Skip to main content
SpringerLink
Log in

High-Order Local Time Stepping on Moving DG Spectral Element Meshes

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

Abstract

We derive and evaluate an explicit local time stepping (LTS) integration for the discontinuous Galerkin spectral element method on moving meshes. The LTS procedure is derived from Adams–Bashforth multirate time integration methods. We also present speedup and memory estimates, which show that the explicit LTS integration scales well with problem size. Time-step refinement studies with static and moving meshes show that the approximations are spectrally accurate in space and have design temporal accuracy. The numerical tests validate theoretical estimates that the LTS procedure can reduce computational cost by as much as an order of magnitude for time accurate problems.

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

Similar content being viewed by others

References

  1. Acosta Minoli, C.A., Kopriva, D.A.: Discontinuous Galerkin spectral element approximations on moving meshes. J. Comput. Phys. 230, 1876–1902 (2010)

    Article  MathSciNet  Google Scholar 

  2. Akay, Y., Akay, M., Welkowitz, W., Semmlow, J., Kostis, J.: Noninvasive acoustical detection of coronary artery disease: a comparative study of signal processing methods. IEEE Trans. Biomed. Eng. 40, 571–578 (1993)

    Article  Google Scholar 

  3. Al-Jumaily, A., Alizad, A. (eds.): Biomedical Applications of Vibration and Acoustics in Therapy, Bioeffect and Modeling. ASME Press, New York (2008)

    Google Scholar 

  4. Ascoli, A., Bernasconi, C., Cavalleri, G.: Refraction and reflection of a nonrelativistic wave when the interface and the media are moving. Phys. Rev. E 54, 6291–6296 (1996)

    Article  Google Scholar 

  5. Birken, P., Gassner, G., Haas, M., Munz, C.: Efficient time integration for discontinuous Galerkin method for the unsteady 3D Navier–Stokes equations (2012)

  6. Canuto, C., Hussaini, M., Quarteroni, A., Zang, T.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)

    Google Scholar 

  7. Canuto, C., Hussaini, M., Quarteroni, A., Zang, T.: Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer, Berlin (2007)

    Google Scholar 

  8. Censor, D.: Non-relativistic scattering by time-varying bodies and media. Prog. Electromagn. Res. 48, 249–278 (2004)

    Article  Google Scholar 

  9. Cockburn, B., Shu, C.W.: The Runge–Kutta local projection \({P}^1\)-discontinuous Galerkin method for scalar conservation laws. Modélisation Mathématique et Analyse Numérique 25, 337–361 (1991)

    MATH  MathSciNet  Google Scholar 

  10. Cockburn, B., Shu, C.W.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Constantinescu, E., Sandu, A.: Multirate timestepping methods for hyperbolic conservation laws. J. Sci. Comput. 33, 239–278 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Constantinescu, E., Sandu, A.: Multirate explicit adams methods for time integration of conservation laws. J. Sci. Comput. 38, 229–249 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dawson, C., Kirby, R.: High resolution schemes for conservation laws with locally varying time steps. SIAM J. Sci. Comput. 22, 2256–2281 (2000)

    Article  MathSciNet  Google Scholar 

  14. De Zutter, D.: Reflections from linearly vibrating objects: plane mirror at oblique incidence. IEEE Trans. Antennas Propag. 30, 898–903 (1982)

    Article  Google Scholar 

  15. Dennie, J., Mandeville, J.B., Boxerman, J.L., Packard, S.D., Rosen, B.R., Weisskoff, R.M.: NMR imaging of changes in vascular morphology due to tumor angiogenesis. Magn. Reson. Med. 40, 793–799 (1998)

    Article  Google Scholar 

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

    Article  Google Scholar 

  17. Étienne, S., Garon, A., Pelletier, D.: Perspective on the geometric conservation law and finite element methods for ALE simulations of incompressible flow. J. Comput. Phys. 228, 2313–2333 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ezziani, A., Joly, P.: Local time stepping and discontinuous Galerkin methods for symmetric first order hyperbolic systems. J. Comput. Appl. Math. 234, 1886–1895 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gassner, G., Hindenlang, F., Munz, C.: A Runge–Kutta based discontinuous Galerkin method with time accurate local time stepping. Adapt. High-Order Methods Comput. Fluid Dyn. 2, 95–118 (2011)

    Article  MathSciNet  Google Scholar 

  20. Gear, C.W., Wells, D.R.: Multirate linear multistep methods. BIT 24, 484–502 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  21. Grote, M., Mitkova, T.: High-order explicit local time-stepping methods for damped wave equations. J. Comput. Appl. Math. 239, 270–289 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  22. Harfoush, F., Taflove, A., Kriegsmann, G.: A numerical technique for analyzing electromagnetic wave scattering from moving surfaces in one and two dimensions. IEEE Trans. Antennas Propag. 37, 55–63 (1989)

    Article  Google Scholar 

  23. Ho, M.: Scattering of electromagnetic waves from vibrating perfect surfaces: simulation using relativistic boundary conditions. J. Electromag. Waves Appl. 20, 425–433 (2006)

    Article  Google Scholar 

  24. Jing, F., Xuemin, W., Mingshi, W., Wei, L.: Noninvasive acoustical analysis system of coronary heart disease. In: Sixteenth southern biomedical, engineering conference, pp. 239–241 (1997)

  25. Kompis, M., Pasterkamp, H., Wodicka, G.R.: Acoustic imaging of the human chest. CHEST: Am. Coll. Chest Phys. 120, 1309–1321 (2001)

    Article  Google Scholar 

  26. Kopriva, D.A.: Metric identities and the discontinuous spectral element method on curvilinear meshes. J. Sci. Comput. 26, 301–327 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kopriva, D.A.: Implementing Spectral Methods for Partial Differential Equations. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  28. Kopriva, D.A., Jimenez, E.: An assessment of the efficiency of nodal discontinuous Galerkin spectral element methods. In: Ansorge, R., Bijl, H., Meister, A., Sonar, T. (eds.) Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws, vol. 120 of Notes on Numerical Fluid Mechanics and Multidisciplinary Design, pp. 223–235. Springer, Berlin (2013)

    Chapter  Google Scholar 

  29. Krivodonova, L.: An efficient local time-stepping scheme for solution of nonlinear conservation laws. J. Comput. Phys. 229, 8537–8551 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. Kunz, K.: Plane electromagnetic waves in moving media and reflections from moving interfaces. J. Appl. Phys. 51, 873–884 (1980)

    Article  Google Scholar 

  31. Lomtev, I., Kirby, R.M., Karniadakis, G.E.: A discontinuous Galerkin ALE method for compressible viscous flows in moving domains. J. Comput. Phys. 155, 128–159 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  33. Luo, G., Kaufman, J.J., Chiabrera, A., Bianco, B., Kinney, J.H., Haupt, D., Ryaby, J.T., Siffert, R.S.: Computational methods for ultrasonic bone assessment. Ultrasound Med. Biol. 25, 823–830 (1999)

    Article  Google Scholar 

  34. Mansy, H., Royston, T., Balk, R., Sandler, R.: Pneumothorax detection using pulmonary acoustic transmission measurements. Med. Biol. Eng. Comput. 40, 520–525 (2002)

    Article  Google Scholar 

  35. Mavriplis, D.J., Yang, Z.: Construction of the discrete geometric conservation law for high-order time-accurate simulations on dynamic meshes. J. Comput. Phys. 213, 557–573 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  36. Morse, P., Ingard, K.: Theoretical Acoustics. Princeton University Press, Princeton (1987)

    Google Scholar 

  37. Mujica, N., Wunenburger, R., Fauve, S.: Scattering of a sound wave by a vibrating surface. Eur. Phys. J. B Condens. Matter Complex Syst. 33, 209–213 (2003)

    Article  Google Scholar 

  38. Ostashev, V.: Acoustics in Moving Inhomogeneous Media. Taylor & Francis, London (1998)

    Google Scholar 

  39. Piquette, J.C., Van Buren, A.L., Rogers, P.H.: Censor’s acoustical Doppler effect analysis—is it a valid method? J. Acoust. Soc. Am. 83, 1681–1682 (1988)

    Article  Google Scholar 

  40. Schneiders, R.: Refining quadrilateral and hexahedral meshes. In 5th International conference on grid generation in computational field simulations, pp. 679–688 (1996).

  41. Stock, A.: Development and application of a multirate multistep AB method to a discontinuous Galerkin method based particle in cell scheme, diploma thesis, Institut für Aerodynamik und Gasdynamik Universität Stuttgart and Brown University (2009)

  42. Tam, C.K.W.: Computational aeroacoustics: an overview of computational challenges and applications. Int. J. Comput. Fluid Dyn. 18, 547–567 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  43. Taube, A., Dumbser, M., Munz, C.-D., Schneider, R.: A high-order discontinuous Galerkin method with time-accurate local time stepping for the Maxwell equations. Int. J. Numer. Model. Electron. Netw. Devices Fields 22, 77–103 (2009)

    Article  MATH  Google Scholar 

  44. Topping, B.H.V., Muylle, J., Iványi, P., Putanowicz, R., Cheng, B.: Finite Element Mesh Generation. Saxe-Coburg Publications, UK (2004)

    Google Scholar 

  45. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, Berlin (2009)

    Book  Google Scholar 

  46. Torres-Jimenez, A., Charleston-Villalobos, S., Gonzalez-Camarena, R., Chi-Lem, G., Aljama-Corrales, T.: Respiratory acoustic thoracic imaging (RATHI): Assessing intrasubject variability. in Engineering in Medicine and Biology Society, 2008, EMBS 2008. In: 30th Annual international conference of the IEEE, pp. 4793–4796 (2008)

  47. Toulorge, T., Desmet, W.: CFL conditions for Runge–Kutta discontinuous Galerkin methods on triangular grids. J. Comput. Phys. 230, 4657–4678 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  48. Wang, Z.: High-order methods for the Euler and Navier–Stokes equations on unstructured grids. Prog. Aerosp. Sci. 43, 1–41 (2007)

    Article  Google Scholar 

  49. Williamson, J.H.: Low-storage Runge–Kutta schemes. J. Comput. Phys. 35, 48–56 (1980)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work and the authors were supported in part by the NSF Grant DMS-0810925.

Author information

Authors and Affiliations

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

    Andrew R. Winters & David A. Kopriva

Authors
  1. Andrew R. Winters
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David A. Kopriva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrew R. Winters.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Winters, A.R., Kopriva, D.A. High-Order Local Time Stepping on Moving DG Spectral Element Meshes. J Sci Comput 58, 176–202 (2014). https://doi.org/10.1007/s10915-013-9730-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-013-9730-z

Keywords

Search

Navigation