Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

A Staggered Semi-implicit Discontinuous Galerkin Scheme with a Posteriori Subcell Finite Volume Limiter for the Euler Equations of Gasdynamics

  • Review
  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper we propose a novel semi-implicit Discontinuous Galerkin (DG) finite element scheme on staggered meshes with a posteriori subcell finite volume limiting for the one and two dimensional Euler equations of compressible gasdynamics. We therefore extend the strategy adopted by Dumbser and Casulli (Appl Math Comput 272:479–497, 2016), where the Euler equations have been solved solved using a semi-implicit finite volume scheme based on the flux-vector splitting method recently proposed by Toro and Vázquez-Cendón (Comput Fluids 70:1–12, 2012). In our scheme, the nonlinear convective terms are discretized explicitly, while the pressure terms are discretized implicitly. As a consequence, the time step is restricted only by a mild CFL condition based on the fluid velocity, which makes this method particularly suitable for simulations in the low Mach number regime. However, the conservative formulation of the scheme, together with the novel subcell finite volume limiter allows also the numerical simulation of high Mach number flows with shock waves. Inserting the discrete momentum equation into the discrete total energy conservation law yields a mildly nonlinear system with the scalar pressure as the only unknown. Due to the use of staggered meshes, the resulting pressure system has the most compact stencil possible and can be efficiently solved with modern iterative methods. In order to deal with shock waves or steep gradients, the new semi-implicit DG scheme proposed in this paper includes an a posteriori subcell finite volume limiting technique. This strategy was first proposed by Dumbser et al. (J Comput Phys 278:47–75, 2014) for explicit DG schemes on collocated grids and is based on the a posteriori MOOD algorithm of Clain, Loubère and Diot. Recently, this methodology was also extended to semi-implicit DG schemes on staggered meshes for the shallow water equations in Ioriatti and Dumbser (Appl Numer Math 135:443–480, 2019). Within the MOOD approach, an unlimited DG scheme first produces a so-called candidate solution for the next time level \(t^{n+1}\). Later on, the control volumes with a non-admissible candidate solution are identified by using physical and numerical detection criteria, such as the positivity of the solution, the absence of floating point errors and the satisfaction of a relaxed discrete maximum principle (DMP). Then, in the detected troubled cells a more robust first order semi-implicit finite volume (FV) method is applied on a sub-grid composed of \(2P + 1\) subcells, where P denotes the polynomial degree used in the DG scheme. For that purpose, the nonlinear convective terms are recomputed in the troubled cells using an explicit finite volume scheme on the subcell level. Also the linear system for the pressure needs to be assembled and solved again, but where now a low order semi-implicit finite volume scheme is used on the sub-cell level in all troubled DG elements, instead of the original high order DG method. Finally, the higher order DG polynomials are reconstructed from the piecewise constant subcell finite volume averages and the scheme proceeds to the next time step. In this paper we present, discuss and test this novel family of methods and simulate a set of classical numerical benchmark problems of compressible gasdynamics. Great attention is dedicated to 1D and 2D Riemann problems and we also show that for these test cases the scheme responds appropriately in the presence of shock waves and does not produce non-physical spurious numerical oscillations.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Similar content being viewed by others

Notes

  1. When the index i is omitted in the vector of degrees of freedom we intend the entire set of all degrees of freedom of all elements.

References

  1. Bassi, F., Crivellini, A., Di Pietro, D.A., Rebay, S.: An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier–Stokes equations. J. Comput. Phys. 218, 208–221 (2006)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. Baumann, C., Oden, J.: A discontinuous hp finite element method for convection–diffusion problems. Comput. Methods Appl. Mech. Eng. 175, 311–341 (1999)

    MathSciNet  MATH  Google Scholar 

  5. Baumann, C., Oden, J.: A discontinuous hp finite element method for the Euler and Navier–Stokes equation. Int. J. Numer. Methods Fluids 31, 79–95 (1999)

    MathSciNet  MATH  Google Scholar 

  6. Bell, J.B., Colella, P., Glaz, H.M.: A second-order projection method for the incompressible Navier–Stokes equations. J. Comput. Phys. 85(2), 257–283 (1989)

    MathSciNet  MATH  Google Scholar 

  7. Boscheri, W., Dumbser, M.: Arbitrary-Lagrangian–Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes. J. Comput. Phys. 346, 449–479 (2017)

    MathSciNet  MATH  Google Scholar 

  8. Boscheri, W., Dumbser, M., Loubère, R.: Cell centered direct arbitrary-Lagrangian–Eulerian ADER-WENO finite volume schemes for nonlinear hyperelasticity. Comput. Fluids 134–135, 111–129 (2016)

    MathSciNet  MATH  Google Scholar 

  9. Busto, S., Chiocchetti, S., Dumbser, M., Gaburro, E., Peshkov, I.: High order ADER schemes for continuum mechanics. Front. Phys. arXiv:1912.01964

  10. Busto, S., Ferrín, J.L., Toro, E.F., Vázquez-Cendón, M.E.: A projection hybrid high order finite volume/finite element method for incompressible turbulent flows. J. Comput. Phys. 353, 169–192 (2018)

    MathSciNet  MATH  Google Scholar 

  11. Busto, S., Toro, Eleuterio F., Elena Vázquez-Cendón, M.: Design and analysis of ADER-type schemes for model advection–diffusion–reaction equations. J. Comput. Phys. 327, 553–575 (2016)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  15. Casulli, V., Cattani, E.: Stability, accuracy and efficiency of a semi-implicit method for three-dimensional shallow water flow. Comput. Math. Appl. 27(4), 99–112 (1994)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  19. Casulli, V., Stelling, S.: Semi-implicit subgrid modelling of three-dimensional free-surface flows. Int. J. Numer. Methods Fluids 67, 441–449 (2010)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  21. Casulli, V., Zanolli, P.: A nested Newton-type algorithm for finite volume methods solving Richards’ equation in mixed form. SIAM J. Sci. Comput. 32, 2255–2273 (2009)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  24. Cockburn, B., Lin, S.Y., Shu, C.W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J. Comput. Phys. 84, 90–113 (1989)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  27. Cockburn, B., Shu, C.W.: The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws. Math. Modell. Numer. Anal. 25, 337–361 (1991)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  29. Diot, S., Clain, S., Loubère, R.: Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials. Comput. Fluids 64(Supplement C), 43–63 (2012)

    MathSciNet  MATH  Google Scholar 

  30. Diot, S., Loubère, R., Clain, S.: The multidimensional optimal order detection method in the three-dimensional case: very high-order finite volume method for hyperbolic systems. Int. J. Numer. Methods Fluids 73(4), 362–392 (2013)

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  32. Dumbser, M.: Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier–Stokes equations. Comput. Fluids 39(1), 60–76 (2010)

    MathSciNet  MATH  Google Scholar 

  33. Dumbser, M., Balsara, D.S., Tavelli, M., Fambri, F.: A divergence-free semi-implicit finite volume scheme for ideal, viscous and resistive magnetohydrodynamics. Int. J. Numer. Methods Fluids 89, 16–42 (2018)

    MathSciNet  Google Scholar 

  34. Dumbser, M., Balsara, D.S., Toro, E.F., Munz, C.D.: A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys. 227(18), 8209–8253 (2008)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  38. Dumbser, M., Käser, M.: Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J. Comput. Phys. 221(2), 693–723 (2007)

    MathSciNet  MATH  Google Scholar 

  39. Dumbser, M., Loubère, R.: A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes. J. Comput. Phys. 319(Supplement C), 163–199 (2016)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  41. Dumbser, M., Peshkov, I., Romenski, E., Zanotti, O.: 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 (2016)

    MathSciNet  MATH  Google Scholar 

  42. Dumbser, M., Zanotti, O., Loubère, R., Diot, S.: A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws. J. Comput. Phys. 278, 47–75 (2014)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  44. Fambri, F., Dumbser, M.: 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 (2016)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  47. Gassner, G., Lörcher, F., Munz, C.-D.: A discontinuous Galerkin scheme based on a space–time expansion II. Viscous flow equations in multi dimensions. J. Sci. Comput. 34(3), 260–286 (2008)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  49. Godunov, S.K.: Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics. Math. USSR Sb. 47, 271–306 (1959)

    MATH  Google Scholar 

  50. Gottlieb, S., Shu, C.W.: Total variation diminishing Runge-Kutta schemes. Math. Comput. 67(221), 73–85 (1998)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  53. Hu, C., Shu, C.W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150, 97–127 (1999)

    MathSciNet  MATH  Google Scholar 

  54. Ioriatti, M., Dumbser, M.: Semi-implicit staggered discontinuous Galerkin schemes for axially symmetric viscous compressible flows in elastic tubes. Comput. Fluids 167, 166–179 (2018)

    MathSciNet  MATH  Google Scholar 

  55. Ioriatti, M., Dumbser, M.: A posteriori sub-cell finite volume limiting of staggered semi-implicit discontinuous Galerkin schemes for the shallow water equations. Appl. Numer. Math. 135, 443–480 (2019)

    MathSciNet  MATH  Google Scholar 

  56. Ioriatti, M., Dumbser, M., Iben, U.: A comparison of explicit and semi-implicit finite volume schemes for viscous compressible flows in elastic pipes in fast transient regime. Zeitschrift fuer Angewandte Mathematik und Mechanik 97, 1358–1380 (2017)

    MathSciNet  Google Scholar 

  57. Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)

    MathSciNet  MATH  Google Scholar 

  58. Käser, M., Iske, A.: ADER schemes on adaptive triangular meshes for scalar conservation laws. J. Comput. Phys. 205(2), 486–508 (2005)

    MathSciNet  MATH  Google Scholar 

  59. Klaij, C.M., van der Vegt, J.J.W., van der Ven, H.: Space-time discontinuous Galerkin method for the compressible Navier–Stokes equations. J. Comput. Phys. 217(2), 589–611 (2006)

    MathSciNet  MATH  Google Scholar 

  60. Kramer, S.C., Stelling, G.S.: A conservative unstructured scheme for rapidly varied flows. Int. J. Numer. Methods Fluids 58, 183–212 (2008)

    MathSciNet  MATH  Google Scholar 

  61. Kurganov, A., Tadmor, E.: Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. Numer. Methods Partial Differ. Equ. 18(5), 584–608 (2002)

    MathSciNet  MATH  Google Scholar 

  62. Lax, Peter, Wendroff, Burton: Systems of conservation laws. Commun. Pure Appl. Math. 13(2), 217–237 (1960)

    MATH  Google Scholar 

  63. Liu, Y.J., Shu, C.W., Tadmor, E., Zhang, M.: Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction. SIAM J. Numer. Anal. 45, 2442–2467 (2007)

    MathSciNet  MATH  Google Scholar 

  64. Liu, Y.J., Shu, C.W., Tadmor, E., Zhang, M.: L2-stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods. Math. Model. Numer. Anal. 42, 593–607 (2008)

    MathSciNet  MATH  Google Scholar 

  65. Loubère, R., Dumbser, M., Diot, S.: A new family of high order unstructured MOOD and ADER finite volume schemes for multidimensional systems of hyperbolic conservation laws. Commun. Comput. Phys. 16(3), 718–763 (2014)

    MathSciNet  MATH  Google Scholar 

  66. Neumann, J., Richtmyer, R.D.: A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys. 21, 232–237 (1950)

    MathSciNet  MATH  Google Scholar 

  67. Park, J.H., Munz, C.D.: Multiple pressure variables methods for fluid flow at all Mach numbers. Int. J. Numer. Methods Fluids 49, 905–931 (2005)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  70. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Technical report LA-UR-73-479, Los Alamos Scientific Laboratory (1973)

  71. Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372 (1981)

    MathSciNet  MATH  Google Scholar 

  72. Schulz-Rinne, C.: Classification of the Riemann problem for two-dimensional gas dynamics. SIAM J. Math. Anal. 24(1), 76–88 (1993)

    MathSciNet  MATH  Google Scholar 

  73. Sod, Gary A.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27(1), 1–31 (1978)

    MathSciNet  MATH  Google Scholar 

  74. Stelling, G.S., Duynmeyer, S.P.A.: A staggered conservative scheme for every Froude number in rapidly varied shallow water flows. Int. J. Numer. Methods Fluids 43, 1329–1354 (2003)

    MathSciNet  MATH  Google Scholar 

  75. Tavelli, M., Dumbser, M.: 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 (2014)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  78. Tavelli, M., Dumbser, M.: 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 (2016)

    MathSciNet  MATH  Google Scholar 

  79. Tavelli, M., Dumbser, M.: A pressure-based semi-implicit space–time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier–Stokes equations at all Mach numbers. J. Comput. Phys. 341, 341–376 (2017)

    MathSciNet  MATH  Google Scholar 

  80. Tavelli, M., Dumbser, M.: Arbitrary high order accurate space–time discontinuous Galerkin finite element schemes on staggered unstructured meshes for linear elasticity. J. Comput. Phys. 366, 386–414 (2018)

    MathSciNet  MATH  Google Scholar 

  81. Tavelli, M., Dumbser, M., Casulli, V.: High resolution methods for scalar transport problems in compliant systems of arteries. Appl. Numer. Math. 74, 62–82 (2013)

    MathSciNet  MATH  Google Scholar 

  82. Toro, E.F.: Riemann Sovlers and Numerical Methods for Fluid Dynamics. Springer, Berlin (2009)

    Google Scholar 

  83. Toro, E.F., Vázquez-Cendón, M.E.: Flux splitting schemes for the Euler equations. Comput. Fluids 70, 1–12 (2012)

    MathSciNet  MATH  Google Scholar 

  84. Tumolo, G., Bonaventura, L.: A semi-implicit, semi-Lagrangian discontinuous Galerkin framework for adaptive numerical weather prediction. Q. J. R. Meteorol. Soc. 141(692), 2582–2601 (2015)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  86. van der Vegt, J.J.W., van der Ven, H.: Space–time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows: I. General formulation. J. Comput. Phys. 182(2), 546–585 (2002)

    MathSciNet  MATH  Google Scholar 

  87. van der Ven, H., van der Vegt, J.J.W.: 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(41), 4747–4780 (2002)

    MathSciNet  MATH  Google Scholar 

  88. van Leer, B.: Toward the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136 (1979)

    MATH  Google Scholar 

  89. Zanotti, O., Dumbser, M., Fambri, F.: Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement. Mon. Not. R. Astron. Soc. 452, 3010–3029 (2015)

    Google Scholar 

  90. Zanotti, O., Fambri, F., Dumbser, M., Hidalgo, A.: Space–time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting. Comput. Fluids 118, 204–224 (2015)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors acknowledge funding from the Istituto Nazionale di Alta Matematica (INdAM) via the GNCS group and the program Young Researchers Funding 2018 via the research project Semi-implicit structure preserving schemes for continuum mechanics. MD acknowledges the financial support received from the Italian Ministry of Education, University and Research (MIUR) in the frame of the Departments of Excellence Initiative 2018–2022 attributed to DICAM of the University of Trento (Grant L. 232/2016) and in the frame of the PRIN 2017 Project Innovative numerical methods for evolutionary partial differential equations and applications. MD has also received funding from the University of Trento via the Strategic Initiative Modeling and Simulation.

Author information

Authors and Affiliations

  1. Department of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123, Trento, Italy

    Matteo Ioriatti & Michael Dumbser

  2. Institut de Mathèmatiques de Bordeaux (IMB), 33405, Talence, France

    Raphaël Loubère

Authors
  1. Matteo Ioriatti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Dumbser
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Raphaël Loubère
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matteo Ioriatti.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Matrices and Tensors for the 1D Semi-implicit Schemes

Appendix A: Matrices and Tensors for the 1D Semi-implicit Schemes

1.1 Appendix A.1: Matrices and Tensors for the 1D Semi-implicit DG Method

$$\begin{aligned} \mathbf {M}= & {} \int \limits _0^1{\varvec{\varphi }}(\xi )\varvec{\varphi }(\xi )d\xi \end{aligned}$$
(A.1)
$$\begin{aligned} \mathbf {K}= & {} \int \limits _0^1{\varvec{\varphi }}'(\xi )\varvec{\varphi }(\xi )d\xi \end{aligned}$$
(A.2)
$$\begin{aligned} \mathbf {R}_{\mathbf {e}}^{\mathbf {DG}}= & {} \varvec{\varphi }(1)\varvec{\varphi }\left( \frac{1}{2}\right) \varvec{\varphi }\left( \frac{1}{2}\right) - \int \limits _{\frac{1}{2}}^{1}\varvec{\varphi }'(\xi ) \varvec{\varphi }\left( \xi -\frac{1}{2}\right) \varvec{\varphi }\left( \xi -\frac{1}{2}\right) d \xi \end{aligned}$$
(A.3)
$$\begin{aligned} \mathbf {L}_{\mathbf {e}}^{\mathbf {DG}}= & {} \varvec{\varphi }(0)\varvec{\varphi }\left( \frac{1}{2}\right) \varvec{\varphi }\left( \frac{1}{2}\right) + \int \limits _{0}^{\frac{1}{2}} \varvec{\varphi }'(\xi )\varvec{\varphi }\left( \xi +\frac{1}{2}\right) \varvec{\varphi }\left( \xi +\frac{1}{2}\right) d \xi \end{aligned}$$
(A.4)
$$\begin{aligned} \mathbf {R}_{\mathbf {p}}^{\mathbf {DG}}= & {} \varvec{\varphi }\left( \frac{1}{2}\right) \varvec{\varphi }(0) + \int \limits _{\frac{1}{2}}^{1} \varvec{\varphi }(\xi )\varvec{\varphi }'\left( \xi -\frac{1}{2}\right) d \xi \end{aligned}$$
(A.5)
$$\begin{aligned} \mathbf {L}_{\mathbf {p}}^{\mathbf {DG}}= & {} \varvec{\varphi }\left( \frac{1}{2}\right) \varvec{\varphi }(1) - \int \limits _{0}^{\frac{1}{2}} \varvec{\varphi }(\xi )\varvec{\varphi }'\left( \xi +\frac{1}{2}\right) d \xi \end{aligned}$$
(A.6)
$$\begin{aligned} \mathbf {M}_{\mathbf {L}}^{\mathbf {DG}}= & {} \int \limits _{0}^{\frac{1}{2}} \varvec{\varphi }(\xi )\varvec{\varphi }\left( \xi +\frac{1}{2}\right) d \xi \end{aligned}$$
(A.7)
$$\begin{aligned} \mathbf {M}_{\mathbf {R}}^{\mathbf {DG}}= & {} \int \limits _{\frac{1}{2}}^1 \varvec{\varphi }(\xi )\varvec{\varphi }\left( \xi -\frac{1}{2}\right) d \xi \end{aligned}$$
(A.8)

1.2 Appendix A.2: Matrices and Tensors for the 1D Semi-implicit Sub-cell FV Method for \(\hbox {P}=2\)

$$\begin{aligned} \mathbf {R}^{\mathbf {FV}}= & {} \begin{pmatrix} 0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0\\ 5&{}0&{}0&{}0&{}0\\ -5&{}5&{}0&{}0&{}0\\ 0&{}-5&{}5&{}0&{}0\\ \end{pmatrix} \quad \mathbf {L}^{\mathbf {FV}}=\begin{pmatrix} 0&{}0&{}5&{}-5&{}0\\ 0&{}0&{}0&{}5&{}-5\\ 0&{}0&{}0&{}0&{}5\\ 0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0\\ \end{pmatrix} \end{aligned}$$
(A.9)
$$\begin{aligned} \mathbf {M}_{\mathbf {R}}^{\mathbf {FV}}= & {} \begin{pmatrix} 0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0\\ 0.5&{}0&{}0&{}0&{}0\\ 0.5&{}0.5&{}0&{}0&{}0\\ 0&{}0.5&{}0.5&{}0&{}0\\ \end{pmatrix} \quad \mathbf {M}_{\mathbf {L}}^{\mathbf {FV}}=\begin{pmatrix} 0&{}0&{}0.5&{}0.5&{}0\\ 0&{}0&{}0&{}0.5&{}0.5\\ 0&{}0&{}0&{}0&{}0.5\\ 0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0\\ \end{pmatrix} \end{aligned}$$
(A.10)

1.3 Appendix A.3: Tensors for the Limited 1D Semi-implicit DG Method

$$\begin{aligned} \mathbf {R}^{\prime }{_{\mathbf {e}}^{\mathbf {DG}}}= & {} \varvec{\varphi }(1)\varvec{\varphi }\left( \frac{1}{2}\right) - \int \limits _{\frac{1}{2}}^{1}\varvec{\varphi }'(\xi ) \varvec{\varphi }\left( \xi -\frac{1}{2}\right) d \xi \end{aligned}$$
(A.11)
$$\begin{aligned} \mathbf {L}^{\prime }{_{\mathbf {e}}^{\mathbf {DG}}}= & {} \varvec{\varphi }(0)\varvec{\varphi }\left( \frac{1}{2}\right) + \int \limits _{0}^{\frac{1}{2}} \varvec{\varphi }'(\xi )\varvec{\varphi }\left( \xi +\frac{1}{2}\right) d \xi \end{aligned}$$
(A.12)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ioriatti, M., Dumbser, M. & Loubère, R. A Staggered Semi-implicit Discontinuous Galerkin Scheme with a Posteriori Subcell Finite Volume Limiter for the Euler Equations of Gasdynamics. J Sci Comput 83, 27 (2020). https://doi.org/10.1007/s10915-020-01209-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01209-w

Keywords