Skip to main content
SpringerLink
Log in

General relaxation methods for initial-value problems with application to multistep schemes

Numerische Mathematik volume 146, pages 875–906 (2020)Cite this article

Abstract

Recently, an approach known as relaxation has been developed for preserving the correct evolution of a functional in the numerical solution of initial-value problems, using Runge–Kutta methods. We generalize this approach to multistep methods, including all general linear methods of order two or higher, and many other classes of schemes. We prove the existence of a valid relaxation parameter and high-order accuracy of the resulting method, in the context of general equations, including but not limited to conservative or dissipative systems. The theory is illustrated with several numerical examples.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

References

  1. Arévalo, C., Führer, C., Söderlind, G.: Regular and singular \(\beta \)-blocking of difference corrected multistep methods for nonstiff index-2 DAEs. Appl. Numer. Math. 35(4), 293–305 (2000). https://doi.org/10.1016/S0168-9274(99)00142-7

    Article  MathSciNet  MATH  Google Scholar 

  2. Arévalo, C., Söderlind, G.: Grid-independent construction of multistep methods. J. Comput. Math. 35(5), 672–692 (2017). https://doi.org/10.4208/jcm.1611-m2015-0404

    Article  MathSciNet  MATH  Google Scholar 

  3. Bashforth, F., Adams, J.C.: An Attempt to Test the Theories of Capillary Action by Comparing the Theoretical and Measured Forms of Drops of Fluid with an Explanation of the Method of Integration Employed in Constructing the Tables Which Give the Theoretical Forms of Such Drops. Cambridge University Press, Cambridge (1883)

    Google Scholar 

  4. Boom, P.D., Zingg, D.W.: High-order implicit time-marching methods based on generalized summation-by-parts operators. SIAM J. Sci. Comput. 37(6), A2682–A2709 (2015). https://doi.org/10.1137/15M1014917

    Article  MathSciNet  MATH  Google Scholar 

  5. Burchard, H., Deleersnijder, E., Meister, A.: A high-order conservative Patankar-type discretisation for stiff systems of production-destruction equations. Appl. Numer. Math. 47(1), 1–30 (2003). https://doi.org/10.1016/S0168-9274(03)00101-6

    Article  MathSciNet  MATH  Google Scholar 

  6. Burrage, K., Butcher, J.C.: Stability criteria for implicit Runge–Kutta methods. SIAM J. Numer. Anal. 16(1), 46–57 (1979). https://doi.org/10.1137/0716004

    Article  MathSciNet  MATH  Google Scholar 

  7. Burrage, K., Butcher, J.C.: Non-linear stability of a general class of differential equation methods. BIT Numer. Math. 20(2), 185–203 (1980). https://doi.org/10.1007/BF01933191

    Article  MATH  Google Scholar 

  8. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. John Wiley & Sons Ltd, Chichester (2016). 10.1002/9781119121534

    Book  Google Scholar 

  9. Calvo, M., Hernández-Abreu, D., Montijano, J.I., Rández, L.: On the preservation of invariants by explicit Runge–Kutta methods. SIAM J. Sci. Comput. 28(3), 868–885 (2006). https://doi.org/10.1137/04061979X

    Article  MathSciNet  MATH  Google Scholar 

  10. Calvo, M., Laburta, M., Montijano, J., Rández, L.: Projection methods preserving Lyapunov functions. BIT Numer. Math. 50(2), 223–241 (2010). https://doi.org/10.1007/s10543-010-0259-3

    Article  MathSciNet  MATH  Google Scholar 

  11. Chan, J.: On discretely entropy conservative and entropy stable discontinuous Galerkin methods. J. Comput. Phys. 362, 346–374 (2018). https://doi.org/10.1016/j.jcp.2018.02.033

    Article  MathSciNet  MATH  Google Scholar 

  12. Curtiss, C.F., Hirschfelder, J.O.: Integration of stiff equations. Proc. Natl. Acad. Sci. U. S. A. 38(3), 235 (1952). https://doi.org/10.1073/pnas.38.3.235

    Article  MathSciNet  MATH  Google Scholar 

  13. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer-Verlag, Berlin (2010). https://doi.org/10.1007/978-3-642-04048-1

    Book  MATH  Google Scholar 

  14. De Frutos, J., Sanz-Serna, J.M.: Accuracy and conservation properties in numerical integration: the case of the Korteweg–de Vries equation. Numer. Math. 75(4), 421–445 (1997). https://doi.org/10.1007/s002110050247

    Article  MathSciNet  MATH  Google Scholar 

  15. Dekker, K., Verwer, J.G.: Stability of Runge–Kutta methods for stiff nonlinear differential equations, CWI Monographs, vol. 2. North-Holland, Amsterdam (1984)

    MATH  Google Scholar 

  16. Eich, E.: Convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints. SIAM J. Numer. Anal. 30(5), 1467–1482 (1993). https://doi.org/10.1137/0730076

    Article  MathSciNet  MATH  Google Scholar 

  17. Fisher, T.C., Carpenter, M.H.: High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains. J. Comput. Phys. 252, 518–557 (2013). https://doi.org/10.1016/j.jcp.2013.06.014

    Article  MathSciNet  MATH  Google Scholar 

  18. Friedrich, L., Schnücke, G., Winters, A.R., Fernández, D.C.D.R., Gassner, G.J., Carpenter, M.H.: Entropy stable space-time discontinuous Galerkin schemes with summation-by-parts property for hyperbolic conservation laws. J. Sci. Comput. 80(1), 175–222 (2019). https://doi.org/10.1007/s10915-019-00933-2. arxiv: 1808.08218 [math.NA]

    Article  MathSciNet  MATH  Google Scholar 

  19. Friedrich, L., Winters, A.R., Fernández, D.C.D.R., Gassner, G.J., Parsani, M., Carpenter, M.H.: An entropy stable h/p non-conforming discontinuous Galerkin method with the summation-by-parts property. J. Sci. Comput. 77(2), 689–725 (2018). https://doi.org/10.1007/s10915-018-0733-7

    Article  MathSciNet  MATH  Google Scholar 

  20. Gear, C.W.: Maintaining solution invariants in the numerical solution of ODEs. SIAM J.Sci. Stat. Comput. 7(3), 734–743 (1986). https://doi.org/10.1137/0907050

    Article  MathSciNet  MATH  Google Scholar 

  21. Glaubitz, J., Öffner, P., Ranocha, H., Sonar, T.: Artificial viscosity for correction procedure via reconstruction using summation-by-parts operators. In: C. Klingenberg, M. Westdickenberg (eds.) Theory, Numerics and Applications of Hyperbolic Problems II, In: Springer Proceedings in Mathematics and Statistics, vol. 237, pp. 363–375. Springer International Publishing, Cham (2018). https://doi.org/10.1007/978-3-319-91548-7_28

  22. Gottlieb, S., Ketcheson, D.I., Shu, C.W.: Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations. World Scientific, Singapore (2011)

    Book  Google Scholar 

  23. Grimm, V., Quispel, G.: Geometric integration methods that preserve Lyapunov functions. BIT Numer. Math. 45(4), 709–723 (2005). https://doi.org/10.1007/s10543-005-0034-z

    Article  MathSciNet  MATH  Google Scholar 

  24. Hadjimichael, Y., Ketcheson, D.I., Lóczi, L., Németh, A.: Strong stability preserving explicit linear multistep methods with variable step size. SIAM J. Numer. Anal. 54(5), 2799–2832 (2016). https://doi.org/10.1137/15M101717X

    Article  MathSciNet  MATH  Google Scholar 

  25. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, vol. 31. Springer-Verlag, Berlin (2006). https://doi.org/10.1007/3-540-30666-8

  26. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems, Springer Series in Computational Mathematics, vol. 8. Springer-Verlag, Berlin (2008). https://doi.org/10.1007/978-3-540-78862-1

  27. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics, vol. 14. Springer-Verlag, Berlin (2010). https://doi.org/10.1007/978-3-642-05221-7

  28. Higueras, I.: Monotonicity for Runge–Kutta methods: inner product norms. J. Sci. Comput. 24(1), 97–117 (2005). https://doi.org/10.1007/s10915-004-4789-1

    Article  MathSciNet  MATH  Google Scholar 

  29. Hunter, J.D.: Matplotlib: a 2D graphics environment. Comput. Sci. Eng. 9(3), 90–95 (2007). https://doi.org/10.1109/MCSE.2007.55

    Article  Google Scholar 

  30. Jüngel, A., Schuchnigg, S.: Entropy-dissipating semi-discrete Runge–Kutta schemes for nonlinear diffusion equations. Commun. Math. Sci. 15(1), 27–53 (2017). https://doi.org/10.4310/CMS.2017.v15.n1.a2

    Article  MathSciNet  MATH  Google Scholar 

  31. Ketcheson, D.I.: Relaxation Runge–Kutta methods: conservation and stability for inner-product norms. SIAM J. Numer. Anal. 57(6), 2850–2870 (2019). https://doi.org/10.1137/19M1263662. arxiv:1905.09847 [math.NA]

    Article  MathSciNet  MATH  Google Scholar 

  32. Kojima, H.: Invariants preserving schemes based on explicit Runge–Kutta methods. BIT Numer. Math. 56(4), 1317–1337 (2016). https://doi.org/10.1007/s10543-016-0608-y

    Article  MathSciNet  MATH  Google Scholar 

  33. Laburta, M., Montijano, J.I., Rández, L., Calvo, M.: Numerical methods for non conservative perturbations of conservative problems. Comput. Phys. Commun. 187, 72–82 (2015). https://doi.org/10.1016/j.cpc.2014.10.012

    Article  MathSciNet  MATH  Google Scholar 

  34. LeFloch, P.G., Mercier, J.M., Rohde, C.: Fully discrete, entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal. 40(5), 1968–1992 (2002). https://doi.org/10.1137/S003614290240069X

    Article  MathSciNet  MATH  Google Scholar 

  35. LeVeque, R.J.: Finite Difference Methods for Ordinary and Partial Differential Equations: steady-state and time-dependent problems. SIAM, Philadelphia (2007)

    Book  Google Scholar 

  36. Lozano, C.: Entropy production by explicit Runge–Kutta schemes. J. Sci. Comput. 76(1), 521–565 (2018). https://doi.org/10.1007/s10915-017-0627-0

    Article  MathSciNet  MATH  Google Scholar 

  37. Lozano, C.: Entropy production by implicit Runge–Kutta schemes. J.Sci. Comput. (2019). https://doi.org/10.1007/s10915-019-00914-5

    Article  MathSciNet  MATH  Google Scholar 

  38. Mohammadi, F., Arévalo, C., Führer, C.: A polynomial formulation of adaptive strong stability preserving multistep methods. SIAM J. Numer. Anal. 57(1), 27–43 (2019). https://doi.org/10.1137/17M1158811

    Article  MathSciNet  MATH  Google Scholar 

  39. Nordström, J., La Cognata, C.: Energy stable boundary conditions for the nonlinear incompressible Navier–Stokes equations. Math. Comput. 88(316), 665–690 (2019). https://doi.org/10.1090/mcom/3375

    Article  MathSciNet  MATH  Google Scholar 

  40. Nyström, E.J.: Über die numerische integration von differentialgleichungen. Acta Soc. Sci. Fennicae 50(13), 1–56 (1925)

    MATH  Google Scholar 

  41. Öffner, P., Glaubitz, J., Ranocha, H.: Analysis of artificial dissipation of explicit and implicit time-integration methods. Int. J. Numer. Anal. Model. 17(3), 332–349 (2020). arxiv:1609.02393 [math.NA]

    MathSciNet  Google Scholar 

  42. Ranocha, H.: Comparison of some entropy conservative numerical fluxes for the Euler equations. J. Sci. Comput. 76(1), 216–242 (2018). https://doi.org/10.1007/s10915-017-0618-1. arxiv:1701.02264 [math.NA]

    Article  MathSciNet  MATH  Google Scholar 

  43. Ranocha, H.: Generalised summation-by-parts operators and entropy stability of numerical methods for hyperbolic balance laws. Ph.D. thesis, TU Braunschweig (2018)

  44. Ranocha, H.: Some notes on summation by parts time integration methods. Results Appl. Math. 1, 100,004 (2019). https://doi.org/10.1016/j.rinam.2019.100004. arxiv:1901.08377 [math.NA]

    Article  MATH  Google Scholar 

  45. Ranocha, H.: On strong stability of explicit Runge–Kutta methods for nonlinear semibounded operators. IMA Journal of Numerical Analysis (2020). https://doi.org/10.1093/imanum/drz070. arxiv:1811.11601 [math.NA]

  46. Ranocha, H., Dalcin, L., Parsani, M.: Fully-discrete explicit locally entropy-stable schemes for the compressible Euler and Navier–Stokes equations. Comput. Math. Appl. 80(5), 1343–1359 (2020). https://doi.org/10.1016/j.camwa.2020.06.016

    Article  MathSciNet  MATH  Google Scholar 

  47. Ranocha, H., Glaubitz, J., Öffner, P., Sonar, T.: Stability of artificial dissipation and modal filtering for flux reconstruction schemes using summation-by-parts operators. Appl. Numer. Math. 128, 1–23 (2018). https://doi.org/10.1016/j.apnum.2018.01.019. arXiv:1606.00995 [math.NA] and arXiv:1606.01056 [math.NA]

  48. Ranocha, H., Ketcheson, D.I.: Energy stability of explicit Runge-Kutta methods for non-autonomous or nonlinear problems (2020). Accepted in SIAM Journal on Numerical Analysis. arxiv:1909.13215 [math.NA]

  49. Ranocha, H., Ketcheson, D.I.: Relaxation-LMM-notebooks. General relaxation methods for initial-value problems with application to multistep schemes. https://github.com/ranocha/Relaxation-LMM-notebooks (2020). https://doi.org/10.5281/zenodo.3697836

  50. Ranocha, H., Ketcheson, D.I.: Relaxation Runge–Kutta methods for Hamiltonian problems. J. Sci. Comput. 84(1), (2020). https://doi.org/10.1007/s10915-020-01277-yarxiv:2001.04826 [math.NA]

  51. Ranocha, H., Lóczi, L., Ketcheson, D.I.: General relaxation methods for initial-value problems with application to multistep schemes (2020). arxiv:2003.03012 [math.NA]

  52. Ranocha, H., Öffner, P.: \(L_2\) stability of explicit Runge–Kutta schemes. J. Sci. Comput. 75(2), 1040–1056 (2018). https://doi.org/10.1007/s10915-017-0595-4

    Article  MathSciNet  MATH  Google Scholar 

  53. Ranocha, H., Sayyari, M., Dalcin, L., Parsani, M., Ketcheson, D.I.: Relaxation Runge–Kutta methods: fully-discrete explicit entropy-stable schemes for the compressible Euler and Navier–Stokes equations. SIAM J. Sci. Comput. 42(2), A612–A638 (2020). https://doi.org/10.1137/19M1263480. arxiv:1905.09129 [math.NA]

    Article  MathSciNet  MATH  Google Scholar 

  54. Rosenlicht, M.: Introduction to Analysis. Dover Publications Inc, New York (1986)

    MATH  Google Scholar 

  55. Ruuth, S.J., Hundsdorfer, W.: High-order linear multistep methods with general monotonicity and boundedness properties. J. Comput. Phys. 209(1), 226–248 (2005). https://doi.org/10.1016/j.jcp.2005.02.029

    Article  MathSciNet  MATH  Google Scholar 

  56. Sanz-Serna, J.M.: An explicit finite-difference scheme with exact conservation properties. J. Comput. Phys. 47(2), 199–210 (1982). https://doi.org/10.1016/0021-9991(82)90074-2

    Article  MathSciNet  MATH  Google Scholar 

  57. Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems, Applied Mathematics and Mathematical Computation, vol. 7. Chapman & Hall, London (1994)

    MATH  Google Scholar 

  58. Sanz-Serna, J.M., Manoranjan, V.: A method for the integration in time of certain partial differential equations. J. Comput. Phys. 52(2), 273–289 (1983). https://doi.org/10.1016/0021-9991(83)90031-1

    Article  MathSciNet  MATH  Google Scholar 

  59. Shampine, L.F.: Conservation laws and the numerical solution of ODEs. Comput. Math. Appl. 12(5–6), 1287–1296 (1986). https://doi.org/10.1016/0898-1221(86)90253-1

    Article  MathSciNet  MATH  Google Scholar 

  60. Shampine, L.F.: Conservation laws and the numerical solution of ODEs, II. Comput. Math. Appl. 38(2), 61–72 (1999). https://doi.org/10.1016/S0898-1221(99)00183-2

    Article  MathSciNet  MATH  Google Scholar 

  61. Sjögreen, B., Yee, H.: High order entropy conservative central schemes for wide ranges of compressible gas dynamics and MHD flows. J. Comput. Phys. 364, 153–185 (2018). https://doi.org/10.1016/j.jcp.2018.02.003

    Article  MathSciNet  MATH  Google Scholar 

  62. Söderlind, G., Fekete, I., Faragó, I.: On the zero-stability of multistep methods on smooth nonuniform grids. BIT Numer. Math. 58(4), 1125–1143 (2018). https://doi.org/10.1007/s10543-018-0716-y

    Article  MathSciNet  MATH  Google Scholar 

  63. Sun, Z., Shu, C.W.: Stability of the fourth order Runge–Kutta method for time-dependent partial differential equations. Ann. Math. Sci. Appl. 2(2), 255–284 (2017). https://doi.org/10.4310/AMSA.2017.v2.n2.a3

    Article  MathSciNet  MATH  Google Scholar 

  64. Sun, Z., Shu, C.W.: Enforcing strong stability of explicit Runge–Kutta methods with superviscosity (2019). arxiv:1912.11596 [math.NA]

  65. Sun, Z., Shu, C.W.: Strong stability of explicit Runge–Kutta time discretizations. SIAM J. Numer. Anal. 57(3), 1158–1182 (2019). https://doi.org/10.1137/18M122892X. arxiv:1811.10680 [math.NA]

    Article  MathSciNet  MATH  Google Scholar 

  66. Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014). https://doi.org/10.1016/j.jcp.2014.02.031

    Article  MathSciNet  MATH  Google Scholar 

  67. Tadmor, E.: From semidiscrete to fully discrete: stability of Runge–Kutta schemes by the energy method II. In: Estep, D.J., Tavener, S. (eds.) Collected Lectures on the Preservation of Stability under Discretization, Proceedings in Applied Mathematics, vol. 109, pp. 25–49. Society for Industrial and Applied Mathematics, Philadelphia (2002)

    Google Scholar 

  68. Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451–512 (2003). https://doi.org/10.1017/S0962492902000156

    Article  MathSciNet  MATH  Google Scholar 

  69. Virtanen, P., Gommers, R., Oliphant, T.E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S.J., Brett, M., Wilson, J., Jarrod Millman, K., Mayorov, N., Nelson, A.R.J., Jones, E., Kern, R., Larson, E., Carey, C., Polat, İ., Feng, Y., Moore, E.W., Vand erPlas, J., Laxalde, D., Perktold, J., Cimrman, R., Henriksen, I., Quintero, E.A., Harris, C.R., Archibald, A.M., Ribeiro, A.H., Pedregosa, F., van Mulbregt, P., SciPy 1.0 Contributors: SciPy 1.0 – fundamental algorithms for scientific computing in python (2019). arxiv:1907.10121 [cs.MS]

  70. Zakerzadeh, H., Fjordholm, U.S.: High-order accurate, fully discrete entropy stable schemes for scalar conservation laws. IMA J. Numer. Anal. 36(2), 633–654 (2016). https://doi.org/10.1093/imanum/drv020

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Research reported in this publication was supported by the King Abdullah University of Science and Technology (KAUST). The project “Application-domain-specific highly reliable IT solutions” has been implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, and financed under the scheme Thematic Excellence Programme no. 2020-4.1.1-TKP2020 (National Challenges Subprogramme).

Author information

Authors and Affiliations

  1. Extreme Computing Research Center (ECRC), Computer Electrical and Mathematical Science and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia

    Hendrik Ranocha & David I. Ketcheson

  2. Department of Numerical Analysis, Eötvös Loránd University, Budapest, Hungary

    Lajos Lóczi

  3. Department of Differential Equations, Budapest University of Technology and Economics, Budapest, Hungary

    Lajos Lóczi

Authors
  1. Hendrik Ranocha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lajos Lóczi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David I. Ketcheson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hendrik Ranocha.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ranocha, H., Lóczi, L. & Ketcheson, D.I. General relaxation methods for initial-value problems with application to multistep schemes. Numer. Math. 146, 875–906 (2020). https://doi.org/10.1007/s00211-020-01158-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-020-01158-4

Mathematics Subject Classification

search

Navigation