zbMATH — the first resource for mathematics

Analysis of adaptive mesh refinement for IMEX discontinuous Galerkin solutions of the compressible Euler equations with application to atmospheric simulations. (English) Zbl 1349.76226
Summary: The resolutions of interests in atmospheric simulations require prohibitively large computational resources. Adaptive mesh refinement (AMR) tries to mitigate this problem by putting high resolution in crucial areas of the domain. We investigate the performance of a tree-based AMR algorithm for the high order discontinuous Galerkin method on quadrilateral grids with non-conforming elements. We perform a detailed analysis of the cost of AMR by comparing this to uniform reference simulations of two standard atmospheric test cases: density current and rising thermal bubble. The analysis shows up to 15 times speed-up of the AMR simulations with the cost of mesh adaptation below 1% of the total runtime. We pay particular attention to the implicit-explicit (IMEX) time integration methods and show that the ARK2 method is more robust with respect to dynamically adapting meshes than BDF2. Preliminary analysis of preconditioning reveals that it can be an important factor in the AMR overhead. The compiler optimizations provide significant runtime reduction and positively affect the effectiveness of AMR allowing for speed-ups greater than it would follow from the simple performance model.

76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
76N15 Gas dynamics (general theory)
86A10 Meteorology and atmospheric physics
Full Text: DOI
[1] Nair, R. D.; Choi, H.-W.; Tufo, H. M., Computational aspects of a scalable high-order discontinuous Galerkin atmospheric dynamical core, Comput. Fluids, 38, 2, 309-319, (2009) · Zbl 1237.76129
[2] Kelly, J. F.; Giraldo, F. X., Continuous and discontinuous Galerkin methods for a scalable 3D nonhydrostatic atmospheric model: limited area mode, J. Comput. Phys., 231, 2, 7988-8008, (2012) · Zbl 1284.65134
[3] Hartmann, R.; Houston, P., Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations, J. Comput. Phys., 183, 2, 508-532, (2002) · Zbl 1057.76033
[4] Bastian, P.; Blatt, M.; Dedner, A.; Engwer, C.; Klöfkorn, R.; Kornhuber, R.; Ohlberger, M.; Sander, O., A generic grid interface for parallel and adaptive scientific computing. part II: implementation and tests in DUNE, Computing, 82, 2-3, 121-138, (2008) · Zbl 1151.65088
[5] Burstedde, C.; Ghattas, O.; Gurnis, M.; Stadler, G.; Tan, E.; Tu, T.; Wilcox, L. C.; Zhong, S., Scalable adaptive mantle convection simulation on petascale supercomputers, (Proc. 2008 ACM/IEEE Supercomputing, (2008), IEEE Press), 62
[6] Restelli, M.; Giraldo, F. X., A conservative discontinuous Galerkin semi-implicit formulation for the Navier-Stokes equations in non-hydrostatic mesoscale modeling, SIAM J. Sci. Comput., 31, 2231-2257, (2009) · Zbl 1405.65127
[7] Jablonowski, C., Adaptive grids in weather and climate modeling, (2004), The University of Michigan, Ph.D. thesis
[8] Behrens, J., Adaptive atmospheric modeling: key techniques in grid generation, data structures, and numerical operations with applications, (2006), Springer · Zbl 1138.86002
[9] Ley, G. W.; Elsberry, R. L., Forecasts of typhoon irma using a nested grid model, Mon. Weather Rev., 104, 1154, (1976)
[10] Kurihara, Y.; Bender, M. A., Use of a movable nested-mesh model for tracking a small vortex, Mon. Weather Rev., 108, 1792-1809, (1980)
[11] Clark, T. L.; Farley, R., Severe downslope windstorm calculations in two and three spatial dimensions using anelastic interactive grid nesting: a possible mechanism for gustiness, J. Atmos. Sci., 41, 3, 329-350, (1984)
[12] Zhang, D.-L.; Chang, H.-R.; Seaman, N. L.; Warner, T. T.; Fritsch, J. M., A two-way interactive nesting procedure with variable terrain resolution, Mon. Weather Rev., 114, 7, 1330-1339, (1986)
[13] Dietachmayer, G. S.; Droegemeier, K. K., Application of continuous dynamic grid adaptation techniques to meteorological modeling. I: basic formulation and accuracy, Mon. Weather Rev., 120, 8, 1675-1706, (1992)
[14] Prusa, J. M.; Smolarkiewicz, P. K., An all-scale anelastic model for geophysical flows: dynamic grid deformation, J. Comput. Phys., 190, 2, 601-622, (2003) · Zbl 1076.86001
[15] Budd, C. J.; Huang, W.; Russell, R. D., Adaptivity with moving grids, Acta Numer., 18, 1, 111-241, (2009) · Zbl 1181.65122
[16] Skamarock, W. C.; Oliger, J.; Street, R. L., Adaptive grid refinement for numerical weather prediction, J. Comput. Phys., 80, 1, 27-60, (1989) · Zbl 0661.76021
[17] Skamarock, W. C.; Klemp, J. B., Adaptive grid refinement for two-dimensional and three-dimensional non-hydrostatic atmospheric flow, Mon. Weather Rev., 121, 3, 788-804, (1993)
[18] Berger, M. J.; Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 53, 3, 484-512, (1984) · Zbl 0536.65071
[19] Berger, M. J.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82, 1, 64-84, (1989) · Zbl 0665.76070
[20] LeVeque, R. J., Wave propagation algorithms for multidimensional hyperbolic systems, J. Comput. Phys., 131, 2, 327-353, (1997) · Zbl 0872.76075
[21] Nikiforakis, N., AMR for global atmospheric modelling, (Adaptive Mesh Refinement—Theory and Applications, (2005), Springer), 505-526 · Zbl 1115.86300
[22] Bacon, D. P.; Ahmad, N. N.; Boybeyi, Z.; Dunn, T. J.; Hall, M. S.; Lee, P. C.S.; Sarma, R. A.; Turner, M. D.; Waight, K. T.; Young, S. H., A dynamically adapting weather and dispersion model: the operational multiscale environment model with grid adaptivity (OMEGA), Mon. Weather Rev., 128, 7, 2044-2076, (2000)
[23] Smolarkiewicz, P. K., A fully multidimensional positive definite advection transport algorithm with small implicit diffusion, J. Comput. Phys., 54, 2, 325-362, (1984)
[24] Kühnlein, C.; Smolarkiewicz, P. K.; Dörnbrack, A., Modelling atmospheric flows with adaptive moving meshes, J. Comput. Phys., 231, 7, 2741-2763, (2012) · Zbl 1426.76390
[25] Iselin, J. P.; Prusa, J. M.; Gutowski, W. J., Dynamic grid adaptation using the MPDATA scheme, Mon. Weather Rev., 130, 4, 1026-1039, (2002)
[26] Giraldo, F. X., The Lagrange-Galerkin method for the two-dimensional shallow water equations on adaptive grids, Int. J. Numer. Methods Fluids, 33, 6, 789-832, (2000) · Zbl 0989.76047
[27] Behrens, J., Atmospheric and Ocean modeling with an adaptive finite element solver for the shallow-water equations, Appl. Numer. Math., 26, 1, 217-226, (1998) · Zbl 0897.76046
[28] Behrens, J.; Rakowsky, N.; Hiller, W.; Handorf, D.; Läuter, M.; Päpke, J.; Dethloff, K., Amatos: parallel adaptive mesh generator for atmospheric and oceanic simulation, Ocean Model., 10, 1, 171-183, (2005)
[29] Taylor, M. A.; Fournier, A., A compatible and conservative spectral element method on unstructured grids, J. Comput. Phys., 229, 17, 5879-5895, (2010) · Zbl 1425.76177
[30] St-Cyr, A.; Jablonowski, C.; Dennis, J. M.; Tufo, H. M.; Thomas, S. J., A comparison of two shallow water models with non-conforming adaptive grids: classical tests, preprint
[31] Kubatko, E. J.; Bunya, S.; Dawson, C.; Westerink, J. J., Dynamic p-adaptive Runge-Kutta discontinuous Galerkin methods for the shallow water equations, Comput. Methods Appl. Mech. Eng., 198, 21, 1766-1774, (2009) · Zbl 1227.76032
[32] Kopriva, D. A., A conservative staggered-grid Chebyshev multidomain method for compressible flows. II. A semi-structured method, J. Comput. Phys., 128, 2, 475-488, (1996) · Zbl 0866.76064
[33] Maday, Y.; Mavriplis, C.; Patera, A. T., Nonconforming mortar element methods: application to spectral discretizations, (1988), Institute for Computer Applications in Science and Engineering, NASA Langley Research Center
[34] Rosenberg, D.; Fournier, A.; Fischer, P.; Pouquet, A., Geophysical-astrophysical spectral-element adaptive refinement (gaspar): object-oriented h-adaptive fluid dynamics simulation, J. Comput. Phys., 215, 1, 59-80, (2006) · Zbl 1140.86300
[35] Mueller, A.; Behrens, J.; Giraldo, F. X.; Wirth, V., An adaptive discontinuous Galerkin method for modeling atmospheric convection, J. Comput. Phys., 235, 1, 371-393, (2012)
[36] Brdar, S.; Baldauf, M.; Dedner, A.; Klöfkorn, R., Comparison of dynamical cores for NWP models: comparison of COSMO and dune, Theor. Comput. Fluid Dyn., 1-20, (2012)
[37] Brdar, S., A higher order locally adaptive discontinuous Galerkin approach for atmospheric simulations, (2012), Universitätsbibliothek Freiburg, Ph.D. thesis · Zbl 1255.86006
[38] Eskilsson, C., An hp-adaptive discontinuous Galerkin method for shallow water flows, Int. J. Numer. Methods Fluids, 67, 11, 1605-1623, (2011) · Zbl 1381.76165
[39] Blaise, S.; St-Cyr, A., A dynamic hp-adaptive discontinuous Galerkin method for shallow-water flows on the sphere with application to a global tsunami simulation, Mon. Weather Rev., 140, 3, 978-996, (2012)
[40] Giraldo, F. X.; Restelli, M., A study of spectral element and discontinuous Galerkin methods for the Navier-Stokes equations in non-hydrostatic mesoscale atmospheric modeling: equation sets and test cases, J. Comput. Phys., 227, 1, 3849-3877, (2008) · Zbl 1194.76189
[41] Burstedde, C.; Ghattas, O.; Gurnis, M.; Isaac, T.; Stadler, G.; Warburton, T.; Wilcox, L. C., Extreme-scale AMR, (Proc. 2010 ACM/IEEE Int. Conference for High Performance Computing, Networking, Storage and Analysis, vol. 1, (2010)), 1-12
[42] Bader, M., Space-filling curves: an introduction with applications in scientific computing, vol. 9, (2012), Springer
[43] Sundar, H.; Sampath, R. S.; Biros, G., Bottom-up construction and 2:1 balance refinement of linear octrees in parallel, SIAM J. Sci. Comput., 30, 5, 2675-2708, (2008) · Zbl 1186.68554
[44] Bassi, F.; Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131, 2, 267-279, (1997) · Zbl 0871.76040
[45] Straka, J. M.; Wilhelmson, R. B.; Anderson, J. R.; Droegemeier, K. K., Numerical solutions of a non-linear density current: a benchmark solution and comparisons, Int. J. Numer. Methods Fluids, 17, 1-22, (1993)
[46] Giraldo, F. X.; Kelly, J. F.; Constantinescu, E. M., Implicit-explicit formulations of a three-dimensional nonhydrostatic unified model of the atmosphere (NUMA), SIAM J. Sci. Comput., 35, 5, B1162-B1194, (2013) · Zbl 1280.86008
[47] De Luca, T. J., Performance of hybrid eulerian-Lagrangian semi-implicit time-integrators for nonhydrostatic mesoscale atmospheric modeling, (2007), Tech. rep., DTIC Document
[48] Giraldo, F. X.; Restelli, M., High-order semi-implicit time-integrators for a triangular discontinuous Galerkin oceanic shallow water model, Int. J. Numer. Methods Fluids, 63, 9, 1077-1102, (2010) · Zbl 1267.76010
[49] Higham, N. J., The accuracy of floating point summation, SIAM J. Sci. Comput., 14, 4, 783-799, (1993) · Zbl 0788.65053
[50] L.E. Carr, C.F. Borges, F.X. Giraldo, Polynomial-based nonlinear least squares optimized precoditioning and its application to continuous and discontinuous element-based discretizations of the Euler equations, J. Sci. Comput. (submitted 2014).
[51] Ruuth, S. J., Global optimization of explicit strong-stability-preserving Runge-Kutta methods, Math. Comput., 75, 253, 183-207, (2006) · Zbl 1080.65088
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.