Skip to main content
SpringerLink
Log in

Discontinuous Galerkin Methods for Compressible and Incompressible Flows on Space–Time Adaptive Meshes: Toward a Novel Family of Efficient Numerical Methods for Fluid Dynamics

  • Original Paper
  • Published:
Archives of Computational Methods in Engineering Aims and scope Submit manuscript

Abstract

In this work the numerical discretization of the partial differential governing equations for compressible and incompressible flows is dealt within the discontinuous Galerkin (DG) framework along space–time adaptive meshes. Two main numerical frameworks can be distinguished: (1) fully explicit ADER-DG methods on collocated grids for compressible fluids (2) spectral semi-implicit and spectral space–time DG methods on edge-based staggered grids for the incompressible Navier–Stokes equations. In this work, the high-resolution properties of the aforementioned numerical methods are significantly enhanced within a 'cell-by-cell' Adaptive Mesh Refinement (AMR) implementation together with time accurate local time stepping (LTS). It is a well known fact that a major weakness of high order DG methods lies in the difficulty of limiting discontinuous solutions, which generate spurious oscillations, namely the so-called 'Gibbs phenomenon'. Over the years, several attempts have been made to cope with this problem and different kinds of limiters have been proposed. In this work the nonlinear stabilization of the scheme is sequentially and locally introduced only for troubled cells on the basis of a multidimensional optimal order detection (MOOD) criterion. ADER-DG is a novel, communication-avoiding family of algorithms, which achieves high order of accuracy in time not via the standard multi-stage Runge–Kutta (RK) time discretization like most other DG schemes, but at the aid of an element-local predictor stage. In practice the method first produces a so-called candidate solution by using a high order accurate unlimited DG scheme. Then, in those cells where at least one of the chosen admissibility criteria is violated the computed candidate solution is detected as troubled and is locally rejected. Next, the numerical solution of the previous time step is scattered onto cell averages on a suitable sub-grid in order to preserve the natural sub-cell resolution of the DG scheme. Then, a more reliable numerical solution is recomputed a posteriori by employing a more robust but still very accurate ADER-WENO finite volume scheme on the sub-grid averages within that troubled cell. Finally, a high order DG polynomial is reconstructed back from the evolved sub-cell averages. In the ADER-DG framework several PDE system are investigated, ranging from the Euler equations of compressible gas dynamics, over the viscous and resistive magneto-hydrodynamics (MHD), to special and general relativistic MHD. Indeed, the adopted formalism is quite general, leading to a novel family of adaptive ADER-DG schemes suitable for hyperbolic systems of partial differential equations in which the numerical fluxes also depend on the gradient of the state vector because of the parabolic nature of diffusive terms. The presented results show clearly that the high-resolution and shock-capturing capabilities of the news schemes are significantly enhanced within the cell-by-cell AMR implementation together with time accurate LTS. A special treatment has been followed for the incompressible Navier–Stokes equations. In fact, the elliptic character of the incompressible Navier–Stokes equations introduces an important difficulty in their numerical solution: whenever the smallest physical or numerical perturbation arises in the fluid flow then it will instantaneously affect the entire computational domain. Thus, a semi-implicit approach has been used. The main advantage of making use of a semi-implicit discretization is that the numerical stability can be obtained for large time-steps without leading to an excessive computational demand. In this context, we derived two new families of spectral semi-implicit and spectral space–time DG methods for the solution of the two and three dimensional Navier–Stokes equations on edge-based adaptive staggered Cartesian grids. The discrete solutions of pressure and velocity are expressed in the form of piecewise polynomials along different meshes. While the pressure is defined on the control volumes of the main grid, the velocity components are defined on edge-based dual control volumes, leading to a spatially staggered mesh. In the first family, high order of accuracy is achieved only in space, while a simple semi-implicit time discretization is derived by introducing an implicitness factor \(\theta \in [0.5,1]\) for the pressure gradient in the momentum equation. The real advantages of the staggering arise after substituting the discrete momentum equation into the weak form of the continuity equation. In fact, the resulting linear system for the pressure is symmetric and positive definite and either block penta-diagonal (in 2D) or block hepta-diagonal (in 3D). As a consequence, the pressure system can be solved very efficiently by means of a classical matrix-free conjugate gradient method. The resulting algorithm is stable, computationally very efficient, and at the same time arbitrary high order accurate in both space and time. This new numerical method has been thoroughly validated for approximation polynomials of degree up to \(N = 12\), using a large set of non-trivial test problems in two and three space dimensions, for which either analytical, numerical or experimental reference solutions exist.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35
Fig. 36
Fig. 37
Fig. 38
Fig. 39
Fig. 40
Fig. 41
Fig. 42
Fig. 43
Fig. 44
Fig. 45
Fig. 46
Fig. 47
Fig. 48
Fig. 49
Fig. 50
Fig. 51
Fig. 52
Fig. 53
Fig. 54
Fig. 55
Fig. 56
Fig. 57

Similar content being viewed by others

Notes

  1. Here \(\circ\) denotes the interior operator, i.e. only the boundary surfaces of the elements overlap, not the volumes.

  2. \(\circ\) denotes the interior operator, i.e. with \(T^{\circ } = T {\setminus} \partial T\).

References

  1. Dumbser M, Fambri F, Tavelli M, Bader M, Weinzierl T (2018) Efficient implementation of ADER discontinuous Galerkin schemes for a scalable hyperbolic PDE engine. Axioms 7(3):63. https://doi.org/10.3390/axioms7030063. http://www.mdpi.com/2075-1680/7/3/63

    Google Scholar 

  2. Ritz W (1909) Über eine neue methode zur lösung gewisser variationsprobleme der mathematischen physik. Journal fur die Reine und Angewandte Mathematik 1909(135):1–61

    MATH  Google Scholar 

  3. Galerkin BG (1915) On electrical circuits for the approximate solution of the laplace equation. Vestnik Inzhenerov i Tekhnikov 19:897–908

    Google Scholar 

  4. Reed WH, Hill TR (1973) Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory

  5. Cockburn B, Lin S-Y, Shu C-W (1989) TVB Runge–Kutta local projection discontinuous galerkin finite element method for conservation laws III: one-dimensional systems. J Comput Phys 84:90

    MathSciNet  MATH  Google Scholar 

  6. Cockburn B, How S, Shu C-W (1990) TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math Comput 54:545

    MATH  Google Scholar 

  7. Cockburn B, Shu CW (1998a) The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J Comput Phys 141(2):199–224

    MathSciNet  MATH  Google Scholar 

  8. Cockburn B, Karniadakis GE, Shu C-W (2000) Discontinuous Galerkin methods: theory computation and applications. Lecture notes on computational science and engineering. Springer, Berlin

    Google Scholar 

  9. Cockburn B, Shu CW (2001) Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J Sci Comput 16(3):173

    MathSciNet  MATH  Google Scholar 

  10. Jiang G, Shu CW (1994) On a cell entropy inequality for discontinuous Galerkin methods. Math Comput 62:531–538

    MathSciNet  MATH  Google Scholar 

  11. Barth T, Charrier P (2001) Energy stable flux formulas for the discontinuous Galerkin discretization of first-order nonlinear conservation laws. Technical Report NAS-01-001, NASA

  12. Hou S, Liu XD (2007) Solutions of multi-dimensional hyperbolic systems of conservation laws by square entropy condition satisfying discontinuous Galerkin method. J Sci Comput 31:127–151

    MathSciNet  MATH  Google Scholar 

  13. Cockburn B, Shu CW (1989) TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math Comput 52:411–435

    MathSciNet  MATH  Google Scholar 

  14. Cockburn B, Lin SY, Shu CW (1989) TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J Comput Phys 84:90–113

    MathSciNet  MATH  Google Scholar 

  15. Cockburn B, Hou S, Shu CW (1990) The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math Comput 54:545–581

    MathSciNet  MATH  Google Scholar 

  16. Cockburn B, Shu CW (1998b) The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J Comput Phys 141:199–224

    MathSciNet  MATH  Google Scholar 

  17. Gottlieb S, Shu CW (1998) Total variation diminishing Runge–Kutta schemes. Math Comput 67:73–85

    MathSciNet  MATH  Google Scholar 

  18. Toro EF, Millington RC, Nejad LAM (2001) Towards very high order Godunov schemes. In: Toro EF (ed) Godunov methods. Theory and applications. Kluwer, New York, pp 905–938

    Google Scholar 

  19. Titarev VA, Toro EF (2002) ADER: arbitrary high order Godunov approach. J Sci Comput 17(1–4):609–618

    MathSciNet  MATH  Google Scholar 

  20. Toro EF, Titarev VA (2002) Solution of the generalized Riemann problem for advection–reaction equations. Proc R Soc Lond, pp 271–281. http://rspa.royalsocietypublishing.org/content/458/2018/271.abstract

    MathSciNet  MATH  Google Scholar 

  21. Titarev VA, Toro EF (2005) ADER schemes for three-dimensional nonlinear hyperbolic systems. J Comput Phys 204:715–736

    MathSciNet  MATH  Google Scholar 

  22. Toro EF, Titarev VA (2006) Derivative Riemann solvers for systems of conservation laws and ADER methods. J Comput Phys 212(1):150–165

    MathSciNet  MATH  Google Scholar 

  23. Toro E F, Titarev V A (2002) Solution of the generalized Riemann problem for advection–reaction equations. Proc R Soc Lond Ser A Math Phys Eng Sci 458 (2018):271–281. http://rspa.royalsocietypublishing.org/content/458/2018/271.abstract

    MathSciNet  MATH  Google Scholar 

  24. Castro CC, Toro EF (2008) Solvers for the high-order Riemann problem for hyperbolic balance laws. J Comput Phys 227:2481–2513

    MathSciNet  MATH  Google Scholar 

  25. Montecinos G, Castro CE, Dumbser M, Toro EF (2012) Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms. J Comput Phys 231(19):6472–6494

    MathSciNet  MATH  Google Scholar 

  26. Goetz CR, Iske A (2016) Approximate solutions of generalized Riemann problems for nonlinear systems of hyperbolic conservation laws. Math Comput 85:35–62

    MathSciNet  MATH  Google Scholar 

  27. Dumbser M, Enaux C, Toro EF (2008) Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J Comput Phys 227:3971–4001

    MathSciNet  MATH  Google Scholar 

  28. Tavelli M, Dumbser M (2016) A staggered space–time discontinuous Galerkin method for the three-dimensional incompressible Navier–Stokes equations on unstructured tetrahedral meshes. J Comput Phys 319:294–323

    MathSciNet  MATH  Google Scholar 

  29. Fambri F, Dumbser M (2016) Spectral semi-implicit and space–time discontinuous Galerkin methods for the incompressible Navier–Stokes equations on staggered Cartesian grids. Appl Numer Math 110:41–74

    MathSciNet  MATH  Google Scholar 

  30. Fambri F, Dumbser M (2017) Semi-implicit discontinuous Galerkin methods for the incompressible Navier–Stokes equations on adaptive staggered Cartesian grids. Comput Methods Appl Mech Eng 324:170–203. https://arxiv.org/abs/1612.09558

    MathSciNet  MATH  Google Scholar 

  31. Dumbser M, Fambri F, Furci I, Mazza M, Serra-Capizzano S, Tavelli M (2018) Staggered discontinuous Galerkin methods for the incompressible Navier–Stokes equations: spectral analysis and computational results. Numer Linear Algebra Appl 0(0):e2151. https://doi.org/10.1002/nla.2151. https://onlinelibrary.wiley.com/doi/abs/10.1002/nla.2151. e2151 nla.2151

    MathSciNet  MATH  Google Scholar 

  32. Dumbser M, Zanotti O (2009) Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations. J Comput Phys 228:6991–7006

    MathSciNet  MATH  Google Scholar 

  33. Hidalgo A, Dumbser M (2011) ADER schemes for nonlinear systems of stiff advection–diffusion–reaction equations. J Sci Comput 48:173–189

    MathSciNet  MATH  Google Scholar 

  34. Dumbser M (2010) Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier–Stokes equations. Comput Fluids 39:60–76

    MathSciNet  MATH  Google Scholar 

  35. Dumbser M, Zanotti O, Hidalgo A, Balsara DS (2013) ADER-WENO finite volume schemes with space–time adaptive mesh refinement. J Comput Phys 248:257–286

    MathSciNet  MATH  Google Scholar 

  36. Dumbser M, Hidalgo A, Zanotti O (2014) High order space–time adaptive ADER-WENO finite volume schemes for non-conservative hyperbolic systems. Comput Methods Appl Mech Eng 268:359–387

    MathSciNet  MATH  Google Scholar 

  37. Balsara DS, Dumbser M, Abgrall R (2014) Multidimensional HLLC Riemann solver for unstructured meshes—with application to Euler and MHD flows. J Comput Phys 261:172–208

    MathSciNet  MATH  Google Scholar 

  38. Dumbser M, Munz CD (2006) Building blocks for arbitrary high order discontinuous Galerkin schemes. J Sci Comput 27:215–230

    MathSciNet  MATH  Google Scholar 

  39. Qiu J, Dumbser M, Shu CW (2005) The discontinuous Galerkin method with Lax–Wendroff type time discretizations. Comput Methods Appl Mech Eng 194:4528–4543

    MathSciNet  MATH  Google Scholar 

  40. Dumbser M, Käser M, Toro EF (2007) An arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes V: local time stepping and \(p\)-adaptivity. Geophys J Int 171:695–717

    Google Scholar 

  41. 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

    MathSciNet  MATH  Google Scholar 

  42. Zanotti O, Fambri F, Dumbser M (2015) Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement. Mon Notoices R Astron Soc 452:3010–3029

    Google Scholar 

  43. 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. https://doi.org/10.1016/j.cpc.2017.08.001. http://www.sciencedirect.com/science/article/pii/S0010465517302448

    MathSciNet  Google Scholar 

  44. Fambri F, Dumbser M, Köppel S, Rezzolla L, Zanotti O (2018) ADER discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics. Mon Notices R Astron Soc 477:4543–4564

    Google Scholar 

  45. Hartmann R, Houston P (2002) Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations. J Comput Phys 183(2):508–532

    MathSciNet  MATH  Google Scholar 

  46. Persson P-O, Peraire J (2006) Sub-cell shock capturing for discontinuous Galerkin methods. AIAA paper 2006-112

  47. Cesenek J, Feistauer M, Horacek J, Kucera V, Prokopova J (2013) Simulation of compressible viscous flow in time-dependent domains. Appl Math Comput 219:7139–7150

    MathSciNet  MATH  Google Scholar 

  48. Radice D, Rezzolla L (2011) Discontinuous Galerkin methods for general-relativistic hydrodynamics: formulation and application to spherically symmetric spacetimes. Phys Rev D 84(2):024010

    Google Scholar 

  49. Qiu J, Shu CW (2005a) Runge–Kutta discontinuous Galerkin method using WENO limiters. SIAM J Sci Comput 26:907–929

    MathSciNet  MATH  Google Scholar 

  50. Qiu J, Shu C-W (2004) Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method: One-dimensional case. J Comput Phys 193(1):115–135

    MathSciNet  MATH  Google Scholar 

  51. Balsara DS, Altmann C, Munz C-D, Dumbser M (2007) A sub-cell based indicator for troubled zones in RKDG schemes and a novel class of hybrid RKDG + HWENO schemes. J Comput Phys 226:586–620

    MathSciNet  MATH  Google Scholar 

  52. Zhu J, Qiu J, Shu C-W, Dumbser M (2008) Runge–Kutta discontinuous Galerkin method using WENO limiters ii: unstructured meshes. J Comput Phys 227(9):4330–4353

    MathSciNet  MATH  Google Scholar 

  53. Shu CW, Zhu J, Zhong X, Qiu J (2013) Runge–Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes. J Comput Phys 248:200–220

    MathSciNet  MATH  Google Scholar 

  54. Luo H, Baum JD, Löhner R (2007) A hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids. J Comput Phys 225(1):686–713

    MathSciNet  MATH  Google Scholar 

  55. Krivodonova L (2007) Limiters for high-order discontinuous Galerkin methods. J Comput Phys 226:879–896

    MathSciNet  MATH  Google Scholar 

  56. Kuzmin D (2014) Hierarchical slope limiting in explicit and implicit discontinuous Galerkin methods. J Comput Phys, 257, Part B(0):1140 – 1162. ISSN 0021-9991. Physics-compatible numerical methods

    MathSciNet  MATH  Google Scholar 

  57. Clain S, Diot S, Loubère R (2011) A high-order finite volume method for systems of conservation lawsmulti-dimensional optimal order detection (MOOD). J Comput Phys 230(10):4028–4050

    MathSciNet  MATH  Google Scholar 

  58. 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

    MathSciNet  MATH  Google Scholar 

  59. 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

    Google Scholar 

  60. Loubère 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 16:718–763

    MathSciNet  MATH  Google Scholar 

  61. 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. https://doi.org/10.1016/j.jcp.2015.03.015

    Article  MathSciNet  MATH  Google Scholar 

  62. 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

    MathSciNet  MATH  Google Scholar 

  63. Baumann CE, Oden JT (1999a) A discontinuous hp finite element method for convection–diffusion problems. Comput Methods Appl Mech Eng 175:311–341

    MathSciNet  MATH  Google Scholar 

  64. Baumann CE, Oden JT (1999b) A discontinuous hp finite element method for the Euler and Navier–Stokes equations. Int J Numer Methods Fluids 31:79–95

    MathSciNet  MATH  Google Scholar 

  65. Houston P, Schwab C, Sli E (2002) Discontinuous hp-finite element methods for advection–diffusion–reaction problems*. SIAM J Numer Anal 39(6):2133–2163

    MathSciNet  MATH  Google Scholar 

  66. Houston P, Schwab C, Sli E (2000) Stabilized hp-finite element methods for first-order hyperbolic problems. SIAM J Numer Anal 37(5):1618–1643

    MathSciNet  MATH  Google Scholar 

  67. Houston P, Sli E (2002) hp-adaptive discontinuous Galerkin finite element methods for first-order hyperbolic problems. SIAM J Sci Comput 23(4):1226–1252

    MathSciNet  Google Scholar 

  68. Leicht T, Hartmann R (2008) Anisotropic mesh refinement for discontinuous galerkin methods in two-dimensional aerodynamic flow simulations. Int J Numer Methods Fluids 56(11):2111–2138

    MathSciNet  MATH  Google Scholar 

  69. Luo H, Baum JD, Löhner R (2008) A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids. J Comput Phys 227:8875–8893

    MathSciNet  MATH  Google Scholar 

  70. Yu Y, Wu D, Xu Y (2011) Three dimensional discontinuous galerkin methods for euler equations on adaptive conforming meshes. Comput Phys Commun 182(9):1771–1775

    MathSciNet  MATH  Google Scholar 

  71. Kopera MA, Giraldo FX (2014) Analysis of adaptive mesh refinement for IMEX discontinuous Galerkin solutions of the compressible Euler equations with application to atmospheric simulations. J Comput Phys 275:92–117

    MathSciNet  MATH  Google Scholar 

  72. Georgoulis EH, Hall E, Houston P (2009) Discontinuous galerkin methods on hp-anisotropic meshes ii: a posteriori error analysis and adaptivity. Appl Numer Math 59(9):2179–2194

    MathSciNet  MATH  Google Scholar 

  73. Lu H, Sun Q (2014) A straightforward hp-adaptivity strategy for shock-capturing with high-order discontinuous galerkin methods. Adv Appl Math Mech 6(1):135–144

    MathSciNet  Google Scholar 

  74. Wilson JR (1975) Some magnetic effects in stellar collapse and accretion. Ann N Y Acad Sci 262:123

    Google Scholar 

  75. Font JA (2008) Numerical hydrodynamics and magnetohydrodynamics in general relativity. Living Rev Relativ 6:4. http://www.livingreviews.org/lrr--2008--7, http://www.livingreviews.org/lrr-2008-7

  76. Martí JM, Müller E (2015) Grid-based methods in relativistic hydrodynamics and magnetohydrodynamics. Living Rev Comput Astrophys, vol 1, December 2015. https://doi.org/10.1007/lrca-2015-3

  77. Komissarov SS (1999) A Godunov-type scheme for relativistic magnetohydrodynamics. Mon Notices R Astron Soc 303:343–366

    Google Scholar 

  78. Balsara D (2001a) Total variation diminishing scheme for relativistic magnetohydrodynamics. Astrophys J Suppl Ser 132:83–101

    Google Scholar 

  79. Mignone A, Bodo G (2006) An HLLC Riemann solver for relativistic flows-II. Magnetohydrodynamics. Mon Notices R Astron Soc 368:1040–1054

    Google Scholar 

  80. Honkkila V, Janhunen P (2007) HLLC solver for ideal relativistic MHD. J Comput Phys 223:643–656

    MathSciNet  MATH  Google Scholar 

  81. Mignone A, Ugliano M, Bodo G (2009) A five-wave Harten–Lax–van Leer Riemann solver for relativistic magnetohydrodynamics. Mon Notices R Astron Soc 393(4):1141–1156

    Google Scholar 

  82. Kim J, Balsara DS (2014) A stable HLLC Riemann solver for relativistic magnetohydrodynamics. J Comput Phys 270:634–639

    MathSciNet  MATH  Google Scholar 

  83. Baumgarte TW, Shapiro SL (2003) General relativistic magnetohydrodynamics for the numerical construction of dynamical spacetimes. Astrophys J 585:921–929

    Google Scholar 

  84. Luca B, Ian H, Montero Pedro J, Frank L, Luciano R, Nikolaos S, Font José A, Seidel E (2005) Three-dimensional relativistic simulations of rotating neutron star collapse to a Kerr black hole. Phys Rev D 71:024035

    Google Scholar 

  85. Duez Matthew D, Tung Liu Yuk, Shapiro Stuart L, Stephens Branson C (2005) Relativistic magnetohydrodynamics in dynamical spacetimes: numerical methods and tests. Phys Rev D 72:024028 arXiv:astro-ph/0503420

    Google Scholar 

  86. Peter Anninos, Chris Fragile P, Salmonson Jay D (2005) Cosmos++: relativistic magnetohydrodynamics on unstructured grids with local adaptive refinement. Astrophys J 635:723

    Google Scholar 

  87. Luis Antón, Olindo Zanotti, Miralles Joan A, Martí José M, Ibáñez José M, Font José A, Pons José A (2006) Numerical 3 + 1 general relativistic magnetohydrodynamics: a local characteristic approach. Astrophys J 637:296

    Google Scholar 

  88. Del Zanna L, Zanotti O, Bucciantini N, Londrillo P (2007) ECHO: a Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics. Astron Astrophys 473:11–30

    Google Scholar 

  89. Bruno Giacomazzo, Luciano Rezzolla (2007) WhiskyMHD: a new numerical code for general relativistic magnetohydrodynamics. Class Quantum Gravity 24:S235

    MathSciNet  MATH  Google Scholar 

  90. Anderson M, Hirschmann EW, Lehner L, Liebling SL, Motl PM, Neilsen D, Palenzuela C, Tohline JE (2008) Magnetized neutron star mergers and gravitational wave signals. Phys Rev Lett 100:191101

    Google Scholar 

  91. Kiuchi K, Sekiguchi Y, Shibata M, Taniguchi K (2009) Longterm general relativistic simulation of binary neutron stars collapsing to a black hole. Phys Rev D 80:064037

    Google Scholar 

  92. Bucciantini N, Del Zanna L (2011) General relativistic magnetohydrodynamics in axisymmetric dynamical spacetimes: the X-ECHO code. Astron Astrophys 528:A101

    Google Scholar 

  93. Radice D, Rezzolla L (2012) THC: a new high-order finite-difference high-resolution shock-capturing code for special-relativistic hydrodynamics. Astron Astrophys 547:A26

    Google Scholar 

  94. Dionysopoulou K, Alic D, Palenzuela C, Rezzolla L, Giacomazzo B (2013) General-relativistic resistive magnetohydrodynamics in three dimensions: formulation and tests. Phys Rev D 88:044020

    Google Scholar 

  95. Radice D, Rezzolla L, Galeazzi F (2013) Beyond second-order convergence in simulations of binary neutron stars in full general-relativity. arXiv:1306.6052

  96. White CJ, Stone JM, Gammie CF (2016) An extension of the athena++ code framework for grmhd based on advanced riemann solvers and staggered-mesh constrained transport. Astrophys J Suppl Ser 225(2):22. http://stacks.iop.org/0067-0049/225/i=2/a=22

    Google Scholar 

  97. Porth O, Olivares H, Mizuno Y, Younsi Z, Rezzolla L, Moscibrodzka M, Falcke H, Kramer M (2017) The black hole accretion code. Comput Astrophys Cosmol 4:1. https://doi.org/10.1186/s40668-017-0020-2

    Article  Google Scholar 

  98. Takahashi R, Umemura M (2017) General relativistic radiative transfer code in rotating black hole space–time: ARTIST. Mon Notices R Astron Soc 464:4567–4585. https://doi.org/10.1093/mnras/stw2479

    Article  Google Scholar 

  99. Palenzuela C, Lehner L, Reula O, Rezzolla L (2009) Beyond ideal MHD: towards a more realistic modelling of relativistic astrophysical plasmas. Mon Notices R Astron Soc 394:1727–1740

    Google Scholar 

  100. Bucciantini N, Del Zanna L (2013) A fully covariant mean-field dynamo closure for numerical 3 + 1 resistive GRMHD. Mon Notices R Astron Soc 428:71–85

    Google Scholar 

  101. Bugli M, Del Zanna L, Bucciantini N, High-order resistive GRMHD simulations (2014) Dynamo action in thick discs around Kerr black holes. Mon Notices R Astron Soc Lett 440:L41–L45. https://doi.org/10.1093/mnrasl/slu017

    Article  Google Scholar 

  102. Aloy MA, Cordero-Carrión I (2016) Minimally implicit Runge–Kutta methods for resistive relativistic MHD. J Phys Confer Ser 719:12015. https://doi.org/10.1088/1742-6596/719/1/012015

    Article  Google Scholar 

  103. Komissarov SS (2007) Multidimensional numerical scheme for resistive relativistic magnetohydrodynamics. Mon Notices R Astron Soc 382:995–1004

    Google Scholar 

  104. Zenitani S, Hesse M, Klimas A (2010) Resistive magnetohydrodynamic simulations of relativistic magnetic reconnection. Astrophys J Lett 716:L214–L218

    Google Scholar 

  105. Takamoto M, Inoue T (2011) A new numerical scheme for resistive relativistic magnetohydrodynamics using method of characteristics. Astrophys J 735:113

    Google Scholar 

  106. Bucciantini N, Del Zanna L (2013) A fully covariant mean-field dynamo closure for numerical 3 + 1 resistive GRMHD. Mon Notices R Astron Soc 428:71–85

    Google Scholar 

  107. Del Zanna L, Bucciantini N, Londrillo P (2003) An efficient shock-capturing central-type scheme for multidimensional relativistic flows. II. Magnetohydrodynamics. Astron Astrophys 400:397–413

    MATH  Google Scholar 

  108. Matthew Anderson, Eric Hirschmann, Liebling Steven L, David Neilsen (2006) Relativistic MHD with adaptive mesh refinement. Class Quantum Gravity 23:6503–6524

    MathSciNet  MATH  Google Scholar 

  109. Zenitani S, Hesse M, Klimas A (2009) Two-fluid magnetohydrodynamic simulations of relativistic magnetic reconnection. Astrophys J 696:1385–1401

    Google Scholar 

  110. Barkov M, Komissarov SS, Korolev V, Zankovich A (2014) A multidimensional numerical scheme for two-fluid relativistic magnetohydrodynamics. Mon Notices R Astron Soc 438:704–716

    Google Scholar 

  111. Balsara DS (2001b) Divergence-free adaptive mesh refinement for magnetohydrodynamics. J Comput Phys 174:614–648

    MATH  Google Scholar 

  112. Neilsen D, Hirschmann EW, Millward RS (2006) Relativistic MHD and excision: formulation and initial tests. Class Quantum Gravity 23:505

    MathSciNet  MATH  Google Scholar 

  113. Etienne Zachariah B, Tung Liu Yuk, Shapiro Stuart L (2010) Relativistic magnetohydrodynamics in dynamical spacetimes: a new AMR implementation. Phys Rev D 82:084031

    Google Scholar 

  114. Mignone A, Zanni C, Tzeferacos P, van Straalen B, Colella P, Bodo G (2012) The PLUTO code for adaptive mesh computations in astrophysical fluid dynamics. Astrophys J Suppl Ser 198:7

    Google Scholar 

  115. Keppens R, Meliani Z, van Marle AJ, Delmont P, Vlasis A, van der Holst B (2012) Parallel, grid-adaptive approaches for relativistic hydro and magnetohydrodynamics. J Comput Phys 231:718–744

    MathSciNet  MATH  Google Scholar 

  116. Zanotti O, Dumbser M (2015) A high order special relativistic hydrodynamic and magnetohydrodynamic code with space–time adaptive mesh refinement. Comput Phys Commun 188:110–127

    MathSciNet  MATH  Google Scholar 

  117. Dumbser M, Balsara DS, Toro EF, Munz C-D (2008) A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. J Comput Phys 227:8209–8253

    MathSciNet  MATH  Google Scholar 

  118. Zumbusch G (2009) Finite element, discontinuous Galerkin, and finite difference evolution schemes in spacetime. Class Quantum Gravity 26(17):175011

    MathSciNet  MATH  Google Scholar 

  119. Zanotti O, Dumbser M (2011) Numerical simulations of high Lundquist number relativistic magnetic reconnection. Mon Notices R Astron Soc 418:1004–1011

    Google Scholar 

  120. Bugner M, Dietrich T, Bernuzzi S, Weyhausen A, Brügmann B (2016) Solving 3d relativistic hydrodynamical problems with weighted essentially nonoscillatory discontinuous Galerkin methods. Phys Rev D 94:084004

    Google Scholar 

  121. Miller JM, Schnetter E (2017) An operator-based local discontinuous galerkin method compatible with the bssn formulation of the einstein equations. Class Quantum Gravity 34(1):015003

    MathSciNet  MATH  Google Scholar 

  122. Dumbser M, Guercilena F, Köppel S, Rezzolla L, Zanotti O (2018c) Conformal and covariant Z4 formulation of the Einstein equations: strongly hyperbolic first-order reduction and solution with discontinuous Galerkin schemes. Phys Rev D 97:084053

    MathSciNet  Google Scholar 

  123. Kidder LE, Field SE, Foucart F, Schnetter E, Teukolsky SA, Bohn A, Deppe N, Diener P, Hbert F, Lippuner J, Miller J, Ott CD, Scheel MA, Vincent T (2017) Spectre: a task-based discontinuous Galerkin code for relativistic astrophysics. J Comput Phys 335:84–114

    MathSciNet  MATH  Google Scholar 

  124. Anninos P, Bryant C, Fragile PC, Holgado AM, Lau C, Nemergut D (2017) Cosmosdg: an hp-adaptive discontinuous Galerkin code for hyper-resolved relativistic MHD. Astrophys J Suppl Ser 231(2):17

    Google Scholar 

  125. Bermúdez A, Vázquez ME (1994) Upwind methods for hyperbolic conservation laws with source terms. Comput Fluids 23:1049–1071

    MathSciNet  MATH  Google Scholar 

  126. Parés C (2006) Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 44(1):300–321

    MathSciNet  MATH  Google Scholar 

  127. Castro MJ, Gallardo JM, 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

    MathSciNet  MATH  Google Scholar 

  128. Castro MJ, Pardo C, Parés A, Toro EF (2010) On some fast well-balanced first order solvers for nonconservative systems. Math Comput 79:1427–1472

    MathSciNet  MATH  Google Scholar 

  129. Gaburro E, Dumbser M, Castro M (2017) Direct Arbitrary–Lagrangian–Eulerian finite volume schemes on moving nonconforming unstructured meshes. Comput Fluids 159:254–275. https://doi.org/10.1016/j.compfluid.2017.09.022

    Article  MathSciNet  MATH  Google Scholar 

  130. Gaburro E, Castro M, Dumbser M(2018) Well balanced Arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming meshes for the Euler equations of gasdynamics with gravity. MNRAS. https://doi.org/10.1093/mnras/sty542. arXiv:1712.07765

    Google Scholar 

  131. van der Vegt JJW, van der Ven H (2002) Space–time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows I. General formulation. J Comput Phys 182:546–585

    MathSciNet  MATH  Google Scholar 

  132. van der Ven H, van der Vegt JJW (2002) Space–time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows II. Efficient flux quadrature. Comput Methods Appl Mech Eng 191:4747–4780

    MathSciNet  MATH  Google Scholar 

  133. Klaij C, Van der Vegt JJW, Van der Ven H (2006) Space–time discontinuous Galerkin method for the compressible Navier–Stokes equations. J Comput Phys 217:589–611

    MathSciNet  MATH  Google Scholar 

  134. Rhebergen S, Cockburn B, van der Vegt Jaap JW (2013) A space–time discontinuous Galerkin method for the incompressible Navier–Stokes equations. J Comput Phys 233:339–358

    MathSciNet  MATH  Google Scholar 

  135. Bassi F, Botti L, Colombo A, Ghidoni A, Massa F (2015) Linearly implicit Rosenbrock-type Runge–Kutta schemes applied to the discontinuous Galerkin solution of compressible and incompressible unsteady flows. Comput Fluids 118:305–320

    MathSciNet  MATH  Google Scholar 

  136. Bassi F, Rebay S (1997) A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J Comput Phys 131:267–279

    MathSciNet  MATH  Google Scholar 

  137. Cockburn B, Shu CW (1998c) The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J Numer Anal 35(6):2440–2463

    MathSciNet  MATH  Google Scholar 

  138. Yan J, Shu CW (2002) A local discontinuous Galerkin method for KdV type equations. SIAM J Numer Anal 40(2):769–791

    MathSciNet  MATH  Google Scholar 

  139. Bassi F, Crivellini A, Di Pietro DA, Rebay S (2007) An implicit high-order discontinuous Galerkin method for steady and unsteady incompressible flows. Comput Fluids 36(10):1529–1546

    MathSciNet  MATH  Google Scholar 

  140. Gassner G, Lörcher F, Munz CD (2007) A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes. J Comput Phys 224:1049–1063

    MathSciNet  MATH  Google Scholar 

  141. Gassner G, Lörcher F, Munz CD (2008) A discontinuous Galerkin scheme based on a space–time expansion II. Viscous flow equations in multi dimensions. J Sci Comput 34:260–286

    MathSciNet  MATH  Google Scholar 

  142. Hartmann R, Houston P (2006) Symmetric interior penalty DG methods for the compressible Navier–Stokes equations I: method formulation. Int J Numer Anal Model 3:1–20

    MathSciNet  MATH  Google Scholar 

  143. Hartmann R, Houston P (2008) An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier–Stokes equations. J Comput Phys 227:9670–9685

    MathSciNet  MATH  Google Scholar 

  144. Crivellini A, D’Alessandro V, Bassi F (2013) High-order discontinuous Galerkin solutions of three-dimensional incompressible RANS equations. Comput Fluids 81:122–133

    MathSciNet  MATH  Google Scholar 

  145. Klein B, Kummer F, Oberlack M (2013) A SIMPLE based discontinuous Galerkin solver for steady incompressible flows. J Comput Phys 237:235–250

    MathSciNet  MATH  Google Scholar 

  146. Casulli V, Greenspan D (1984) Pressure method for the numerical solution of transient, compressible fluid flows. Int J Numer Methods Fluids 4(11):1001–1012

    MATH  Google Scholar 

  147. Casulli V, Walters RA (2000) An unstructured grid, three-dimensional model based on the shallow water equations. Int J Numer Methods Fluids 32:331–348

    MATH  Google Scholar 

  148. Casulli V (2009) A high-resolution wetting and drying algorithm for free-surface hydrodynamics. Int J Numer Methods Fluids 60:391–408

    MathSciNet  MATH  Google Scholar 

  149. Casulli V (2014) A semi-implicit numerical method for the free-surface Navier–Stokes equations. Int J Numer Methods Fluids 74:605–622

    MathSciNet  Google Scholar 

  150. Casulli V, Cattani E (1994) Stability, accuracy and efficiency of a semi implicit method for three-dimensional shallow water flow. Comput Math Appl 27:99–112

    MathSciNet  MATH  Google Scholar 

  151. Brugnano L, Casulli V (2008) Iterative solution of piecewise linear systems. SIAM J Sci Comput 30:463–472

    MathSciNet  MATH  Google Scholar 

  152. Brugnano L, Casulli V (2009) Iterative solution of piecewise linear systems and applications to flows in porous media. SIAM J Sci Comput 31:1858–1873

    MathSciNet  MATH  Google Scholar 

  153. Casulli V, Zanolli P (2012) Iterative solutions of mildly nonlinear systems. J Comput Appl Math 236:3937–3947

    MathSciNet  MATH  Google Scholar 

  154. Casulli V, Dumbser M, Toro EF (2012) Semi-implicit numerical modeling of axially symmetric flows in compliant arterial systems. Int J Numer Methods Biomed Eng 28:257–272

    MathSciNet  MATH  Google Scholar 

  155. Fambri F, Dumbser M, Casulli V (2014) An efficient semi-implicit method for three-dimensional non-hydrostatic flows in compliant arterial vessels. Int J Numer Methods Biomed Eng 30(11):1170–1198

    MathSciNet  Google Scholar 

  156. Dumbser M, Iben U, Ioriatti M (2015) An efficient semi-implicit finite volume method for axially symmetric compressible flows in compliant tubes. Appl Numer Math 89:24–44

    MathSciNet  MATH  Google Scholar 

  157. Dumbser M, Casulli V (2016) A conservative, weakly nonlinear semi-implicit finite volume scheme for the compressible Navier–Stokes equations with general equation of state. Appl Math Comput 272(Part 2):479–497

    MathSciNet  MATH  Google Scholar 

  158. Dumbser M, Casulli V (2013) A staggered semi-implicit spectral discontinuous Galerkin scheme for the shallow water equations. Appl Math Comput 219(15):8057–8077

    MathSciNet  MATH  Google Scholar 

  159. Tavelli M, Dumbser M (2014) A high order semi-implicit discontinuous Galerkin method for the two dimensional shallow water equations on staggered unstructured meshes. Appl Math Comput 234:623–644

    MathSciNet  MATH  Google Scholar 

  160. Tavelli M, Dumbser M (2014b) A staggered semi-implicit discontinuous Galerkin method for the two dimensional incompressible Navier–Stokes equations. Appl Math Comput 248:70–92

    MathSciNet  MATH  Google Scholar 

  161. Tavelli M, Dumbser M (2015) A staggered space–time discontinuous Galerkin method for the incompressible Navier–Stokes equations on two-dimensional triangular meshes. Comput Fluids 119:235–249

    MathSciNet  MATH  Google Scholar 

  162. Serra-Capizzano S (1998) Asymptotic results on the spectra of block Toeplitz preconditioned matrices. SIAM J Matrix Anal Appl 20(1):31–44

    MathSciNet  MATH  Google Scholar 

  163. Grenander U, Szegö G (1984) Toeplitz forms and their applications, vol 321, 2nd edn. Chelsea, New York

    MATH  Google Scholar 

  164. Serra-Capizzano S (2003) Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations. Linear Algebra Appl. 366:371–402

    MathSciNet  MATH  Google Scholar 

  165. Dolejsi V, Feistauer M (2004) A semi-implicit discontinuous Galerkin finite element method for the numerical solution of inviscid compressible flow. J Comput Phys 198(2):727–746

    MathSciNet  MATH  Google Scholar 

  166. Dolejsi V, Feistauer M, Hozman J (2007) Analysis of semi-implicit DGFEM for nonlinear convection–diffusion problems on nonconforming meshes. Comput Methods Appl Mech Eng 196(29–30):2813–2827

    MathSciNet  MATH  Google Scholar 

  167. Dolejsi V (2008) Semi-implicit interior penalty discontinuous Galerkin method for viscous compressible flows. Commun Comput Phys 4(2):231–274

    MathSciNet  MATH  Google Scholar 

  168. Giraldo FX, Restelli M (2010) High-order semi-implicit time-integrators for a triangular discontinuous Galerkin oceanic shallow water model. Int J Numer Methods Fluids 63(9):1077–1102

    MathSciNet  MATH  Google Scholar 

  169. Tumolo G, Bonaventura L, Restelli M (2013) A semi-implicit, semi-Lagrangian, p-adaptive discontinuous Galerkin method for the shallow water equations. J Comput Phys 232(1):46–67

    MathSciNet  MATH  Google Scholar 

  170. Chung ET, Lee CS (2012) A staggered discontinuous Galerkin method for the convection–diffusion equation. J Numer Math 20(1):1–32

    MathSciNet  MATH  Google Scholar 

  171. Cheung SW, Chung E, Kim HH, Qian Y (2015) Staggered discontinuous Galerkin methods for the incompressible Navier–Stokes equations. J Comput Phys 302:251–266

    MathSciNet  MATH  Google Scholar 

  172. Liu Y, Shu CW, Tadmor E, Zhang M (2007) Central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction. SIAM J Numer Anal 45(6):2442–2467. https://doi.org/10.1137/060666974

    Article  MathSciNet  MATH  Google Scholar 

  173. Liu C, Shu CW, Tdmor E, Zhang M (2008) L2 stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods. ESAIM: Math Model Numer Anal 42(04):593–607

    MATH  Google Scholar 

  174. Dumbser M, Balsara DS (2009) High-order unstructured one-step pnpm schemes for the viscous and resistive mhd equations. CMES 52(2):301–332

    MATH  Google Scholar 

  175. Rusanov VV (1961) Calculation of interaction of non-steady shock waves with obstacles. J Comput Math Phys USSR 1:267–279

    Google Scholar 

  176. Dal Maso G, LeFloch PG, Murat F (1995) Definition and weak stability of nonconservative products. Journal de mathmatiques pures et appliques 74:483–548

    MathSciNet  MATH  Google Scholar 

  177. Dumbser M, Castro M, Parés C, Toro EF (2009) ADER schemes on unstructured meshes for non-conservative hyperbolic systems: applications to geophysical flows. Comput Fluids 38:1731–1748

    MathSciNet  MATH  Google Scholar 

  178. Dumbser M, Toro EF (2011a) A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems. J Sci Comput 48:70–88

    MathSciNet  MATH  Google Scholar 

  179. Dumbser M, Balsara DS (2016) A new efficient formulation of the HLLEM riemann solver for general conservative and non-conservative hyperbolic systems. J Comput Phys 304:275–319

    MathSciNet  MATH  Google Scholar 

  180. Einfeldt B, Munz CD, Roe PL, Sjögreen B (1991) On godunov-type methods near low densities. J Comput Phys 92(2):273–295

    MathSciNet  MATH  Google Scholar 

  181. Lörcher F, Gassner G (2007) A discontinuous Galerkin scheme based on a space–time expansion. I. Inviscid compressible flow in one space dimension. J Sci Comput 32:175–199

    MathSciNet  MATH  Google Scholar 

  182. Sonntag M, Munz CD (2014) Shock capturing for discontinuous Galerkin methods using finite volume subcells. In: Fuhrmann J, Ohlberger M, Rohde C (eds) Finite volumes for complex applications VII. Springer, Berlin, pp 945–953

    MATH  Google Scholar 

  183. Casoni E, Peraire J, Huerta A (2013) One-dimensional shock-capturing for high-order discontinuous Galerkin methods. Int J Numer Methods Fluids 71(6):737–755

    MathSciNet  MATH  Google Scholar 

  184. Huerta A, Casoni E, Peraire J (2012) A simple shock-capturing technique for high-order discontinuous Galerkin methods. Int J Numer Methods Fluids 69(10):1614–1632

    MathSciNet  MATH  Google Scholar 

  185. Fechter S, Munz C-D (2015) A discontinuous Galerkin-based sharp-interface method to simulate three-dimensional compressible two-phase flow. Int J Numer Methods Fluids 78(7):413–435

    MathSciNet  Google Scholar 

  186. Meister A, Ortleb S (2016) A positivity preserving and well-balanced DG scheme using finite volume subcells in almost dry regions. Appl Math Comput 272:259–273 (in press)

    MathSciNet  MATH  Google Scholar 

  187. Zanotti O, Dumbser M (2016) Efficient conservative ader schemes based on weno reconstruction and space–time predictor in primitive variables. Comput Astrophys Cosmol 3(1):1. arXiv:1707.09910

  188. Berger MJ, Oliger J (1984) Adaptive mesh refinement for hyperbolic partial differential equations. J Comput Phys 53:484

    MathSciNet  MATH  Google Scholar 

  189. Berger MJ, Colella P (1989) Local adaptive mesh refinement for shock hydrodynamics. J Comput Phys 82:64–84

    MATH  Google Scholar 

  190. Baeza A, Mulet P (2006) Adaptive mesh refinement techniques for high-order shock capturing schemes for multi-dimensional hydrodynamic simulations. Int J Numer Methods Fluids 52:455–471

    MathSciNet  MATH  Google Scholar 

  191. Khokhlov AM (1998) Fully threaded tree algorithms for adaptive refinement fluid dynamics simulations. J Comput Phys 143(2):519–543

    MathSciNet  MATH  Google Scholar 

  192. Donat R, Mart MC, Martnez-Gavara A, Mulet P (2014) Well-balanced adaptive mesh refinement for shallow water flows. J Comput Phys 257, Part A(0):937 – 953

    MathSciNet  MATH  Google Scholar 

  193. Agbaglah G, Delaux S, Fuster D, Hoepffner J, Josserand C, Popinet S, Ray P, Scardovelli R, Zaleski S (2011) Parallel simulation of multiphase flows using octree adaptivity and the volume-of-fluid method. Compte-rendus de l’Acadmie des Sciences, Paris 339:194–207

    MATH  Google Scholar 

  194. Tsai C-C, Hou T-H, Popinet S, Chao Y-Y (2013) Prediction of waves generated by tropical cyclones with a quadtree-adaptive model. Coast Eng 77:108–119. https://doi.org/10.1016/j.coastaleng.2013.02.011

    Article  Google Scholar 

  195. Lucian Ivan, Groth Clinton PT (2009) High-order central eno finite-volume scheme with adaptive mesh refinement for the advection–diffusion equation. Comput Fluid Dyn 2008:443–449

    Google Scholar 

  196. Ivan Lucian, Groth Clinton PT (2014) High-order solution-adaptive central essentially non-oscillatory (ceno) method for viscous flows. J Comput Phys 257, Part A(0):830–862

    MathSciNet  MATH  Google Scholar 

  197. Godunov SK (1959) Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics. Math USSR Sbornik 47:271–306

    MATH  Google Scholar 

  198. Toro EF (2009) Riemann solvers and numerical methods for fluid dynamics, 3rd edn. Springer, Berlin

    MATH  Google Scholar 

  199. Harten A, Engquist B, Osher S, Chakravarthy S (1987) Uniformly high order essentially non-oscillatory schemes, III. J Comput Phys 71:231–303

    MathSciNet  MATH  Google Scholar 

  200. Jiang G-S, Shu CW (1996) Efficient implementation of weighted ENO schemes. J Comput Phys 126:202–228

    MathSciNet  MATH  Google Scholar 

  201. Balsara D, Shu CW (2000) Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J Comput Phys 160:405–452

    MathSciNet  MATH  Google Scholar 

  202. Titarev VA, Toro EF (2004) Finite-volume weno schemes for three-dimensional conservation laws. J Comput Phys 201:238–260

    MathSciNet  MATH  Google Scholar 

  203. Feistauer M, Kucera V, Prokopová J (2010) Discontinuous Galerkin solution of compressible flow in time-dependent domains. Math Comput Simul 80(8):1612–1623

    MathSciNet  MATH  Google Scholar 

  204. Dolejsi V, Feistauer M, Schwab C (2003) On some aspects of the discontinuous Galerkin finite element method for conservation laws. Math Comput Simul 61(3–6):333–346

    MathSciNet  MATH  Google Scholar 

  205. Feistauer M, Dolejsi V, Kucera V (2007) On the discontinuous Galerkin method for the simulation of compressible flow with wide range of mach numbers. Comput Vis Sci 10(1):17–27

    MathSciNet  Google Scholar 

  206. Qiu J, Shu CW (2005b) Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method II: two dimensional case. Comput Fluids 34:642–663

    MathSciNet  MATH  Google Scholar 

  207. Biswas R, Devine KD, Flaherty JE (1994) Parallel, adaptive finite element methods for conservation laws. Appl Numer Math 14:255–283

    MathSciNet  MATH  Google Scholar 

  208. Burbeau A, Sagaut P, Bruneau CH (2001) A problem-independent limiter for high-order Runge–Kutta discontinuous Galerkin methods. J Comput Phys 169:111–150

    MathSciNet  MATH  Google Scholar 

  209. Yang M, Wang Z (2009) A parameter-free generalized moment limiter for high-order methods on unstructured grids. In: 47th AIAA aerospace sciences meeting including the New Horizons Forum and Aerospace Exposition. Orlando, Florida. https://doi.org/10.2514/6.2009-605

  210. Löhner R (1987) An adaptive finite element scheme for transient problems in CFD. Comput Methods Appl Mech Eng 61:323–338

    MATH  Google Scholar 

  211. Shu CW (1997) Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic Conservation Laws. NASA/CR-97-206253 ICASE Report No. 97-65, November

  212. Dumbser M, Toro EF (2011) On universal Osher-type schemes for general nonlinear hyperbolic conservation laws. Commun Comput Phys 10:635–671

    MathSciNet  MATH  Google Scholar 

  213. Woodward P, Colella P (1984) The numerical simulation of two-dimensional fluid flow with strong shocks. J Comput Phys 54:115–173

    MathSciNet  MATH  Google Scholar 

  214. Kurganov A, Tadmor E (2002) Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. Numer Methods Partial Differ Equ 18:584–608

    MathSciNet  MATH  Google Scholar 

  215. Ghia U, Ghia KN, Shin CT (1982) High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. J Comput Phys 48(3):387–411

    MATH  Google Scholar 

  216. Balsara D, Spicer D (1999) A staggered mesh algorithm using high order godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. J Comput Phys 149:270–292

    MathSciNet  MATH  Google Scholar 

  217. Orszag SA, Tang CM (1979) Small-scale structure of two-dimensional magnetohydrodynamic turbulence. J Fluid Mech 90:129

    Google Scholar 

  218. Picone JM, Dahlburg RB (1991) Evolution of the Orszag–Tang vortex system in a compressible medium. II. Supersonic flow. Phys. Fluids B 3:29–44

    Google Scholar 

  219. Dahlburg RB, Picone JM (1989) Evolution of the Orszag–Tang vortex system in a compressible medium. I. Initial average subsonic flow. Phys Fluids B 1:2153–2171

    MathSciNet  Google Scholar 

  220. Jiang GS, Wu CC (1999) A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics. J Comput Phys 150:561–594

    MathSciNet  MATH  Google Scholar 

  221. Colonius T, Lele SK, Moin P (1997) Sound generation in a mixing layer. J Fluid Mech 330:375–409

    MATH  Google Scholar 

  222. Babucke A, Kloker M, Rist U (2008) DNS of a plane mixing layer for the investigation of sound generation mechanisms. Comput Fluids 37:360–368

    MATH  Google Scholar 

  223. Dumbser M, Peshkov I, Romenski E, Zanotti O (2016) High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids. J Comput Phys 314:824–862

    MathSciNet  MATH  Google Scholar 

  224. Landau LD, Lifshitz EM (2004) Fluid mechanics, course of theoretical physics, vol 6. Elsevier, Oxford

    Google Scholar 

  225. Dumbser M, Kaeser M, Titarev VA, Toro EF (2007) Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys 226:204–243

    MathSciNet  MATH  Google Scholar 

  226. Rault A, Chiavassa G, Donat R (2003) Shock–Vortex interactions at high mach numbers. J Sci Comput 19:347–371

    MathSciNet  MATH  Google Scholar 

  227. Woodward P, Colella P (1984) The numerical simulation of two-dimensional fluid flow with strong shocks. J Comput Phys 54:115–173

    MathSciNet  MATH  Google Scholar 

  228. Beckwith Kris, Stone James M (2011) A second-order godunov method for multi-dimensional relativistic magnetohydrodynamics. Astrophys J Suppl Ser 193(1):6

    Google Scholar 

  229. Biskamp D (1986) Magnetic reconnection via current sheets. Phys Fluids 29:1520–1531

    MATH  Google Scholar 

  230. Loureiro NF, Schekochihin AA, Cowley SC (2007) Instability of current sheets and formation of plasmoid chains. Phys Plasm 14(10):100703

    Google Scholar 

  231. Samtaney R, Loureiro NF, Uzdensky DA, Schekochihin AA, Cowley SC (2009) Formation of plasmoid chains in magnetic reconnection. Phys Rev Lett 103(10):105004

    Google Scholar 

  232. Landi S, Del Zanna L, Papini E, Pucci F, Velli M (2015) Resistive magnetohydrodynamics simulations of the ideal tearing mode. Astrophys J 806:131

    Google Scholar 

  233. Brachet ME, Meiron DI, Orszag SA, Nickel BG, Morf RH, Frisch U (1983) Small-scale structure of the Taylor–Green vortex. J Fluid Mech 130:411–452

    MATH  Google Scholar 

  234. Morf RH, Orszag SA, Frisch U (1980) Spontaneous singularity in three-dimensional inviscid, incompressible flow. Phys Rev Lett 44:572–575

    MathSciNet  MATH  Google Scholar 

  235. Thorne Kip S, Macdonald D (1982) Electrodynamics in curved spacetime: 3 + 1 formulation. Mon Notoices R Astron Soc 198:339–343

    MATH  Google Scholar 

  236. Rezzolla L, Zanotti O (2013) Relativistic hydrodynamics. Oxford University Press, Oxford

    MATH  Google Scholar 

  237. Dedner A, Kemm F, Kröner D, Munz C-D, Schnitzer T, Wesenberg M (2002) Hyperbolic divergence cleaning for the MHD equations. J Comput Phys 175:645–673

    MathSciNet  MATH  Google Scholar 

  238. Komissarov SS (1997) On the properties of Alfvén waves in relativistic magnetohydrodynamics. Phys Lett A 232:435–442

    MathSciNet  MATH  Google Scholar 

  239. Martí JM, Müller E (2003) Numerical hydrodynamics in special relativity. Living Rev Relativ 6:7. http://www.livingreviews.org/lrr--2003--7, http://www.livingreviews.org/lrr-2003-7

  240. Zhang W, MacFadyen A, Wang P (2009) Three-dimensional relativistic magnetohydrodynamic simulations of the Kelvin–Helmholtz instability: magnetic field amplification by a turbulent dynamo. Astrophys J 692:L40–L44

    Google Scholar 

  241. Zrake J, MacFadyen AI (2012) Numerical simulations of driven relativistic magnetohydrodynamic turbulence. Astrophys J 744(1):32

    Google Scholar 

  242. Biskamp D (2008) Magnetohydrodynamic turbulence. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  243. Michel FC (1972) Accretion of matter by condensed objects. Astrophys Space Sci 15:153

    Google Scholar 

  244. Bruno Giacomazzo, Luciano Rezzolla (2006) The exact solution of the Riemann problem in relativistic MHD. J Fluid Mech 562:223–259

    MathSciNet  MATH  Google Scholar 

  245. Balsara D (1998) Total variation diminishing scheme for adiabatic and isothermal magnetohydrodynamics. Astrophys J Suppl Ser 116:133–153

    Google Scholar 

  246. Balsara DS, Spicer D (1999) Maintaining pressure positivity in magnetohydrodynamic simulations. J Comput Phys 148:133–148

    MathSciNet  MATH  Google Scholar 

  247. Abramowicz M, Jaroszynski M, Sikora M (1978) Relativistic, accreting disks. Astron Astrophys 63:221–224

    MathSciNet  MATH  Google Scholar 

  248. Kozlowski M, Jaroszynski M, Abramowicz MA (1978) The analytic theory of fluid disks orbiting the Kerr black hole. Astron Astrophys 63:209–220

    MathSciNet  MATH  Google Scholar 

  249. Kutta W (1901) Beitrag zur näherungsweisen Integration totaler differentialgleichungen. Zeit Math Physik 46:435–452

    MATH  Google Scholar 

  250. Fehlberg E (1969) Klassische Runge–Kutta–Formeln fünfter und siebenter Ordnung mit Schrittweiten–Kontrolle. Computing 4(2):93–106

    MathSciNet  MATH  Google Scholar 

  251. Butcher JC (1964) On Runge–Kutta processes of high order. J Aust Math Soc 4(2):179–194

    MathSciNet  MATH  Google Scholar 

  252. Arakawa A, Lamb VR (1977) Computational design of the basic dynamical processes of the UCLA general circulation model. Methods Comput Phys 17:173–265

    Google Scholar 

  253. Bermudez A, Dervieux A, Desideri JA, Vázquez Cendón ME (1998) Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes. Comput Methods Appl Mech Eng 155:49–72

    MathSciNet  MATH  Google Scholar 

  254. Bermúdez A, Ferrín JL, Saavedra L, Vázquez Cendón ME (2014) A projection hybrid finite volume/element method for low-Mach number flows. J Comput Phys 271:360–378

    MathSciNet  MATH  Google Scholar 

  255. Toro EF, Hidalgo A, Dumbser M (2009) FORCE schemes on unstructured meshes I: conservative hyperbolic systems. J Comput Phys 228:3368–3389

    MathSciNet  MATH  Google Scholar 

  256. Hestenes MR, Stieffel E (1952) Methods of conjugate gradients for solving linear systems. J Res Natl Bur Stand 49:409–417

    MathSciNet  MATH  Google Scholar 

  257. Toro EF (2009) Riemann solvers and numerical methods for fluid dynamics. Springer, Berlin

    MATH  Google Scholar 

  258. Saad Y, Schultz MH (1986) GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7(3):856–869

    MATH  Google Scholar 

  259. Karniadakis GE, Israeli M, Orszag SA (1991) High-order splitting methods for the incompressible Navier–Stokes equations. J Comput Phys 97(2):414–443

    MathSciNet  MATH  Google Scholar 

  260. Zang TA, Hussaini MY (1986) On spectral multigrid methods for the time-dependent Navier–Stokes equations. Appl Math Comput 19(1–4):359–372

    MathSciNet  MATH  Google Scholar 

  261. Kim J, Moin P (1985) Application of a fractional-step method to incompressible Navier–Stokes equations. J Comput Phys 59(2):308–323

    MathSciNet  MATH  Google Scholar 

  262. Marcus PS (1984) Simulation of Taylor–Couette flow. Part 1. Numerical methods and comparison with experiment. J Fluid Mech 146:45–64. http://journals.cambridge.org/article_S0022112084001762

    MATH  Google Scholar 

  263. Layton AT (2008) On the choice of correctors for semi-implicit Picard deferred correction methods. Appl Numer Math 58(6):845–858

    MathSciNet  MATH  Google Scholar 

  264. Minion ML (2003) Semi-implicit spectral deferred correction methods for ordinary differential equations. Commun Math Sci 1(3):471–500

    MathSciNet  MATH  Google Scholar 

  265. Minion ML (2003) Higher-order semi-implicit projection methods. In: Hafez M (ed) Numerical simulations of incompressible flows: proceedings of a conference held at Half Moon Bay, CA, June 18–20, 2001

  266. 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 72:607–631

    MathSciNet  Google Scholar 

  267. Casulli V (1999) A semi-implicit finite difference method for non-hydrostatic, free-surface flows. Int J Numer Methods Fluids 30:425–440

    MATH  Google Scholar 

  268. Casulli V, Cheng RT (1992) Semi-implicit finite difference methods for three-dimensional shallow water flow. Int J Numer Methods Fluids 15:629–648

    MATH  Google Scholar 

  269. Casulli V (1990) Semi-implicit finite difference methods for the two-dimensional shallow water equations. J Comput Phys 86:56–74

    MathSciNet  MATH  Google Scholar 

  270. Casulli V, Zanolli P (2002) Semi-implicit numerical modeling of nonhydrostatic free-surface flows for environmental problems. Math Comput Model 36:1131–1149

    MathSciNet  MATH  Google Scholar 

  271. Casulli V, Stelling GS (2011) Semi-implicit subgrid modelling of three-dimensional free-surface flows. Int J Numer Methods Fluids 67:441–449

    MathSciNet  MATH  Google Scholar 

  272. Kurzweg UH (1985) Enhanced heat conduction in oscillating viscous flows within parallel-plate channels. J Fluid Mech 156:291–300. http://journals.cambridge.org/article_S0022112085002105

    MATH  Google Scholar 

  273. Loudon C, Tordesillas A (1998) The use of the dimensionless womersley number to characterize the unsteady nature of internal flow. J Theor Biol 191(1):63–78

    Google Scholar 

  274. Womersley JR (1955) Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J Physiol 127(3):553–563

    Google Scholar 

  275. Arnold VI (1965) Sur la topologic des écoulements stationnaires des fluides parfaits. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 261:17–20

    MathSciNet  Google Scholar 

  276. Childress S (1970) New solutions of the kinematic dynamo problem. J Math Phys 11:3063–3076

    MathSciNet  Google Scholar 

  277. Prandtl L (1904) Über Flüssigkeitsbewegung bei sehr kleiner Reibung. Verhandlg III Int Math Kongr. Heidelberg, pp 484–491

  278. Schlichting H, Gersten K (2005) Grenzschichttheorie. Springer, Berlin

    Google Scholar 

  279. Lee T, Mateescu D (1998) Experimental and numerical investigation of 2-d backward-facing step flow. J Fluids Struct 12(6):703–716

    Google Scholar 

  280. Erturk E (2008) Numerical solutions of 2-d steady incompressible flow over a backward-facing step, part I: high reynolds number solutions. Comput Fluids 37(6):633–655

    MATH  Google Scholar 

  281. Tylli N, Kaiktsis L, Ineichen B (2002) Sidewall effects in flow over a backward-facing step: experiments and numerical simulations. Phys Fluids 14(11):3835–3845

    MATH  Google Scholar 

  282. Armaly BF, Durst F, Pereira JCF, Schönung B (1983) Experimental and theoretical investigation of backward-facing step flow. J Fluid Mech 127:473–496. http://journals.cambridge.org/article_S0022112083002839

    Google Scholar 

  283. Mouza AA, Pantzali MN, Paras SV, Tihon J (2005) Experimental and numerical study of backward-facing step flow. In: 5th national chemical engineering conference, Thessaloniki, Greece

  284. Rani HP, Sheu Tony WH, Tsai Eric SF (2007) Eddy structures in a transitional backward-facing step flow. J Fluid Mech 588:43–58. http://journals.cambridge.org/article_S002211200700763X

    MATH  Google Scholar 

  285. Brown DL, Minion ML (1995) Performance of under-resolved two-dimensional incompressible flow simulations. J Comput Phys 122(1):165–183

    MathSciNet  MATH  Google Scholar 

  286. Minion ML, Brown DL (1997) Performance of under-resolved two-dimensional incompressible flow simulations, II. J Comput Phys 138(2):734–765

    MATH  Google Scholar 

  287. Ku HC, Hirsh RS, Taylor TD (1987) A pseudospectral method for solution of the three-dimensional incompressible Navier–Stokes equations. J Comput Phys 70(2):439–462

    MATH  Google Scholar 

  288. Albensoeder S, Kuhlmann HC (2005) Accurate three-dimensional lid-driven cavity flow. J Comput Phys 206(2):536–558

    MATH  Google Scholar 

  289. Kida S, Takaoka M, Hussain F (1991) Collision of two vortex rings. J Fluid Mech 230:583–646

    MathSciNet  MATH  Google Scholar 

  290. Kida S, Takaoka M (1994) Vortex reconnection. Annu Rev Fluid Mech 26:169–189

    MathSciNet  MATH  Google Scholar 

  291. Ghosh D, Baeder JD (2012) High-order accurate incompressible Navier–Stokes algorithm for vortex-ring interactions with solid wall. AIAA J 50(2):2408–2433

    Google Scholar 

  292. Hahn S, Iaccarino G (2009) Towards adaptive vorticity confinement. In: 47th AIAA aerospace sciences meeting, Orlando, FL, AIAA paper 2009–1613

  293. Stanaway SK, Cantwell B, Spalart PR (1988) Ames Research Center. A numerical study of viscous vortex rings using a spectral method [microform]. National Aeronautics and Space Administration, Ames Research Center

  294. Cheng M, Lou J, Lim TT (2015) Leapfrogging of multiple coaxial viscous vortex rings. Phys Fluids 27(3):031702

    Google Scholar 

  295. Levy D, Puppo G, Russo G (1999) Central weno schemes for hyperbolic systems of conservation laws. ESAIM: Math Model Numer Anal 33(3):547–571

    MathSciNet  MATH  Google Scholar 

  296. Puppo G, Semplice M (2011) Numerical entropy and adaptivity for finite volume schemes. Commun Comput Phys 10(5):1132–1160

    MathSciNet  MATH  Google Scholar 

  297. 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

    MathSciNet  MATH  Google Scholar 

  298. Cravero I, Semplice M (2016) On the accuracy of weno and cweno reconstructions of third order on nonuniform meshes. J Sci Comput 67(3):1219–1246

    MathSciNet  MATH  Google Scholar 

  299. Peshkov I, Romenski E (2016) A hyperbolic model for viscous Newtonian flows. Contin Mech Thermodyn 28:85–104

    MathSciNet  MATH  Google Scholar 

  300. Dumbser M, Peshkov I, Romenski E, Zanotti O (2017) High order ADER schemes for a unified first order hyperbolic formulation of Newtonian continuum mechanics coupled with electro-dynamics. J Comput Phys 348:298–342

    MathSciNet  MATH  Google Scholar 

  301. Godunov SK, Romenski EI (1972) Nonstationary equations of the nonlinear theory of elasticity in Euler coordinates. J Appl Mech Tech Phys 13:868–885

    Google Scholar 

  302. Godunov SK (1961) An interesting class of quasilinear systems. Dokl Akad Nauk SSSR 139(3):521–523

    MathSciNet  MATH  Google Scholar 

  303. Godunov SK (1972) Symmetric form of the magnetohydrodynamic equation. Numer Methods Mech Contin Medium 3(1):26–34. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.55.9645&rep=rep1&type=pdf

  304. Romenski EI (1998) Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics. Math comput Model 28(10):115–130

    MathSciNet  Google Scholar 

  305. Godunov SK, Romenski EI (2003) Elements of continuum mechanics and conservation laws. Kluwer Academic, New York

    Google Scholar 

  306. Tavelli M, Dumbser M (2018) Arbitrary high order accurate spacetime discontinuous galerkin finite element schemes on staggered unstructured meshes for linear elasticity. J Comput Phys 366:386–414. https://doi.org/10.1016/j.jcp.2018.03.038

    MathSciNet  MATH  Google Scholar 

  307. Ioriatti M, Dumbser M (2018) A posteriori sub-cell finite volume limiting of staggered semi-implicit discontinuous galerkin schemes for the shallow water equations. Appl Numer Math. https://doi.org/10.1016/j.apnum.2018.08.018

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work has been carried out under the supervision of Prof. Dr.-Ing Michael Dumbser, who is warmly thanked by the author, within the three-years Doctoral Programme in Civil, Environmental and Mechanical Engineering at DICAM Dept. of University of Trento. The author would like to thank also prof. Arturo Hidalgo, prof. Olindo Zanotti, prof. Vincenzo Casulli, prof. Francesco Serra-Capizzano and prof. Luciano Rezzolla for the fruitful collaboration and the valuable discussions. The presented research has been financed mainly 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, and by the European Unions Horizon 2020 Research and Innovation Programme under the project ExaHyPE, Grant No. 671698 (call FETHPC-1-2014). The simulations were performed on the SuperMUC supercomputer at the LRZ in Garching, Germany, on the LOEWE cluster in CSC in Frankfurt, on the HazelHen supercomputer at the HLRS in Stuttgart, Germany, as well as on the local HPC cluster at the University of Trento. Several numerical simulations have been also tested on the MARCONI supercomputer at at the CINECA HPC center, Italy, by means of the class C project ’THOAst’—Towards High Order methods for Astrophysics—(no. IsC58, ISCRA call). We acknowledge the CINECA award under the ISCRA initiative, for the availability of high performance computing resources and support. In particular, the author is grateful for the technical support provided by Dr. Björn Dick (HLRS) for the strong scaling study performed on the Hazel-Hen supercomputer. Moreover, some tests concerning of ADERDG schemes have been performed also on the ’Swan’ CRAY XC50 system, Access to the Cray XC40 supercomputer ’Swan’ was kindly provided through Cray Inc.’s Marketing Partner Network by means of Dr. Stefan Andersson (CRAY). The author is members of the Italian INdAM Research group GNCS.

Author information

Authors and Affiliations

  1. Laboratory of Applied Mathematics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano 77, 38123, Trento, Italy

    Francesco Fambri

Authors
  1. Francesco Fambri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesco Fambri.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fambri, F. Discontinuous Galerkin Methods for Compressible and Incompressible Flows on Space–Time Adaptive Meshes: Toward a Novel Family of Efficient Numerical Methods for Fluid Dynamics. Arch Computat Methods Eng 27, 199–283 (2020). https://doi.org/10.1007/s11831-018-09308-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11831-018-09308-6

Search

Navigation