) A high-order finite volume method for systems of conservation laws-multi-dimensional optimal order detection (MOOD). J Comput Phys 230(10):4028–4050. https://doi.org/10.1016/j.jcp.2011.02.026
Clair G, Després B, Labourasse E (2013) A new method to introduce constraints in cell-centered Lagrangian schemes. Comput Methods Appl Mech Eng 261–262:56–65
Clair G, Després B, Labourasse E (2014) A one-mesh method for the cell-centered discretization of sliding. Comput Methods Appl Mech Eng 269:315–333
Claisse A, Després B, Labourasse E, Ledoux F (2012) A new exceptional points method with application to cell-centered Lagrangian schemes and curved meshes. J Comput Phys 231:4324–4354
Cravero I, Puppo G, Semplice M, Visconti G (2018) Cweno: uniformly accurate reconstructions for balance laws. Math Comput 87(312):1689–1719
Cremonesi M, Frangi A, Perego U (2010) A lagrangian finite element approach for the analysis of fluid-structure interaction problems. Int J Numer Methods Eng 84(5):610–630
Cremonesi M, Frangi A, Perego U (2011) A lagrangian finite element approach for the simulation of water-waves induced by landslides. Comput Struct 89(11–12):1086–1093
Cremonesi M, Meduri S, Perego U, Frangi A (2017) An explicit Lagrangian finite element method for free-surface weakly compressible flows. Comput Part Mech 4(3):357–369
Cavalcanti JR, Dumbser DdMM M, Junior CF (2015) A conservative finite volume scheme with time-accurate local time stepping for scalar transport on unstructured grids. Adv Water Resour 86:217–230
Dal Maso G, LeFloch P, Murat F (1995) Definition and weak stability of nonconservative products. J Math Pures Appl 74:483–548
Dedner A, Kemm F, Kröner D, Munz CD, Schnitzer T, Wesenberg M (2002) Hyperbolic divergence cleaning for the MHD equations. J Comput Phys 175:645–673
Després B (2017) Numerical methods for Eulerian and Lagrangian conservation laws. Birkhäuser, Boston
Diot S, Clain S, Loubère R (2012) Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials. Comput Fluids 64:43–63. https://doi.org/10.1016/j.compfluid.2012.05.004
Diot S, Loubère R, Clain S (2013) The MOOD method in the three-dimensional case: very-high-order finite volume method for hyperbolic systems. Int J Numer Methods Fluids 73:362–392
Dobrev V, Ellis T, Kolev T, Rieben R (2011) Curvilinear finite elements for lagrangian hydrodynamics. Int J Numer Methods Fluids 65:1295–1310
Dobrev V, Ellis T, Kolev T, Rieben R (2013) High-order curvilinear finite elements for axisymmetric lagrangian hydrodynamics. Comput Fluids 83:58–69
Dobrev V, Kolev T, Rieben R (2012) High-order curvilinear finite element methods for lagrangian hydrodynamics. SIAM J Sci Comput 34(5):B606–B641. https://doi.org/10.1137/120864672
Dumbser M (2014) Arbitrary-Lagrangian–Eulerian ADER-WENO finite volume schemes with time-accurate local time stepping for hyperbolic conservation laws. Comput Methods Appl Mech Eng 280:57–83
Dumbser M, Balsara D (2016) A new, efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems. J Comput Phys 304:275–319
Dumbser M, Balsara D, Toro E, Munz C (2008) A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. J Comput Phys 227:8209–8253
Dumbser M, Boscheri W (2013) 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
Dumbser M, Boscheri W, Semplice M, Russo G (2017) Central weighted eno schemes for hyperbolic conservation laws on fixed and moving unstructured meshes. SIAM J Sci Comput 39(6):A2564–A2591
Dumbser M, Enaux C, Toro E (2008) Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J Comput Phys 227:3971–4001
Dumbser M, Käser M (2007) Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J Comput Phys 221:693–723
Dumbser M, Käser M, Titarev V, Toro E (2007) Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys 226:204–243
Dumbser M, Loubère R (2016) A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes. J Comput Phys 319:163–199
Dumbser M, Toro EF (2011) On universal Osher-type schemes for general nonlinear hyperbolic conservation laws. Commun Comput Phys 10:635–671
Dumbser M, Toro EF (2011) A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems. J Sci Comput 48:70–88
Dumbser M, Zanotti O, Loubère R, Diot S (2014) A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws. J Comput Phys 278:47–75
Dumbser M, Fambri F, Gaburro E, Reinarz A (2020) On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations. J Comput Phys 404:109088
Einfeldt B, Munz CD, Roe PL, Sjögreen B (1991) On Godunov-type methods near low densities. J Comput Phys 92:273–295
Fambri F, Dumbser M, Köppel S, Rezzolla L, Zanotti O (2018) ADER discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics. Mon Not R Astron Soc. arXiv:abs/1801.02839
Fambri F, Dumbser M, Zanotti O (2017) Space-time adaptive ADER-DG schemes for dissipative flows: compressible navier-stokes and resistive mhd equations. Comput Phys Commun 220:297–318
Gaburro E (2018) Well balanced Arbitrary-Lagrangian–Eulerian finite volume schemes on moving nonconforming meshes for non-conservative hyperbolic systems. Ph.D. thesis, University of Trento
Gaburro E, Boscheri W, Chiocchetti S, Klingenberg C, Springel V, Dumbser M (2019) High order direct Arbitrary-Lagrangian–Eulerian schemes on moving Voronoi meshes with topology changes. Journal of Computational Physics. In press. https://doi.org/10.1016/j.jcp.2019.109167
Gaburro E, Castro MJ, Dumbser M (2018) Well-balanced Arbitrary-Lagrangian–Eulerian finite volume schemes on moving nonconforming meshes for the euler equations of gas dynamics with gravity. Mon Not R Astron Soc 477(2):2251–2275
Gaburro E, Castro MJ, Dumbser M (2018) A well balanced diffuse interface method for complex nonhydrostatic free surface flows. Comput Fluids 175:180–198
Gaburro E, Dumbser M, Castro MJ (2017) Direct Arbitrary-Lagrangian–Eulerian finite volume schemes on moving nonconforming unstructured meshes. Comput Fluids 159:254–275
Gaburro E, Dumbser M, Castro MJ (2018) Reprint of: direct Arbitrary-Lagrangian–Eulerian finite volume schemes on moving nonconforming unstructured meshes. Comput Fluids
Galera S, Maire P, Breil J (2010) A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction. J Comput Phys 229:5755–5787
Godunov S (1959) Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics. Math USSR Sbornik 47:271–306
Gosse L (2000) A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comput Math Appl 39(9):135–159
Gosse L (2001) A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms. Math Models Methods Appl Sci 11(02):339–365
Greenberg J, Leroux A, Baraille R, Noussair A (1997) Analysis and approximation of conservation laws with source terms. SIAM J Numer Anal 34(5):1980–2007
Greenberg JM, Leroux AY (1996) A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J Numer Anal 33(1):1–16
Harten A, Engquist B, Osher S, Chakravarthy S (1987) Uniformly high order accurate essentially non-oscillatory schemes III. J Comput Phys 71:231–303
Harten A, Engquist B, Osher S, Chakravarthy S (1987) Uniformly high order essentially non-oscillatory schemes. III. J Comput Phys 71:231–303
Hidalgo A, Dumbser M (2011) Ader schemes for nonlinear systems of stiff advection–diffusion–reaction equations. J Sci Comput 48(1–3):173–189
Hu C, Shu C (1999) A high-order weno finite difference scheme for the equations of ideal magnetohydrodynamics. J Comput Phys 150:561–594
Hu C, Shu C (1999) Weighted essentially non-oscillatory schemes on triangular meshes. J Comput Phys 150(1):97–127
Idelsohn S, Mier-Torrecilla M, Oñate E (2009) Multi-fluid flows with the particle finite element method. Comput Methods Appl Mech Eng 198:2750–2767
Idelsohn SR, Oñate E, Pin FD (2004) The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves. Int J Numer Methods Eng 61:964–984
Jackson H (2017) On the eigenvalues of the ader-weno Galerkin predictor. J Comput Phys 333:409–413
Käppeli R, Mishra S (2016) A well-balanced finite volume scheme for the euler equations with gravitation. Astron Astrophys 587:A94
Käser M, Iske A (2005) ADER schemes on adaptive triangular meshes for scalar conservation laws. J Comput Phys 205:486–508
Kemm F, Gaburro E, Thein F, Dumbser M, (2020) A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer-Nunziato model. arXiv preprint arXiv:2001.10326
Knupp P (2000) Achieving finite element mesh quality via optimization of the jacobian matrix norm and associated quantities. Part II—a framework for volume mesh optimization and the condition number of the jacobian matrix. Int J Numer Methods Eng 48:1165–1185
Kucharik M, Breil J, Galera S, Maire P, Berndt M, Shashkov M (2011) Hybrid remap for multi-material ALE. Comput Fluids 46:293–297
Kucharik M, Loubère R, Bednàrik L, Liska R (2013) Enhancement of Lagrangian slide lines as a combined force and velocity boundary condition. Comput Fluids 83:3–14
Kucharik M, Shashkov M (2012) One-step hybrid remapping algorithm for multi-material arbitrary Lagrangian–Eulerian methods. J Comput Phys 231:2851–2864
Larese A, Rossi R, Oñate E, Idelsohn S (2008) Validation of the particle finite element method (PFEM) for simulation of the free-surface flows. Eng Comput 25:385–425
LeVeque RJ (1998) Balancing source terms and flux gradients in high-resolution godunov methods: the quasi-steady wave-propagation algorithm. J Comput Phys 146(1):346–365
Levy D, Puppo G, Russo G (1999) Central WENO schemes for hyperbolic systems of conservation laws. Math Model Numer Anal 33(3):547–571
Levy D, Puppo G, Russo G (2000) A third order central WENO scheme for 2D conservation laws. Appl Numer Math 33:415–421
Levy D, Puppo G, Russo G (2002) A fourth-order central WENO scheme for multidimensional hyperbolic systems of conservation laws. SIAM J Sci Comput 24:480–506
Li Z, Yu X, Jia Z (2014) The cell-centered discontinuous Galerkin method for Lagrangian compressible Euler equations in two dimensions. Comput Fluids 96:152–164
Liska R, Váchal MSP, Wendroff B (2011) Synchronized flux corrected remapping for ALE methods. Comput Fluids 46:312–317
Liu W, Cheng J, Shu C (2009) High order conservative Lagrangian schemes with Lax–Wendroff type time discretization for the compressible Euler equations. J Comput Phys 228:8872–8891
Loubere R, Dumbser M, Diot S (2014) A new family of high order unstructured mood and ader finite volume schemes for multidimensional systems of hyperbolic conservation laws. Commun Comput Phys 16(3):718–763
Loubère R, Maire P, Váchal P (2010) A second-order compatible staggered Lagrangian hydrodynamics scheme using a cell-centered multidimensional approximate Riemann solver. Procedia Comput Sci 1:1931–1939
Loubère R, Maire P, Váchal P (2013) 3D staggered Lagrangian hydrodynamics scheme with cell-centered Riemann solver-based artificial viscosity. Int J Numer Methods Fluids 72:22–42
Loubère R, Maire PH, Váchal P (2010) Staggered Lagrangian hydrodynamics based on cell-centered Riemann solver. Commun Comput Phys 10(4):940–978
Ma R, Chang X, Zhang L, He X, Li M (2015) On the geometric conservation law for unsteady flow simulations on moving mesh. Procedia Eng 126:639–644
Maire P (2009) A high-order cell-centered lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes. J Comput Phys 228:2391–2425
Maire P (2011) A high-order one-step sub-cell force-based discretization for cell-centered lagrangian hydrodynamics on polygonal grids. Comput Fluids 46(1):341–347
Maire P (2011) A unified sub-cell force-based discretization for cell-centered lagrangian hydrodynamics on polygonal grids. Int J Numer Methods Fluids 65:1281–1294
Maire P, Nkonga B (2009) Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics. J Comput Phys 228:799–821
Maso GD, LeFloch P, Murat F (1995) Definition and weak stability of nonconservative products. J Math Pures Appl 74:483–548
Mignone A, Bodo G, Massaglia S, Matsakos T, Tesileanu O, Zanni C, Ferrari A (2007) Pluto: a numerical code for computational astrophysics. Astrophys J Suppl Ser 170(1):228
Mignone A, Zanni C, Tzeferacos P, Van Straalen B, Colella P, Bodo G (2011) The pluto code for adaptive mesh computations in astrophysical fluid dynamics. Astrophys J Suppl Ser 198(1):7
Munz C (1994) On Godunov-type schemes for Lagrangian gas dynamics. SIAM J Numer Anal 31:17–42
Oñate E, Celigueta M, Idelsohn S, Salazar F, Suarez B (2011) Possibilities of the particle finite element method for fluid-soil-structure interaction problems. J Comput Mech 48:307–318
Oñate E, Idelsohn S, Celigueta M, Rossi R (2008) Advances in the particle finite element method for the analysis of fluid-multibody interaction and bed erosion in free-surface flows. Comput Methods Appl Mech Eng 197:1777–1800
Ortega AL, Scovazzi G (2011) A geometrically-conservative, synchronized, flux-corrected remap for arbitrary Lagrangian–Eulerian computations with nodal finite elements. J Comput Phys 230:6709–6741
Osher S, Solomon F (1982) Upwind difference schemes for hyperbolic conservation laws. Math Comput 38:339–374
Pakmor R, Marinacci F, Springel V (2014) Magnetic fields in cosmological simulations of disk galaxies. Astrophys J Lett 783(1):L20
Pakmor R, Springel V, Bauer A, Mocz P, Munoz DJ, Ohlmann ST, Schaal K, Zhu C (2015) Improving the convergence properties of the moving-mesh code arepo. Mon Not R Astron Soc 455(1):1134–1143
Parés C (2006) Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 44:300–321
Pin FD, Idelsohn SR, Oñate E, Aubry R (2007) The ALE/Lagrangian particle finite element method: a new approach to computation of free-surface flows and fluid–object interactions. Comput Fluids 36:27–38
Pino SD (2010) A curvilinear finite-volume method to solve compressible gas dynamics in semi-Lagrangian coordinates. Comptes Rendus de l’Académie des Sciences Series I Mathematics 348:1027–1032
Qiu J, Shu CW (2005) Hermite weno schemes and their application as limiters for Runge–Kutta discontinuous galerkin method II: two dimensional case. Comput Fluids 34(6):642–663
Re B, Dobrzynski C, Guardone A (2017) An interpolation-free ALE scheme for unsteady inviscid flows computations with large boundary displacements over three-dimensional adaptive grids. J Comput Phys 340:26–54
Reed W, Hill T (1973) Triangular mesh methods for neutron transport equation. Tech. Rep. LA-UR-73-479, Los Alamos Scientific Laboratory
Rusanov VV (1961) Calculation of interaction of non-steady shock waves with obstacles. J Comput Math Phys USSR 1:267–279
Sambasivan S, Shashkov M, Burton D (2013) A finite volume cell-centered Lagrangian hydrodynamics approach for solids in general unstructured grids. Int J Numer Methods Fluids 72:770–810
Schwartzkopff T, Munz C, Toro E (2002) ADER: a high order approach for linear hyperbolic systems in 2D. J Sci Comput 17(1–4):231–240
Scovazzi G (2012) Lagrangian shock hydrodynamics on tetrahedral meshes: a stable and accurate variational multiscale approach. J Comput Phys 231:8029–8069
Sedov L (1959) Similarity and dimensional methods in mechanics. Academic Press, New York
Semplice M, Coco A, Russo G (2016) Adaptive mesh refinement for hyperbolic systems based on third-order compact WENO reconstruction. J Sci Comput 66(2):692–724
Springel V (2010) E pur si muove: Galilean-invariant cosmological hydrodynamical simulations on a moving mesh. Mon Not R Astron Soc 401:791–851
Springel V (2010) Moving-mesh hydrodynamics with the arepo code. Proc Int Astron Union 6(S270):203–206
Stroud A (1971) Approximate calculation of multiple integrals. Prentice-Hall, Englewood Cliffs
Tavelli M, Boscheri W A high order parallel Eulerian–Lagrangian algorithm for advection-diffusion problems on unstructured meshes. Int J Numer Methods Fluids
Titarev V, Toro E (2002) ADER: arbitrary high order Godunov approach. J Sci Comput 17(1–4):609–618
Titarev V, Toro E (2005) ADER schemes for three-dimensional nonlinear hyperbolic systems. J Comput Phys 204:715–736
Toro E (1999) Riemann solvers and numerical methods for fluid dynamics, 2nd edn. Springer, Berlin
Toro E, Titarev V (2002) Solution of the generalized Riemann problem for advection-reaction equations. Proc R Soc Lond 458:271–281
Toro EF, Titarev VA (2006) Derivative Riemann solvers for systems of conservation laws and ADER methods. J Comput Phys 212(1):150–165
Vilar F (2012) Cell-centered discontinuous Galerkin discretization for two-dimensional Lagrangian hydrodynamics. Comput Fluids 64:64–73
Vilar F, Maire P, Abgrall R (2010) Cell-centered discontinuous Galerkin discretizations for two-dimensional scalar conservation laws on unstructured grids and for one-dimensional Lagrangian hydrodynamics. Comput Fluids 46(1):498–604
Vilar F, Maire P, Abgrall R (2014) A discontinuous Galerkin discretization for solving the two-dimensional gas dynamics equations written under total Lagrangian formulation on general unstructured grids. J Comput Phys 276:188–234
von Neumann J, Richtmyer R (1950) A method for the calculation of hydrodynamics shocks. J Appl Phys 21:232–237
van Leer B (1974) Towards the ultimate conservative difference scheme II: monotonicity and conservation combined in a second order scheme. J Comput Phys 14:361–370
van Leer B (1979) Towards the ultimate conservative difference scheme V: a second order sequel to Godunov’s method. J Comput Phys 32:101–136
Wilkins ML (1964) Calculation of elastic-plastic flow. Methods Comput Phys 3
Winslow AM (1997) Numerical solution of the quasilinear poisson equation in a nonuniform triangle mesh. J Comput Phys 135(2):128–138
Zanotti O, Fambri F, Dumbser M, Hidalgo A (2015) Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting. Comput Fluids 118:204–224
Acknowledgements
The research presented in this paper has been partially financed by the European Research Council (ERC) under the European Union’s Seventh Framework Programme (FP7/2007-2013) with the research project STiMulUs, ERC Grant Agreement No. 278267. The author has also been supported by a national mobility grant for young researchers in Italy, funded by GNCS-INdAM, and by the UniTN Starting Grant, funded by the University of Trento in Italy.
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Gaburro, E. A Unified Framework for the Solution of Hyperbolic PDE Systems Using High Order Direct Arbitrary-Lagrangian–Eulerian Schemes on Moving Unstructured Meshes with Topology Change. Arch Computat Methods Eng 28, 1249–1321 (2021). https://doi.org/10.1007/s11831-020-09411-7
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DOI: https://doi.org/10.1007/s11831-020-09411-7