Non-linear Boltzmann equation on hybrid-unstructured non-conforming multi-domains. (English) Zbl 07517098
Summary: Adaptivity is crucial for addressing practical challenges of the next decades. In this regard, recent surveys [J. P. Slotnick et al., CFD vision 2030 study: a path to revolutionary computational aerosciences, Techn. Rep., https://ntrs.nasa.gov/citations/20140003093] highlight that it continues to be a major bottleneck in computational fluid dynamics workflow. We introduce mixed non-conforming discontinuous Galerkin discretization for the full Boltzmann equation in 2D/3D. These schemes have been designed for efficiency – motivated in part by spectral and isogeometric weighted collocation methods – and retain an optimal \(O(p + 1)\) convergence for a \(p\) order approximation for non-linear kinetic systems on non-orthogonal grids. In this setting, it is possible to analyze highly complex problems of industrial strength i.e., structured, unstructured, mixed, irregular, multi-domain (multi-block) adaptive geometries at massively parallel scale (ten thousand cores or beyond). Mixed domains permit flexible mesh generation, whereas local nature of discontinuous Galerkin permits construction of adaptive numerical schemes that scale well. To address flows in mixing regime (low/high rarefaction), we couple the scheme with an asymptotic preserving implicit-explicit time discretization. These schemes are iteration free and applicable for a wide range of rarefied flows from free-molecular to continuum. To ensure stability in presence of shocks, we describe a method of constructing limiters on non-conforming grids. Finally, we show that the computational overhead for solving kinetic equations on non-conforming structured/unstructured domains is negligible relative to conforming domains. So, there is no reason to not prefer non-conforming unstructured domains.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 76Mxx | Basic methods in fluid mechanics |
| 65Nxx | Numerical methods for partial differential equations, boundary value problems |
Keywords:
discontinuous Galerkin; non-conforming discretization; Boltzmann equation; implicit-explicit schemesSoftware:
Polly; SPARTA; Nektar++; GiMMiK; PyFR; OCCA; libParanumal; Kokkos; Nek5000; mpmath; Gmsh; FEniCS; SyFi; LLVM; Rhinoceros; PETScReferences:
| [1] | Ainsworth, M., Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods, J. Comput. Phys., 198, 1, 106-130 (2004) · Zbl 1058.65103 |
| [2] | Ainsworth, M.; Oden, J. T., A Posteriori Error Estimation in Finite Element Analysis, vol. 37 (2011), John Wiley & Sons |
| [3] | Alekseenko, A.; Josyula, E., Deterministic solution of the spatially homogeneous Boltzmann equation using discontinuous Galerkin discretizations in the velocity space, J. Comput. Phys., 272, 170-188 (2014) · Zbl 1349.82051 |
| [4] | Ascher, U. M.; Ruuth, S. J.; Spiteri, R. J., Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math., 25, 2-3, 151-167 (1997) · Zbl 0896.65061 |
| [5] | Auricchio, F.; Da Veiga, L. B.; Hughes, T.; Reali, A.; Sangalli, G., Isogeometric collocation methods, Math. Models Methods Appl. Sci., 20, 11, 2075-2107 (2010) · Zbl 1226.65091 |
| [6] | Auricchio, F.; Da Veiga, L. B.; Hughes, T. J.; Reali, A.; Sangalli, G., Isogeometric collocation for elastostatics and explicit dynamics, Comput. Methods Appl. Mech. Eng., 249, 2-14 (2012) · Zbl 1348.74305 |
| [7] | Babuška, I.; Suri, M., The p and h-p versions of the finite element method, basic principles and properties, SIAM Rev., 36, 4, 578-632 (1994) · Zbl 0813.65118 |
| [8] | Balay, S., PETSc web page (2021) |
| [9] | Banks, J. W.; Hagstrom, T., On Galerkin difference methods, J. Comput. Phys., 313, 310-327 (2016) · Zbl 1349.65264 |
| [10] | Beirão da Veiga, L.; Brezzi, F.; Marini, L. D.; Russo, A., The hitchhiker’s guide to the virtual element method, Math. Models Methods Appl. Sci., 24, 08, 1541-1573 (2014) · Zbl 1291.65336 |
| [11] | Berger, M., A Panoramic View of Riemannian Geometry (2012), Springer Science & Business Media |
| [12] | Berger, M.; Aftosmis, M.; Muman, S., Analysis of slope limiters on irregular grids, (43rd AIAA Aerospace Sciences Meeting and Exhibit (2005)), 490 |
| [13] | Berger, M. J.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82, 1, 64-84 (1989) · Zbl 0665.76070 |
| [14] | Bhatnagar, P. L.; Gross, E. P.; Krook, M., A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94, 3, 511 (1954) · Zbl 0055.23609 |
| [15] | Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows (1994), Clarendon |
| [16] | Boscarino, S., Error analysis of IMEX Runge-Kutta methods derived from differential-algebraic systems, SIAM J. Numer. Anal., 45, 4, 1600-1621 (2007) · Zbl 1152.65088 |
| [17] | Cai, W.; Gottlieb, D.; Harten, A., Cell averaging Chebyshev methods for hyperbolic problems, (Upwind and High-Resolution Schemes (1997), Springer), 397-417 |
| [18] | Cantwell, C. D.; Moxey, D.; Comerford, A.; Bolis, A.; Rocco, G.; Mengaldo, G.; De Grazia, D.; Yakovlev, S.; Lombard, J.-E.; Ekelschot, D., Nektar++: an open-source spectral/hp element framework, Comput. Phys. Commun., 192, 205-219 (2015) · Zbl 1380.65465 |
| [19] | Castonguay, P.; Vincent, P. E.; Jameson, A., A new class of high-order energy stable flux reconstruction schemes for triangular elements, J. Sci. Comput., 51, 1, 224-256 (2012) · Zbl 1457.65101 |
| [20] | Cercignani, C., Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations, vol. 21 (2000), Cambridge University Press · Zbl 0961.76002 |
| [21] | Cercignani, C.; Lampis, M., On the H-theorem for polyatomic gases, J. Stat. Phys., 26, 4, 795-801 (1981) |
| [22] | Chalmers, N.; Karakus, A.; Austin, A. P.; Swirydowicz, K.; Warburton, T., libParanumal: A Performance Portable High-Order Finite Element Library (2020), Release 0.3.1 |
| [23] | Chan, J.; Wang, Z.; Modave, A.; Remacle, J.-F.; Warburton, T., GPU-accelerated discontinuous Galerkin methods on hybrid meshes, J. Comput. Phys., 318, 142-168 (2016) · Zbl 1349.65443 |
| [24] | Chapman, S.; Cowling, T. G.; Burnett, D., The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases (1990), Cambridge University Press |
| [25] | Chen, L.; Li, R., An integrated linear reconstruction for finite volume scheme on unstructured grids, J. Sci. Comput., 68, 3, 1172-1197 (2016) · Zbl 1365.65203 |
| [26] | Chen, Q.; Babuška, I., Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle, Comput. Methods Appl. Mech. Eng., 128, 3-4, 405-417 (1995) · Zbl 0862.65006 |
| [27] | Cockburn, B.; Gopalakrishnan, J.; Lazarov, R., Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47, 2, 1319-1365 (2009) · Zbl 1205.65312 |
| [28] | Cockburn, B.; Karniadakis, G. E.; Shu, C.-W., Discontinuous Galerkin Methods: Theory, Computation and Applications, vol. 11 (2012), Springer Science & Business Media |
| [29] | Cockburn, B.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comput., 52, 186, 411-435 (1989) · Zbl 0662.65083 |
| [30] | Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141, 2, 199-224 (1998) · Zbl 0920.65059 |
| [31] | Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 3, 173-261 (2001) · Zbl 1065.76135 |
| [32] | Cottrell, J. A.; Hughes, T. J.; Bazilevs, Y., Isogeometric Analysis: Toward Integration of CAD and FEA (2009), John Wiley & Sons · Zbl 1378.65009 |
| [33] | Demkowicz, L.; Oden, J. T.; Rachowicz, W.; Hardy, O., Toward a universal hp adaptive finite element strategy, Part 1. Constrained approximation and data structure, Comput. Methods Appl. Mech. Eng., 77, 1-2, 79-112 (1989) · Zbl 0723.73074 |
| [34] | Dimarco, G.; Loubère, R.; Narski, J.; Rey, T., An efficient numerical method for solving the Boltzmann equation in multidimensions, J. Comput. Phys., 353, 46-81 (2018) · Zbl 1380.76088 |
| [35] | Dimarco, G.; Pareschi, L., Asymptotic preserving implicit-explicit Runge-Kutta methods for nonlinear kinetic equations, SIAM J. Numer. Anal., 51, 2, 1064-1087 (2013) · Zbl 1268.76055 |
| [36] | Dimarco, G.; Pareschi, L., Implicit-explicit linear multistep methods for stiff kinetic equations, SIAM J. Numer. Anal., 55, 2, 664-690 (2017) · Zbl 1432.65131 |
| [37] | DiPerna, R. J.; Lions, P.-L., On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. Math., 321-366 (1989) · Zbl 0698.45010 |
| [38] | Duczek, S.; Saputra, A. A.; Gravenkamp, H., High order transition elements: the xNy-element concept—part I: statics, Comput. Methods Appl. Mech. Eng., 362, Article 112833 pp. (2020) · Zbl 1439.74408 |
| [39] | Duffy, M. G., Quadrature over a pyramid or cube of integrands with a singularity at a vertex, SIAM J. Numer. Anal., 19, 6, 1260-1262 (1982) · Zbl 0493.65011 |
| [40] | Dumbser, M.; Käser, M.; Toro, E. F., An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes-V. Local time stepping and p-adaptivity, Geophys. J. Int., 171, 2, 695-717 (2007) |
| [41] | Edwards, H. C.; Trott, C. R.; Sunderland, D., Kokkos: enabling manycore performance portability through polymorphic memory access patterns, J. Parallel Distrib. Comput., 74, 12, 3202-3216 (2014) |
| [42] | Evans, J. A.; Hughes, T. J., Isogeometric divergence-conforming B-splines for the steady Navier-Stokes equations, Math. Models Methods Appl. Sci., 23, 08, 1421-1478 (2013) · Zbl 1383.76337 |
| [43] | Filbet, F.; Jin, S., A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., 229, 20, 7625-7648 (2010) · Zbl 1202.82066 |
| [44] | Filbet, F.; Jin, S., An asymptotic preserving scheme for the ES-BGK model of the Boltzmann equation, J. Sci. Comput., 46, 2, 204-224 (2011) · Zbl 1433.82022 |
| [45] | Filbet, F.; Mouhot, C.; Pareschi, L., Solving the Boltzmann equation in N log2 N, SIAM J. Sci. Comput., 28, 3, 1029-1053 (2006) · Zbl 1174.82012 |
| [46] | Gallis, M. A.; Plimpton, S. J., SPARTA web page |
| [47] | Gamba, I. M.; Haack, J. R.; Hauck, C. D.; Hu, J., A fast spectral method for the Boltzmann collision operator with general collision kernels, SIAM J. Sci. Comput., 39, 4, B658-B674 (2017) · Zbl 1394.35310 |
| [48] | Gamba, I. M.; Rjasanow, S., Galerkin-Petrov approach for the Boltzmann equation, J. Comput. Phys., 366, 341-365 (2018) · Zbl 1406.65094 |
| [49] | Gassner, G.; Kopriva, D. A., A comparison of the dispersion and dissipation errors of Gauss and Gauss-Lobatto discontinuous Galerkin spectral element methods, SIAM J. Sci. Comput., 33, 5, 2560-2579 (2011) · Zbl 1255.65089 |
| [50] | Geuzaine, C.; Remacle, J.-F., Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, Int. J. Numer. Methods Eng., 79, 11, 1309-1331 (2009) · Zbl 1176.74181 |
| [51] | Giuliani, A.; Krivodonova, L., A moment limiter for the discontinuous Galerkin method on unstructured triangular meshes, SIAM J. Sci. Comput., 41, 1, A508-A537 (2019) · Zbl 1414.65020 |
| [52] | Giuliani, A.; Krivodonova, L., A moment limiter for the discontinuous Galerkin method on unstructured tetrahedral meshes, J. Comput. Phys., 404, Article 109106 pp. (2020) · Zbl 1453.65318 |
| [53] | Gottlieb, D.; Hesthaven, J. S., Spectral methods for hyperbolic problems, J. Comput. Appl. Math., 128, 1-2, 83-131 (2001) · Zbl 0974.65093 |
| [54] | Gottlieb, D.; Orszag, S. A., Numerical Analysis of Spectral Methods: Theory and Applications (1977), SIAM · Zbl 0412.65058 |
| [55] | Grad, H., Principles of the kinetic theory of gases, (Thermodynamik der Gase/Thermodynamics of Gases (1958), Springer), 205-294 |
| [56] | Greenshields, C. J.; Weller, H. G.; Gasparini, L.; Reese, J. M., Implementation of semi-discrete, non-staggered central schemes in a colocated, polyhedral, finite volume framework, for high-speed viscous flows, Int. J. Numer. Methods Fluids, 63, 1, 1-21 (2010) · Zbl 1425.76163 |
| [57] | Grosser, T.; Groesslinger, A.; Lengauer, C., Polly—performing polyhedral optimizations on a low-level intermediate representation, Parallel Process. Lett., 22, 04, Article 1250010 pp. (2012) |
| [58] | Guermond, J.-L.; Pasquetti, R.; Popov, B., Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., 230, 11, 4248-4267 (2011) · Zbl 1220.65134 |
| [59] | Harten, A., ENO schemes with subcell resolution, J. Comput. Phys., 83, 1, 148-184 (1989) · Zbl 0696.65078 |
| [60] | Harten, A.; Osher, S., Uniformly high-order accurate nonoscillatory schemes. I, SIAM J. Numer. Anal., 24, 2, 279-309 (1987) · Zbl 0627.65102 |
| [61] | Hartmann, R., Adjoint consistency analysis of discontinuous Galerkin discretizations, SIAM J. Numer. Anal., 45, 6, 2671-2696 (2007) · Zbl 1189.76341 |
| [62] | 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 |
| [63] | Hesthaven, J. S., Numerical Methods for Conservation Laws: From Analysis to Algorithms (2017), SIAM |
| [64] | Hesthaven, J. S.; Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (2007), Springer Science & Business Media |
| [65] | Holway, L. H., New statistical models for kinetic theory: methods of construction, Phys. Fluids, 9, 9, 1658-1673 (1966) |
| [66] | Hu, Z.; Cai, Z., Burnett spectral method for high-speed rarefied gas flows (2019), arXiv preprint |
| [67] | Hughes, T. J.; Engel, G.; Mazzei, L.; Larson, M. G., The continuous Galerkin method is locally conservative, J. Comput. Phys., 163, 2, 467-488 (2000) · Zbl 0969.65104 |
| [68] | Hughes, T. J.; Franca, L. P.; Hulbert, G. M., A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations, Comput. Methods Appl. Mech. Eng., 73, 2, 173-189 (1989) · Zbl 0697.76100 |
| [69] | Hughes, T. J.; Scovazzi, G.; Bochev, P. B.; Buffa, A., A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method, Comput. Methods Appl. Mech. Eng., 195, 19-22, 2761-2787 (2006) · Zbl 1124.76027 |
| [70] | Hundsdorfer, W.; Ruuth, S. J., IMEX extensions of linear multistep methods with general monotonicity and boundedness properties, J. Comput. Phys., 225, 2, 2016-2042 (2007) · Zbl 1123.65068 |
| [71] | Huynh, H. T., A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods, (18th AIAA Computational Fluid Dynamics Conference (2007)), 4079 |
| [72] | Jaiswal, S., Isogeometric schemes in the rarefied gas dynamics context, Comput. Methods Appl. Mech. Eng., 381, Article 113926 pp. (2021) · Zbl 1506.76155 |
| [73] | Jaiswal, S.; Alexeenko, A. A.; Hu, J., A discontinuous Galerkin fast spectral method for the full Boltzmann equation with general collision kernels, J. Comput. Phys., 378, 178-208 (2019) · Zbl 1416.76113 |
| [74] | Jaiswal, S.; Hu, J.; Brillon, J. K.; Alexeenko, A. A., A discontinuous Galerkin fast spectral method for multi-species full Boltzmann on streaming multi-processors, (Proceedings of the Platform for Advanced Scientific Computing Conference (2019)), 1-9 |
| [75] | Jasak, H., Error analysis and estimation for the finite volume method with applications to fluid flows (1996), Imperial College London (University of London), PhD thesis |
| [76] | Jawahar, P.; Kamath, H., A high-resolution procedure for Euler and Navier-Stokes computations on unstructured grids, J. Comput. Phys., 164, 1, 165-203 (2000) · Zbl 0992.76063 |
| [77] | Jiang, G. S.; Shu, C.-W., On a cell entropy inequality for discontinuous Galerkin methods, Math. Comput., 62, 206, 531-538 (1994) · Zbl 0801.65098 |
| [78] | Johansson, F., mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 1.1.0) (December 2018) |
| [79] | Karniadakis, G.; Sherwin, S., Spectral/hp Element Methods for CFD (1999), Oxford University Press · Zbl 0954.76001 |
| [80] | Kirby, R. M.; Karniadakis, G. E., De-aliasing on non-uniform grids: algorithms and applications, J. Comput. Phys., 191, 1, 249-264 (2003) · Zbl 1161.76534 |
| [81] | Kitzler, G.; Schöberl, J., A polynomial spectral method for the spatially homogeneous Boltzmann equation, SIAM J. Sci. Comput., 41, 1, B27-B49 (2019) · Zbl 1421.35238 |
| [82] | Klöckner, A.; Warburton, T.; Bridge, J.; Hesthaven, J. S., Nodal discontinuous Galerkin methods on graphics processors, J. Comput. Phys., 228, 21, 7863-7882 (2009) · Zbl 1175.65111 |
| [83] | Koornwinder, T., Two-variable analogues of the classical orthogonal polynomials, (Theory and Application of Special Functions (1975), Elsevier), 435-495 · Zbl 0326.33002 |
| [84] | Kopriva, D. A., A staggered-grid multidomain spectral method for the compressible Navier-Stokes equations, J. Comput. Phys., 143, 1, 125-158 (1998) · Zbl 0921.76121 |
| [85] | Kopriva, D. A.; Kolias, J. H., A conservative staggered-grid Chebyshev multidomain method for compressible flows, J. Comput. Phys., 125, 1, 244-261 (1996) · Zbl 0847.76069 |
| [86] | Kopriva, D. A.; Woodruff, S. L.; Hussaini, M. Y., Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method, Int. J. Numer. Methods Eng., 53, 1, 105-122 (2002) · Zbl 0994.78020 |
| [87] | Krivodonova, L.; Xin, J.; Remacle, J.-F.; Chevaugeon, N.; Flaherty, J. E., Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws, Appl. Numer. Math., 48, 3-4, 323-338 (2004) · Zbl 1038.65096 |
| [88] | Kronbichler, M.; Wall, W. A., A performance comparison of continuous and discontinuous Galerkin methods with fast multigrid solvers, SIAM J. Sci. Comput., 40, 5, A3423-A3448 (2018) · Zbl 1402.65163 |
| [89] | Kurganov, A.; Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160, 1, 241-282 (2000) · Zbl 0987.65085 |
| [90] | Lattner, C.; Adve, V., LLVM: a compilation framework for lifelong program analysis & transformation, (International Symposium on Code Generation and Optimization. International Symposium on Code Generation and Optimization, CGO 2004 (2004)), 75-86 (2004), IEEE |
| [91] | Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115, 1, 200-212 (1994) · Zbl 0811.65076 |
| [92] | Liu, Y.; Vinokur, M.; Wang, Z. J., Spectral difference method for unstructured grids I: basic formulation, J. Comput. Phys., 216, 2, 780-801 (2006) · Zbl 1097.65089 |
| [93] | Liu, Y.; Vinokur, M.; Wang, Z. J., Spectral (finite) volume method for conservation laws on unstructured grids V: extension to three-dimensional systems, J. Comput. Phys., 212, 2, 454-472 (2006) · Zbl 1085.65099 |
| [94] | Logg, A.; Mardal, K.-A.; Wells, G., Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, vol. 84 (2012), Springer Science & Business Media · Zbl 1247.65105 |
| [95] | Mavriplis, C., Nonconforming discretizations and a posteriori error estimators for adaptive spectral element techniques (1989), Massachusetts Institute of Technology, PhD thesis |
| [96] | May, S.; Berger, M., Two-dimensional slope limiters for finite volume schemes on non-coordinate-aligned meshes, SIAM J. Sci. Comput., 35, 5, A2163-A2187 (2013) · Zbl 1281.65116 |
| [97] | Medina, D. S.; St-Cyr, A.; Warburton, T., OCCA: a unified approach to multi-threading languages (2014), arXiv preprint |
| [98] | Mengaldo, G.; De Grazia, D.; Moxey, D.; Vincent, P. E.; Sherwin, S. J., Dealiasing techniques for high-order spectral element methods on regular and irregular grids, J. Comput. Phys., 299, 56-81 (2015) · Zbl 1352.65396 |
| [99] | Mouhot, C.; Pareschi, L., Fast algorithms for computing the Boltzmann collision operator, Math. Comput., 75, 256, 1833-1852 (2006) · Zbl 1105.76043 |
| [100] | Orszag, S. A., Spectral methods for problems in complex geometrics, (Numerical Methods for Partial Differential Equations (1979), Elsevier), 273-305 · Zbl 0448.65072 |
| [101] | Patera, A. T., A spectral element method for fluid dynamics: laminar flow in a channel expansion, J. Comput. Phys., 54, 3, 468-488 (1984) · Zbl 0535.76035 |
| [102] | Fischer, Paul F.; Lottes, J. W.; Kerkemeier, S. G., nek5000 web page (2008) |
| [103] | Persson, P.-O., High-order LES simulations using implicit-explicit Runge-Kutta schemes, (49th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition (2011)), 684 |
| [104] | Perugia, I.; Schötzau, D., The hp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations, Math. Comput., 72, 243, 1179-1214 (2003) · Zbl 1084.78007 |
| [105] | Pieraccini, S.; Puppo, G., Implicit-explicit schemes for BGK kinetic equations, J. Sci. Comput., 32, 1, 1-28 (2007) · Zbl 1115.76057 |
| [106] | Pletcher, R. H.; Tannehill, J. C.; Anderson, D., Computational Fluid Mechanics and Heat Transfer (2012), CRC Press |
| [107] | Plimpton, S.; Moore, S.; Borner, A.; Stagg, A.; Koehler, T.; Torczynski, J.; Gallis, M., Direct simulation Monte Carlo on petaflop supercomputers and beyond, Phys. Fluids, 31, 8, Article 086101 pp. (2019) |
| [108] | Qiu, J.; Shu, C.-W., A comparison of troubled-cell indicators for Runge-Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters, SIAM J. Sci. Comput., 27, 3, 995-1013 (2005) · Zbl 1092.65084 |
| [109] | Qiu, J.; Shu, C.-W., Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: two dimensional case, Comput. Fluids, 34, 6, 642-663 (2005) · Zbl 1134.65358 |
| [110] | Reddy, J. N.; Gartling, D. K., The Finite Element Method in Heat Transfer and Fluid Dynamics (2010), CRC Press · Zbl 1257.80001 |
| [111] | McNeel, Robert; Associates, Rhinoceros 3D web page |
| [112] | Schillinger, D.; Evans, J. A.; Reali, A.; Scott, M. A.; Hughes, T. J., Isogeometric collocation: cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations, Comput. Methods Appl. Mech. Eng., 267, 170-232 (2013) · Zbl 1286.65174 |
| [113] | Shakhov, E., Generalization of the Krook kinetic relaxation equation, Fluid Dyn., 3, 5, 95-96 (1968) |
| [114] | Shen, J.; Tang, T.; Wang, L.-L., Spectral Methods: Algorithms, Analysis and Applications, vol. 41 (2011), Springer Science & Business Media · Zbl 1227.65117 |
| [115] | Shoeybi, M.; Svärd, M.; Ham, F. E.; Moin, P., An adaptive implicit-explicit scheme for the DNS and LES of compressible flows on unstructured grids, J. Comput. Phys., 229, 17, 5944-5965 (2010) · Zbl 1425.76108 |
| [116] | Shunn, L.; Ham, F., Symmetric quadrature rules for tetrahedra based on a cubic close-packed lattice arrangement, J. Comput. Appl. Math., 236, 17, 4348-4364 (2012) · Zbl 1247.65035 |
| [117] | Slotnick, J.; Khodadoust, A.; Alonso, J.; Darmofal, D.; Gropp, W.; Lurie, E.; Mavriplis, D., CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences (2014) |
| [118] | Spiegel, S. C.; Huynh, H.; DeBonis, J. R., A survey of the isentropic Euler vortex problem using high-order methods, (22nd AIAA Computational Fluid Dynamics Conference (2015)), 2444 |
| [119] | Tadmor, E., A review of numerical methods for nonlinear partial differential equations, Bull. Am. Math. Soc., 49, 4, 507-554 (2012) · Zbl 1258.65073 |
| [120] | Tadmor, E.; Tao, T., Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDEs, Commun. Pure Appl. Math.:, 60, 10, 1488-1521 (2007), A Journal Issued by the Courant Institute of Mathematical Sciences · Zbl 1131.35004 |
| [121] | Taylor, M. A.; Wingate, B. A.; Bos, L. P., Several new quadrature formulas for polynomial integration in the triangle (2005), arXiv preprint |
| [122] | Truesdell, C.; Muncaster, R. G., Fundamentals of Maxwel’s Kinetic Theory of a Simple Monatomic Gas: Treated as a Branch of Rational Mechanics, vol. 83 (1980), Academic Press · Zbl 1515.80001 |
| [123] | Villani, C., A review of mathematical topics in collisional kinetic theory, (Handbook of Mathematical Fluid Dynamics, vol. 1 (2002)), 71-305, 3-8 · Zbl 1170.82369 |
| [124] | Vincent, T.; Pfeiffer, H. P.; Fischer, N. L., hp-adaptive discontinuous Galerkin solver for elliptic equations in numerical relativity, Phys. Rev. D, 100, 8, Article 084052 pp. (2019) |
| [125] | Wang, Z. J., Spectral (finite) volume method for conservation laws on unstructured grids. Basic formulation: basic formulation, J. Comput. Phys., 178, 1, 210-251 (2002) · Zbl 0997.65115 |
| [126] | Warburton, T., A low-storage curvilinear discontinuous Galerkin method for wave problems, SIAM J. Sci. Comput., 35, 4, A1987-A2012 (2013) · Zbl 1362.65109 |
| [127] | Weller, H. G.; Tabor, G.; Jasak, H.; Fureby, C., A tensorial approach to computational continuum mechanics using object-oriented techniques, Comput. Phys., 12, 6, 620-631 (1998) |
| [128] | Williams, D. M.; Shunn, L.; Jameson, A., Symmetric quadrature rules for simplexes based on sphere close packed lattice arrangements, J. Comput. Appl. Math., 266, 18-38 (2014) · Zbl 1293.65039 |
| [129] | Winters, A. R.; Moura, R. C.; Mengaldo, G.; Gassner, G. J.; Walch, S.; Peiro, J.; Sherwin, S. J., A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations, J. Comput. Phys., 372, 1-21 (2018) · Zbl 1415.76461 |
| [130] | Witherden, F.; Vincent, P., On nodal point sets for flux reconstruction, J. Comput. Appl. Math., Article 113014 pp. (2020) · Zbl 1486.65183 |
| [131] | Witherden, F. D.; Farrington, A. M.; Vincent, P. E., PyFR: an open source framework for solving advection-diffusion type problems on streaming architectures using the flux reconstruction approach, Comput. Phys. Commun., 185, 11, 3028-3040 (2014) · Zbl 1348.65005 |
| [132] | Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54, 1, 115-173 (1984) · Zbl 0573.76057 |
| [133] | Wozniak, B. D.; Witherden, F. D.; Russell, F. P.; Vincent, P. E.; Kelly, P. H., GiMMiK—generating bespoke matrix multiplication kernels for accelerators: application to high-order computational fluid dynamics, Comput. Phys. Commun., 202, 12-22 (2016) |
| [134] | Xiao, H.; Gimbutas, Z., A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions, Comput. Math. Appl., 59, 2, 663-676 (2010) · Zbl 1189.65047 |
| [135] | Yuan, K.-Y.; Huang, Y.-S.; Pian, T. H., Inverse mapping and distortion measures for quadrilaterals with curved boundaries, Int. J. Numer. Methods Eng., 37, 5, 861-875 (1994) · Zbl 0796.73068 |
| [136] | Zhang, X.; Shu, C.-W., Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms, J. Comput. Phys., 230, 4, 1238-1248 (2011) · Zbl 1391.76375 |
| [137] | Zhu, J.; Qiu, J., Hermite weno schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method, iii: unstructured meshes, J. Sci. Comput., 39, 2, 293-321 (2009) · Zbl 1203.65156 |
| [138] | Zhu, J.; Zhong, X.; Shu, C.-W.; Qiu, J., Runge-Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes, J. Comput. Phys., 248, 200-220 (2013) · Zbl 1349.65501 |
| [139] | Zhu, J.; Zhong, X.; Shu, C.-W.; Qiu, J., Runge-Kutta discontinuous Galerkin method with a simple and compact Hermite WENO limiter, Commun. Comput. Phys., 19, 4, 944-969 (2016) · Zbl 1373.76113 |
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.
