Skip to main content
SpringerLink
Log in

New Multi-implicit Space–Time Spectral Element Methods for Advection–Diffusion–Reaction Problems

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

Abstract

Novel multi-implicit space–time spectral element methods are described for approximating solutions to advection–diffusion–reaction problems characterized by multiple time scales. The new methods are spectrally accurate in space and time and they are designed to be easy to implement and robust. In other words, given an existing stable low order operator split method for approximating solutions to PDEs exhibiting multiple scales, the algorithms described in this article enable one to easily extend a low order method to be a robust space–time spectrally accurate method. In space, two spectrally accurate advective flux reconstructions are proposed: extended element-wise flux reconstruction and non-extended element-wise flux reconstruction. In time, for the hyperbolic term(s), a low-order explicit I-stable building block time integration scheme is introduced in order to obtain a stable and efficient building block for the spectrally accurate space–time scheme. In this article, multiple spectrally accurate space discretization strategies, and multiple spectrally accurate time discretization strategies are compared to one another. It is found that all methods described are spectrally accurate with each method having distinguishing properties.

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

Similar content being viewed by others

References

  1. Abbassi, H., Mashayek, F., Jacobs, G.B.: Shock capturing with entropy-based artificial viscosity for staggered grid discontinuous spectral element method. Comput. Fluids 98, 152–163 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Almgren, A.S., Aspden, A.J., Bell, J.B., Minion, M.L.: On the use of higher-order projection methods for incompressible turbulent flow. SIAM J. Sci. Comput. 35(1), B25–B42 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bao, W., Jin, S.: Weakly compressible high-order i-stable central difference schemes for incompressible viscous flows. Comput. Methods Appl. Mech. Eng. 190(37), 5009–5026 (2001)

    Article  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bourlioux, A., Layton, A.T., Minion, M.L.: High-order multi-implicit spectral deferred correction methods for problems of reactive flow. J. Comput. Phys. 189(2), 651–675 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bourlioux, A., Majda, A.J.: An elementary model for the validation of flamelet approximations in non-premixed turbulent combustion. Combust. Theory Model. 4(2), 189–210 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bruno, O.P., Cubillos, M.: Higher-order in time “quasi-unconditionally stable” ADI solvers for the compressible Navier–Stokes equations in 2D and 3D curvilinear domains. J. Comput. Phys. 307, 476–495 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chorin, A.J.: Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 745–762 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cockburn, B.B., Karniadakis, G., Shu, C.W. (eds.): Discontinuous Galerkin methods: theory, computation, and applications. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2000)

  10. Cole, J.T., Musslimani, Z.H.: Time-dependent spectral renormalization method. Phys. D 358, 15–24 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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). https://doi.org/10.1016/j.apnum.2016.07.014

    Article  MathSciNet  MATH  Google Scholar 

  13. Gottlieb, S.: On high order strong stability preserving Runge–Kutta and multi step time discretizations. J. Sci. Comput. 25(1–2), 105–128 (2005)

    MathSciNet  MATH  Google Scholar 

  14. Gottlieb, S., Grant, Z., Higgs, D.: Optimal explicit strong stability preserving Runge–Kutta methods with high linear order and optimal nonlinear order. Math. Comput. 84(296), 2743–2761 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grooss, J., Hesthaven, J.S.: A level set discontinuous galerkin method for free surface flows. Comput. Methods Appl. Mech. Eng. 195(25), 3406–3429 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195(44–47), 6011–6045 (2006). https://doi.org/10.1016/j.cma.2005.10.010

    Article  MathSciNet  MATH  Google Scholar 

  17. Huynh, H.T.: A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. In: 18th AIAA Computational Fluid Dynamics Conference, p. 4079 (2007)

  18. Jacobs, G.B., Kopriva, D.A., Mashayek, F.: A conservative isothermal wall boundary condition for the compressible Navier–Stokes equations. J. Sci. Comput. 30(2), 177–192 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jemison, M., Sussman, M., Arienti, M.: Compressible, multiphase semi-implicit method with moment of fluid interface representation. J. Comput. Phys. 279, 182–217 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kadioglu, S.Y., Klein, R., Minion, M.L.: A fourth-order auxiliary variable projection method for zero-Mach number gas dynamics. J. Comput. Phys. 227(3), 2012–2043 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. 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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kopriva, D.A.: Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers. Scientific Computation. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  24. Koziol, A.S., Pudykiewicz, J.A.: Global-scale environmental transport of persistent organic pollutants. Chemosphere 45(8), 1181–1200 (2001)

    Article  Google Scholar 

  25. Kwatra, N., Su, J., Grétarsson, J.T., Fedkiw, R.: A method for avoiding the acoustic time step restriction in compressible flow. J. Comput. Phys. 228(11), 4146–4161 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lalanne, B., Rueda Villegas, L., Tanguy, S., Risso, F.: On the computation of viscous terms for incompressible two-phase flows with level set/ghost fluid method. J. Comput. Phys. 301, 289–307 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Layton, A.T.: On the efficiency of spectral deferred correction methods for time-dependent partial differential equations. Appl. Numer. Math. 59(7), 1629–1643 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Liu, Y., Shu, C.W., Zhang, M.: Strong stability preserving property of the deferred correction time discretization. J. Comput. Math. 26(5), 633–656 (2008)

    MathSciNet  MATH  Google Scholar 

  30. Luo, H., Xia, Y., Spiegel, S., Nourgaliev, R., Jiang, Z.: A reconstructed discontinuous Galerkin method based on a hierarchical WENO reconstruction for compressible flows on tetrahedral grids. J. Comput. Phys. 236, 477–492 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Minion, M.L.: Semi-implicit projection methods for incompressible flow based on spectral deferred corrections (Workshop on innovative time integrators for PDEs). Appl. Numer. Math. 48(3–4), 369–387 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Nonaka, A., Bell, J., Day, M., Gilet, C., Almgren, A., Minion, M.: A deferred correction coupling strategy for low mach number flow with complex chemistry. Combust. Theory Model. 16(6), 1053–1088 (2012)

    Article  Google Scholar 

  33. Pazner, W.E., Nonaka, A., Bell, J.B., Day, M.S., Minion, M.L.: A high-order spectral deferred correction strategy for low Mach number flow with complex chemistry. Combust. Theory Model. 20(3), 521–547 (2016)

    Article  MathSciNet  Google Scholar 

  34. Pei, C., Sussman, M., Hussaini, M.Y.: A space-time discontinuous Galerkin spectral element method for nonlinear hyperbolic problems. Int. J. Comput. Methods (2018) (accepted)

  35. Pei, C., Sussman, M., Hussaini, M.Y.: A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete Contin. Dyn. Syst. Ser. B (2017). https://doi.org/10.3934/dcdsb.2017216

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

    Article  MathSciNet  MATH  Google Scholar 

  37. Sollie, W.E.H., Bokhove, O., van der Vegt, J.J.W.: Space–time discontinuous Galerkin finite element method for two-fluid flows. J. Comput. Phys. 230(3), 789–817 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5(3), 506–517 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  39. Temam, R.: On the approximation of the solution of Navier–Stokes equations by the fractional steps method ii. Arch. Ration. Mech. Anal. 32, 377–385 (1969)

    Article  Google Scholar 

  40. van der Vegt, J.J.W., Sudirham, J.J.: A space–time discontinuous Galerkin method for the time-dependent Oseen equations. Appl. Numer. Math. 58(12), 1892–1917 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work and the authors were supported in part by the National Science Foundation under Contract DMS 1418983.

Author information

Authors and Affiliations

  1. Department of Mathematics, Florida State University, Tallahassee, FL, 32306, USA

    Chaoxu Pei, Mark Sussman & M. Yousuff Hussaini

Authors
  1. Chaoxu Pei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark Sussman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Yousuff Hussaini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mark Sussman.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pei, C., Sussman, M. & Hussaini, M.Y. New Multi-implicit Space–Time Spectral Element Methods for Advection–Diffusion–Reaction Problems. J Sci Comput 78, 653–686 (2019). https://doi.org/10.1007/s10915-018-0654-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-018-0654-5

Keywords

Mathematics Subject Classification

Search

Navigation