A scalable well-balanced numerical scheme for the simulation of fast landslides with efficient time stepping. (English) Zbl 07834060
Summary: We consider a single-phase depth-averaged model for the numerical simulation of fast-moving landslides with the goal of constructing a well-balanced, yet scalable and efficient, second-order time-stepping algorithm. We apply a Strang splitting approach to distinguish between parabolic and hyperbolic problems. For the parabolic contribution, we adopt a second-order Implicit-Explicit Runge-Kutta-Chebyshev scheme, while we use a two-stage Taylor discretization combined with a path-conservative strategy, to deal with the purely hyperbolic contribution. The proposed strategy allows to decouple hyperbolic from parabolic-reaction stiff contributions resulting in an overall well-balanced scheme subject just to stability restrictions of the hyperbolic term. The spatial discretization we adopt is based on a standard finite element method, associated with a hierarchically refined Cartesian grid. After providing numerical evidence of the well-balancing property, we demonstrate the capability of the proposed approach to select time steps larger than the ones adopted by a classical Taylor-Galerkin scheme. Finally, we provide some meaningful scaling results on ideal and realistic scenarios.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76T25 | Granular flows |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
Taylor-Galerkin scheme; depth-integrated models; implicit-explicit Runge-Kutta-Chebyshev scheme; C-property; path-conservative methods; parallel simulationsReferences:
| [1] | Donea, J., A Taylor-Galerkin method for convective transport problems. Int. J. Numer. Methods Eng., 1, 101-119 (1984) · Zbl 0524.65071 |
| [2] | Löhner, R.; Morgan, K.; Zienkiewicz, O. C., The solution of non-linear hyperbolic equation systems by the finite element method. Int. J. Numer. Methods Fluids, 11, 1043-1063 (1984) · Zbl 0551.76002 |
| [3] | Sai, B. V.K. S.; Zienkiewicz, O. C.; Manzari, M. T.; Lyra, P. R.M.; Morgan, K., General purpose versus special algorithms for high-speed flows with shocks. Int. J. Numer. Methods Fluids, 1-4, 57-80 (1998) · Zbl 0904.76065 |
| [4] | Quecedo, M.; Pastor, M., A reappraisal of Taylor-Galerkin algorithm for drying-wetting areas in shallow water computations. Int. J. Numer. Methods Fluids, 6, 515-531 (2002) · Zbl 0996.76055 |
| [5] | Quecedo, M.; Pastor, M.; Herreros, M. I.; Fernández Merodo, J. A., Numerical modelling of the propagation of fast landslides using the finite element method. Int. J. Numer. Methods Eng., 6, 755-794 (2004) · Zbl 1047.76051 |
| [6] | Gatti, F.; Fois, M.; Perotto, S.; de Falco, C.; Formaggia, L., Parallel simulations for fast-moving landslides: space-time mesh adaptation and sharp tracking of the wetting front. Int. J. Numer. Methods Fluids, 8, 1286-1309 (2023) · Zbl 07847183 |
| [7] | Marchuk, G. I., Some application of splitting-up methods to the solution of mathematical physics problems. Apl. Mat., 2, 103-132 (1968) · Zbl 0159.44702 |
| [8] | Strang, G., On the construction and comparison of difference schemes. SIAM J. Numer. Anal., 3, 506-517 (1968) · Zbl 0184.38503 |
| [9] | Dal Maso, G.; Lefloch, P. G.; Murat, F., Definition and weak stability of nonconservative products. J. Math. Pures Appl., 6, 483-548 (1995) · Zbl 0853.35068 |
| [10] | Parés, C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal., 1, 300-321 (2006) · Zbl 1130.65089 |
| [11] | Castro, M. J.; Fernández-Nieto, E. D.; Ferreiro, A. M.; García-Rodríguez, J. A.; Parés, C., High order extensions of Roe schemes for two-dimensional nonconservative hyperbolic systems. J. Sci. Comput., 1, 67-114 (2009) · Zbl 1203.65131 |
| [12] | Castro, M.; Gallardo, J. M.; López-García, J. A.; Parés, C., Well-balanced high order extensions of Godunov’s method for semilinear balance laws. SIAM J. Numer. Anal., 2, 1012-1039 (2008) · Zbl 1159.74045 |
| [13] | Gallardo, J. M.; Parés, C.; Castro, M., On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas. J. Comput. Phys., 1, 574-601 (2007) · Zbl 1126.76036 |
| [14] | Muñoz-Ruiz, M. L.; Parés, C., Godunov method for nonconservative hyperbolic systems. ESAIM: Math. Model. Numer. Anal., 1, 169-185 (2007) · Zbl 1124.65077 |
| [15] | Castro, M.; Pardo, A.; Parés, C.; Toro, E., On some fast well-balanced first order solvers for nonconservative systems. Math. Comput., 271, 1427-1472 (2010) · Zbl 1369.65107 |
| [16] | Dumbser, M.; Castro, M.; Parés, C.; Toro, E. F., Ader schemes on unstructured meshes for nonconservative hyperbolic systems: applications to geophysical flows. Comput. Fluids, 9, 1731-1748 (2009) · Zbl 1177.76222 |
| [17] | Dumbser, M.; Balsara, D. S., A new efficient formulation of the hllem Riemann solver for general conservative and non-conservative hyperbolic systems. J. Comput. Phys., 275-319 (2016) · Zbl 1349.76603 |
| [18] | Dumbser, M.; Hidalgo, A.; Castro, M.; Parés, C.; Toro, E. F., Force schemes on unstructured meshes II: non-conservative hyperbolic systems. Comput. Methods Appl. Mech. Eng., 9-12, 625-647 (2010) · Zbl 1227.76043 |
| [19] | Castro, M.; Gallardo, J.; Parés, C., High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Math. Comput., 255, 1103-1134 (2006) · Zbl 1096.65082 |
| [20] | Dumbser, M.; Toro, E. F., A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems. J. Sci. Comput., 1, 70-88 (2011) · Zbl 1220.65110 |
| [21] | Fambri, F.; Dumbser, M.; Köppel, S.; Rezzolla, L.; Zanotti, O., Ader discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics. Mon. Not. R. Astron. Soc., 4, 4543-4564 (2018) |
| [22] | Busto, S.; Dumbser, M.; Gavrilyuk, S.; Ivanova, K., On thermodynamically compatible finite volume methods and path-conservative ader discontinuous Galerkin schemes for turbulent shallow water flows. J. Sci. Comput., 1, 28 (2021) · Zbl 1501.65047 |
| [23] | Verwer, J. G.; Sommeijer, B. P., An implicit-explicit Runge-Kutta-Chebyshev scheme for diffusion-reaction equations. SIAM J. Sci. Comput., 5, 1824-1835 (2004) · Zbl 1061.65090 |
| [24] | Verwer, J. G.; Hundsdorfer, W.; Sommeijer, B. P., Convergence properties of the Runge-Kutta-Chebyshev method. Numer. Math., 1, 157-178 (1990) · Zbl 0697.65072 |
| [25] | Sommeijer, B. P.; Shampine, L. F.; Verwer, J. G., RKC: an explicit solver for parabolic PDEs. J. Comput. Appl. Math., 2, 315-326 (1998) · Zbl 0910.65067 |
| [26] | Bermejo, R.; Carpio, J., An adaptive finite element semi-Lagrangian implicit-explicit Runge-Kutta-Chebyshev method for convection dominated reaction-diffusion problems. Appl. Numer. Math., 1, 16-39 (2008) · Zbl 1128.65073 |
| [27] | Bermejo, R.; del Sastre, P. G., An implicit-explicit Runge-Kutta-Chebyshev finite element method for the nonlinear lithium-ion battery equations. Appl. Math. Comput., 398-420 (2019) · Zbl 1428.78029 |
| [28] | Abbott, M. B., Computational Hydraulics: Elements of the Theory of Free Surface Flows (1979), Pitman: Pitman London · Zbl 0406.76002 |
| [29] | Franci, A.; Cremonesi, M.; Perego, U.; Crosta, G.; Oñate, E., 3D simulation of Vajont disaster. Part 1: numerical formulation and validation. Eng. Geol. (2020) |
| [30] | Pastor, M.; Blanc, T.; Haddad, B.; Drempetic, V.; Mories, M.; Stickle, P.; Mira, M.; Merodo, J., Depth averaged models for fast landslide propagation: mathematical, rheological and numerical aspects. Arch. Comput. Methods Eng., 67-104 (2015) · Zbl 1348.76177 |
| [31] | Papanastasiou, T. C., Flows of materials with yield. J. Rheol., 5, 385-404 (1987) · Zbl 0666.76022 |
| [32] | Franci, A.; Cremonesi, M.; Perego, U.; Oñate, E.; Crosta, G., 3D simulation of Vajont disaster. Part 2: multi-failure scenarios. Eng. Geol. (2020) |
| [33] | LeVeque, R. J., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys., 1, 346-365 (1998) · Zbl 0931.76059 |
| [34] | Gourgue, O.; Comblen, R.; Lambrechts, J.; Kärnä, T.; Legat, V.; Deleersnijder, E., A flux-limiting wetting-drying method for finite-element shallow-water models, with application to the Scheldt estuary. Adv. Water Resour., 12, 1726-1739 (2009) |
| [35] | Qiu, J.; Dumbser, M.; Shu, C.-W., The discontinuous Galerkin method with Lax-Wendroff type time discretizations. Comput. Methods Appl. Mech. Eng., 42-44, 4528-4543 (2005) · Zbl 1093.76038 |
| [36] | Pastor, M.; Quecedo, M.; Fernández Merodo, J.; Herrores, M.; Gonzalez, E.; Mira, P., Modelling tailings dams and mine waste dumps failures. Geotechnique, 8, 579-591 (2002) |
| [37] | Givoli, D., Non-reflecting boundary conditions. J. Comput. Phys., 1, 1-29 (1991) · Zbl 0731.65109 |
| [38] | Zienkiewicz, O. C.; Morgan, K., Finite Elements and Approximation (2006), Courier Corporation · Zbl 1116.65113 |
| [39] | 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. J. Comput. Phys., 17, 8107-8129 (2008) · Zbl 1176.76084 |
| [40] | Rhebergen, S.; Bokhove, O.; van der Vegt, J. J., Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations. J. Comput. Phys., 3, 1887-1922 (2008) · Zbl 1153.65097 |
| [41] | Abgrall, R.; Karni, S., A comment on the computation of non-conservative products. J. Comput. Phys., 8, 2759-2763 (2010) · Zbl 1188.65134 |
| [42] | Peraire, J., A finite element method for convection dominated flows (1986), University College of Swansea: University College of Swansea Swansea, Ph.D. thesis |
| [43] | Zalesak, S. T., Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys., 3, 335-362 (1979) · Zbl 0416.76002 |
| [44] | Boris, J. P.; Book, D. L., Flux-corrected transport. III. Minimal-error FCT algorithms. J. Comput. Phys., 4, 397-431 (1976) · Zbl 0325.76037 |
| [45] | Kuzmin, D.; Möller, M.; Turek, S., High-resolution FEM-FCT schemes for multidimensional conservation laws. Comput. Methods Appl. Mech. Eng., 45-47, 4915-4946 (2004) · Zbl 1112.76393 |
| [46] | Xing, Y.; Zhang, X.; Shu, C.-W., Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Adv. Water Resour., 12, 1476-1493 (2010) |
| [47] | Verwer, J. G., Explicit Runge-Kutta methods for parabolic partial differential equations. Appl. Numer. Math., 1-3, 359-379 (1996) · Zbl 0868.65064 |
| [48] | Wohlmuth, B., Variationally consistent discretization schemes and numerical algorithms for contact problems. Acta Numer., 569-734 (2011) · Zbl 1432.74176 |
| [49] | Berge, R. L.; Berre, I.; Keilegavlen, E.; Nordbotten, J. M.; Wohlmuth, B., Finite volume discretization for poroelastic media with fractures modeled by contact mechanics. Int. J. Numer. Methods Eng., 4, 644-663 (2020) · Zbl 07843216 |
| [50] | Formaggia, L.; Gatti, F.; Zonca, S., An XFEM/DG approach for fluid-structure interaction problems with contact. Appl. Math., 2, 183-211 (2021) · Zbl 1538.65355 |
| [51] | Africa, P. C., Scalable adaptive simulation of organic thin-film transistors, PhD Thesis |
| [52] | Africa, P. C.; de Falco, C.; Perotto, S., Scalable recovery-based adaptation on Cartesian quadtree meshes for advection-diffusion-reaction problems. Adv. Comput. Sci. Eng., 4, 443-473 (2023) |
| [53] | Burstedde, C.; Wilcox, L. C.; Ghattas, O., p4est: scalable algorithms for parallel adaptive mesh refinement on forests of octrees. SIAM J. Sci. Comput., 3, 1103-1133 (2011) · Zbl 1230.65106 |
| [54] | Xing, Y.; Shu, C.-W., 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. Comput. Phys., 1, 100-134 (2006) · Zbl 1115.65096 |
| [55] | Margottini, C.; Canuti, P.; Sassa, K., Landslide Science and Practice: Volume 3: Spatial Analysis and Modelling (2013), Springer Science & Business Media |
| [56] | Secondi, M. M.; Crosta, G.; di Prisco, C.; Frigerio, G.; Frattini, P.; Agliardi, F., Landslide motion forecasting by a dynamic visco-plastic model, 151-159 |
| [57] | Sudret, B., Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf., 7, 964-979 (2008) |
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