High Order Strong Stability Preserving Time Discretizations


Strong stability preserving (SSP) high order time discretizations were developed to ensure nonlinear stability properties necessary in the numerical solution of hyperbolic partial differential equations with discontinuous solutions. SSP methods preserve the strong stability properties—in any norm, seminorm or convex functional—of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the connections between the timestep restrictions for strong stability preservation and contractivity. Numerical examples demonstrate that common linearly stable but not strong stability preserving time discretizations may lead to violation of important boundedness properties, whereas SSP methods guarantee the desired properties provided only that these properties are satisfied with forward Euler timestepping. We review optimal explicit and implicit SSP Runge–Kutta and multistep methods, for linear and nonlinear problems. We also discuss the SSP properties of spectral deferred correction methods.

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

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 199

This is the net price. Taxes to be calculated in checkout.


  1. 1.

    Baiotti, L., Hawke, I., Montero, P.J., Loffler, F., Rezzolla, L., Stergioulas, N., Font, J.A., Seidel, E.: Three-dimensional relativistic simulations of rotating neutron-star collapse to a Kerr black hole. Phys. Rev. D 71 (2005)

  2. 2.

    Balbás, J., Tadmor, E.: A central differencing simulation of the Orszag-Tang vortex system. IEEE Trans. Plasma Sci. 33(2), 470–471 (2005)

  3. 3.

    Bassano, E.: Numerical simulation of thermo-solutal-capillary migration of a dissolving drop in a cavity. Int. J. Numer. Methods Fluids 41, 765–788 (2003)

  4. 4.

    Butcher, J.C.: On the implementation of implicit Runge-Kutta methods. BIT 6, 237–240 (1976)

  5. 5.

    Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, New York (2003)

  6. 6.

    Caiden, R., Fedkiw, R.P., Anderson, C.: A numerical method for two-phase flow consisting of separate compressible and incompressible regions. J. Comput. Phys. 166, 1–27 (2001)

  7. 7.

    Carrillo, J., Gamba, I.M., Majorana, A., Shu, C.-W.: A WENO-solver for the transients of Boltzmann-Poisson system for semiconductor devices: performance and comparisons with Monte Carlo methods. J. Comput. Phys. 184, 498–525 (2003)

  8. 8.

    Chen, M.-H., Cockburn, B., Reitich, F.: High-order RKDG methods for computational electromagnetics. J. Sci. Comput. 22–23, 205–226 (2005)

  9. 9.

    Cheng, L.-T., Liu, H., Osher, S.: Computational high-frequency wave propagation using the level set method, with applications to the semi-classical limit of Schrodinger equations. Commun. Math. Sci. 1(3), 593–621 (2003)

  10. 10.

    Cheruvu, V., Nair, R.D., Tufo, H.M.: A spectral finite volume transport scheme on the cubed-sphere. Appl. Numer. Math. 57, 1021–1032 (2007)

  11. 11.

    Cockburn, B., Li, F., Shu, C.-W.: Locally divergence-free discontinuous Galerkin methods for the Maxwell equations. J. Comput. Phys. 194, 588–610 (2004)

  12. 12.

    Cockburn, B., Qian, J., Reitich, F., Wang, J.: An accurate spectral/discontinuous finite-element formulation of a phase-space-based level set approach to geometrical optics. J. Comput. Phys. 208, 175–195 (2005)

  13. 13.

    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)

  14. 14.

    Dahlquist, G., Jeltsch, R.: Generalized disks of contractivity for explicit and implicit Runge–Kutta methods. Technical Report, Department of Numerical Analysis and Computational Science, Royal Institute of Technology, Stockholm (1979)

  15. 15.

    Dekker, K., Verwer, J.G.: Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations. CWI Monographs, vol. 2. North-Holland, Amsterdam (1984)

  16. 16.

    Del Zanna, L., Bucciantini, N.: An efficient shock-capturing central-type scheme for multidimensional relativistic flows: I. hydrodynamics. Astron. Astrophys. 390, 1177–1186 (2002)

  17. 17.

    Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. BIT 40, 241–266 (2000)

  18. 18.

    Enright, D., Fedkiw, R., Ferziger, J., Mitchell, I.: A hybrid particle level set method for improved interface capturing. J. Comput. Phys. 183, 83–116 (2002)

  19. 19.

    Feng, L.-L., Shu, C.-W., Zhang, M.: A hybrid cosmological hydrodynamic/n-body code based on a weighted essentially nonoscillatory scheme. Astrophys. J. 612, 1–13 (2004)

  20. 20.

    Ferracina, L., Spijker, M.N.: Stepsize restrictions for the total-variation-diminishing property in general Runge-Kutta methods. SIAM J. Numer. Anal. 42, 1073–1093 (2004)

  21. 21.

    Ferracina, L., Spijker, M.N.: Computing optimal monotonicity-preserving Runge-Kutta methods. Technical Report MI2005-07, Mathematical Institute, Leiden University (2005)

  22. 22.

    Ferracina, L., Spijker, M.N.: An extension and analysis of the Shu-Osher representation of Runge-Kutta methods. Math. Comput. 249, 201–219 (2005)

  23. 23.

    Ferracina, L., Spijker, M.N.: Strong stability of singly-diagonally-implicit Runge-Kutta methods. Appl. Numer. Math. (2008). doi:10.1016/j.apnum.2007.10.004

  24. 24.

    Gottlieb, D., Tadmor, E.: The CFL condition for spectral approximations to hyperbolic initial-boundary value problems. Math. Comput. 56, 565–588 (1991)

  25. 25.

    Gottlieb, S.: On high order strong stability preserving Runge-Kutta and multi step time discretizations. J. Sci. Comput. 25, 105–127 (2005)

  26. 26.

    Gottlieb, S., Gottlieb, L.J.: Strong stability preserving properties of Runge-Kutta time discretization methods for linear constant coefficient operators. J. Sci. Comput. 18, 83–109 (2003)

  27. 27.

    Gottlieb, S., Ruuth, S.J.: Optimal strong-stability-preserving time-stepping schemes with fast downwind spatial discretizations. J. Sci. Comput. 27, 289–303 (2006)

  28. 28.

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

  29. 29.

    Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)

  30. 30.

    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics, vol. 14. Springer, Berlin (1991)

  31. 31.

    Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357–393 (1983)

  32. 32.

    Higueras, I.: On strong stability preserving time discretization methods. J. Sci. Comput. 21, 193–223 (2004)

  33. 33.

    Higueras, I.: Representations of Runge-Kutta methods and strong stability preserving methods. SIAM J. Numer. Anal. 43, 924–948 (2005)

  34. 34.

    Higueras, I.: Strong stability for additive Runge-Kutta methods. SIAM J. Numer. Anal. 44, 1735–1758 (2006)

  35. 35.

    Horvath, Z.: On the positivity of matrix-vector products. Linear Algebra Appl. 393, 253–258 (2004)

  36. 36.

    Horvath, Z.: On the positivity step size threshold of Runge-Kutta methods. Appl. Numer. Math. 53, 341–356 (2005)

  37. 37.

    Huang, C.: Strong stability preserving hybrid methods. Appl. Numer. Math. (2008). doi: 10.1016/j.apnum.2008.03.030

  38. 38.

    Hundsdorfer, W., Ruuth, S.J.: On monotonicity and boundedness properties of linear multistep methods. Math. Comput. 75(254), 655–672 (2005)

  39. 39.

    Hundsdorfer, W., Ruuth, S.J.: IMEX extensions of linear multistep methods with general monotonicity and boundedness properties. J. Comput. Phys. 225, 201–2042 (2007)

  40. 40.

    Hundsdorfer, W., Ruuth, S.J., Spiteri, R.J.: Monotonicity-preserving linear multistep methods. SIAM J. Numer. Anal. 41, 605–623 (2003)

  41. 41.

    Jeltsch, R., Nevanlinna, O.: Stability of explicit time discretizations for solving initial value problems. Numer. Math. 37, 61–91 (1981)

  42. 42.

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

  43. 43.

    Jin, S., Liu, H., Osher, S., Tsai, Y.-H.R.: Computing multivalued physical observables for the semiclassical limit of the Schrodinger equation. J. Comput. Phys. 205, 222–241 (2005)

  44. 44.

    Kennedy, C.A., Carpenter, M.H., Lewis, R.M.: Low-storage, explicit Runge-Kutta schemes for the compressible Navier-Stokes equations. Appl. Numer. Math. 35, 177–219 (2000)

  45. 45.

    Ketcheson, D.I.: Highly efficient strong stability preserving Runge-Kutta methods with low-storage implementations. SIAM J. Sci. Comput. (2008, to appear)

  46. 46.

    Ketcheson, D.I.: Computation of optimal contractive general linear methods and strong stability preserving linear multistep methods. Math. Comput. (to appear)

  47. 47.

    Ketcheson, D.I., Macdonald, C.B., Gottlieb, S.: Optimal implicit strong stability preserving Runge-Kutta methods. Appl. Numer. Math. (2008). doi: 10.1016/j.apnum.2008.03.034

  48. 48.

    Ketcheson, D.I., Macdonald, C.B., Gottlieb, S.: Numerically optimal SSP Runge–Kutta methods (website). (2007)

  49. 49.

    Ketcheson, D.I., Robinson, A.C.: On the practical importance of the SSP property for Runge-Kutta time integrators for some common Godunov-type schemes. Int. J. Numer. Methods Fluids 48, 271–303 (2005)

  50. 50.

    Kraaijevanger, J.F.B.M.: Absolute monotonicity of polynomials occurring in the numerical solution of initial value problems. Numer. Math. 48, 303–322 (1986)

  51. 51.

    Kraaijevanger, J.F.B.M.: Contractivity of Runge-Kutta methods. BIT 31, 482–528 (1991)

  52. 52.

    Kubatko, E.J., Westerink, J.J., Dawson, C.: Semi discrete discontinuous Galerkin methods and stage-exceeding-order, strong-stability-preserving Runge-Kutta time discretizations. J. Comput. Phys. 222, 832–848 (2007)

  53. 53.

    Kurganov, A., Tadmor, E.: New high-resolution schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160, 241–282 (2000)

  54. 54.

    Labrunie, S., Carrillo, J., Bertrand, P.: Numerical study on hydrodynamic and quasi-neutral approximations for collisionless two-species plasmas. J. Comput. Phys. 200, 267–298 (2004)

  55. 55.

    Lax, P.D.: Gibbs Phenomena. J. Sci. Comput. 28, 445–449 (2006)

  56. 56.

    Lenferink, H.W.J.: Contractivity-preserving explicit linear multistep methods. Numer. Math. 55, 213–223 (1989)

  57. 57.

    Lenferink, H.W.J.: Contractivity-preserving implicit linear multistep methods. Math. Comput. 56, 177–199 (1991)

  58. 58.

    Levy, D., Tadmor, E.: From semi-discrete to fully discrete: stability of Runge-Kutta schemes by the energy method. SIAM Rev. 40, 40–73 (1998)

  59. 59.

    Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)

  60. 60.

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

  61. 61.

    Macdonald, C.B.: Constructing high-order Runge-Kutta methods with embedded strong-stability-preserving pairs. Master’s Thesis, Simon Fraser University (2003)

  62. 62.

    Macdonald, C.B., Gottlieb, S., Ruuth, S.: A numerical study of diagonally split Runge-Kutta methods for PDEs with discontinuities (2008, submitted)

  63. 63.

    Majda, A., Osher, S.: A systematic approach for correcting nonlinear instabilities. the Lax-Wendroff scheme for scalar conservation laws. Numer. Math. 30, 429–452 (1978)

  64. 64.

    Mignone, A.: The dynamics of radiative shock waves: linear and nonlinear evolution. Astrophys. J. 626, 373–388 (2005)

  65. 65.

    Minion, M.L.: Semi-implicit spectral deferred correction methods for ordinary differential equations. SIAM J. Numer. Anal. 41, 605–623 (2003)

  66. 66.

    Nessyahu, H., Tadmor, E.: Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463 (1990)

  67. 67.

    Osher, S., Chakravarthy, S.: High resolution schemes and the entropy condition. SIAM J. Numer. Anal. 21, 955–984 (1984)

  68. 68.

    Osher, S., Tadmor, E.: On the convergence of difference approximations to scalar conservation laws. Math. Comput. 50, 19–51 (1988)

  69. 69.

    Pantano, C., Deiterding, R., Hill, D.J., Pullin, D.I.: A low numerical dissipation patch-based adaptive mesh refinement method for large-eddy simulation of compressible flows. J. Comput. Phys. 221, 63–87 (2007)

  70. 70.

    Patel, S., Drikakis, D.: Effects of preconditioning on the accuracy and efficiency of incompressible flows. Int. J. Numer. Methods Fluids 47, 963–970 (2005)

  71. 71.

    Peng, D., Merriman, B., Osher, S., Zhao, H., Kang, M.: A PDE-based fast local level set method. J. Comput. Phys. 155, 410–438 (1999)

  72. 72.

    Ruuth, S.J.: Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Math. Comput. 75, 183–207 (2006)

  73. 73.

    Ruuth, S.J., Hundsdorfer, W.: High-order linear multistep methods with general monotonicity and boundedness properties. J. Comput. Phys. 209, 226–248 (2005)

  74. 74.

    Ruuth, S.J., Spiteri, R.J.: Two barriers on strong-stability-preserving time discretization methods. J. Sci. Comput. 17, 211–220 (2002)

  75. 75.

    Ruuth, S.J., Spiteri, R.J.: High-order strong-stability-preserving Runge-Kutta methods with downwind-biased spatial discretizations. SIAM J. Numer. Anal. 42, 974–996 (2004)

  76. 76.

    Sahinidis, N.V., Tawarmalani, M.: BARON 7.2: Global optimization of mixed-integer nonlinear programs, user’s manual. Available at (2004)

  77. 77.

    Shu, C.-W.: Total-variation diminishing time discretizations. SIAM J. Sci. Statist. Comput. 9, 1073–1084 (1988)

  78. 78.

    Shu, C.-W.: A survey of strong stability-preserving high-order time discretization methods. In: Collected Lectures on the Preservation of Stability under Discretization. SIAM, Philadelphia (2002)

  79. 79.

    Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

  80. 80.

    Spijker, M.N.: Contractivity in the numerical solution of initial value problems. Numer. Math. 42, 271–290 (1983)

  81. 81.

    Spijker, M.N.: Stepsize conditions for general monotonicity in numerical initial value problems. SIAM J. Numer. Anal. 45, 1226–1245 (2007)

  82. 82.

    Spiteri, R.J., Ruuth, S.J.: A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM J. Numer. Anal. 40, 469–491 (2002)

  83. 83.

    Spiteri, R.J., Ruuth, S.J.: Nonlinear evolution using optimal fourth-order strong-stability-preserving Runge-Kutta methods. Math. Comput. Simul. 62, 125–135 (2003)

  84. 84.

    Strang, G.: Accurate partial difference methods II: nonlinear problems. Numer. Math. 6, 37–46 (1964)

  85. 85.

    Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations. Cole Mathematics Series. Wadsworth and Brooks, California (1989)

  86. 86.

    Sun, Y., Wang, Z.J., Liu, Y.: Spectral (finite) volume method for conservation laws on unstructured grids VI: Extension to viscous flow. J. Comput. Phys. 215, 41–58 (2006)

  87. 87.

    Sweby, P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21, 995–1011 (1984)

  88. 88.

    Tadmor, E.: Approximate solutions of nonlinear conservation laws. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lectures Notes from CIME Course Cetraro, Italy, 1997. Lecture Notes in Mathematics, vol. 1697, pp. 1–150. Springer, Berlin (1998)

  89. 89.

    Tanguay, M., Colonius, T.: Progress in modeling and simulation of shock wave lithotripsy (swl). In: Fifth International Symposium on Cavitation (CAV2003), number OS-2-1-010 (2003)

  90. 90.

    van de Griend, J.A., Kraaijevanger, J.F.B.M.: Absolute monotonicity of rational functions occurring in the numerical solution of initial value problems. Numer. Math. 49, 413–424 (1986)

  91. 91.

    Wang, R., Spiteri, R.J.: Linear instability of the fifth-order WENO method. SIAM J. Numer. Anal. 45(5), 1871–1901 (2007)

  92. 92.

    Wang, Z.J., Liu, Y.: The spectral difference method for the 2D Euler equations on unstructured grids. In: 17th AIAA Computational Fluid Dynamics Conference. AIAA, Washington (2005)

  93. 93.

    Wang, Z.J., Liu, Y., May, G., Jameson, A.: Spectral difference method for unstructured grids II: Extension to the Euler equations. J. Sci. Comput. 32(1), 45–71 (2007)

  94. 94.

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

  95. 95.

    Xia, Y., Xu, Y., Shu, C.-W.: Efficient time discretization for local discontinuous Galerkin methods. Discrete Continuous Dyn. Syst. Ser B 8, 677–693 (2007)

  96. 96.

    Zhang, W., MacFayden, A.I.: RAM: A relativistic adaptive mesh refinement hydrodynamics code. Astrophys. J. Suppl. Ser. 164, 255–279 (2006)

Download references

Author information

Correspondence to Sigal Gottlieb.

Additional information

The work of S. Gottlieb was supported by AFOSR grant number FA9550-06-1-0255.

The work of D.I. Ketcheson was supported by a US Dept. of Energy Computational Science Graduate Fellowship under grant DE-FG02-97ER25308.

The research of C.-W. Shu is supported in part by NSF grants DMS-0510345 and DMS-0809086.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gottlieb, S., Ketcheson, D.I. & Shu, C. High Order Strong Stability Preserving Time Discretizations. J Sci Comput 38, 251–289 (2009) doi:10.1007/s10915-008-9239-z

Download citation


  • Strong stability preserving
  • Runge–Kutta methods
  • Multistep methods
  • Spectral deferred correction methods
  • High order accuracy
  • Time discretization