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A Strong Stability Preserving Analysis for Explicit Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions

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Abstract

High-order strong stability preserving (SSP) time discretizations are often needed to ensure the nonlinear (and sometimes non-inner-product) strong stability properties of spatial discretizations specially designed for the solution of hyperbolic PDEs. Multi-derivative time-stepping methods have recently been increasingly used for evolving hyperbolic PDEs, and the strong stability properties of these methods are of interest. In our prior work we explored time discretizations that preserve the strong stability properties of spatial discretizations coupled with forward Euler and a second-derivative formulation. However, many spatial discretizations do not satisfy strong stability properties when coupled with this second-derivative formulation, but rather with a more natural Taylor series formulation. In this work we demonstrate sufficient conditions for an explicit two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and Taylor series formulation. We call these strong stability preserving Taylor series (SSP-TS) methods. We also prove that the maximal order of SSP-TS methods is \(p=6\), and define an optimization procedure that allows us to find such SSP methods. Several types of these methods are presented and their efficiency compared. Finally, these methods are tested on several PDEs to demonstrate the benefit of SSP-TS methods, the need for the SSP property, and the sharpness of the SSP time-step in many cases.

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Notes

  1. Note that here we use \({\dot{F}}\) to indicate that these methods are designed for the exact time derivative of F. However, in practice we use the approximation \({\tilde{F}}\) as explained above.

  2. In this work we use \(\odot\) to denote component-wise multiplication.

  3. These efficiency measures do not account for the fact that the methods in [32] are of type SSP-TS M3 and so require fewer funding evaluations. Correcting for this, our methods are still 10%–40% more efficient.

References

  1. Bresten, C., Gottlieb, S., Grant, Z., Higgs, D., Ketcheson, D.I., Németh, A.: Strong stability preserving multistep Runge-Kutta methods. Math. Comput. 86, 747–769 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bunya, S., Kubatko, E.J., Westerink, J.J., Dawson, C.: A wetting and drying treatment for the Runge-Kutta discontinuous Galerkin solution to the shallow water equations. Comput. Methods Appl. Mech. Eng. 198, 1548–1562 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chan, R.P.K., Tsai, A.Y.J.: On explicit two-derivative Runge-Kutta methods. Numer. Algorithms 53, 171–194 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cheng, J.B., Toro, E.F., Jiang, S., Tang, W.: A sub-cell WENO reconstruction method for spatial derivatives in the ADER scheme. J. Comput. Phys. 251, 53–80 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Christlieb, A., Gottlieb, S., Grant, Z., Seal, D.C.: Explicit strong stability preserving multistage two-derivative time-stepping schemes. J. Sci. Comput. 68, 914–942 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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 

  7. Daru, V., Tenaud, C.: High order one-step monotonicity-preserving schemes for unsteady compressible flow calculations. J. Comput. Phys. 193, 563–594 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dormand, J.R., Prince, P.J.: A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  9. Du, Z., Li, J.: A Hermite WENO reconstruction for fourth order temporal accurate schemes based on the GRP solver for hyperbolic conservation laws. J. Comput. Phys. 355, 385–396 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dumbser, M., Zanotti, O., Hidalgo, A., Balsara, D.S.: ADER-WENO finite volume schemes with space-time adaptive mesh refinement. J. Comput. Phys. 248, 257–286 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Gottlieb, S., Ketcheson, D.I., Shu, C.-W.: Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations. World Scientific Press, London (2011)

    Book  MATH  Google Scholar 

  14. Gottlieb, S., Grant, Z.J., Seal, D.C.: Explicit SSP multistage two-derivative methods with Taylor series base conditions. https://github.com/SSPmethods/SSPTSmethods. Accessed 1 Mar 2018

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Higueras, I.: Characterizing strong stability preserving additive Runge-Kutta methods. J. Sci. Comput. 39(1), 115–128 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jeltsch, R.: Reducibility and contractivity of Runge-Kutta methods revisited. BIT Numer. Math. 46(3), 567–587 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Kastlunger, K., Wanner, G.: On Turan type implicit Runge-Kutta methods. Computing (Arch. Elektron. Rechnen) 9, 317–325 (1972)

    MathSciNet  MATH  Google Scholar 

  22. Kastlunger, K.H., Wanner, G.: Runge Kutta processes with multiple nodes. Computing (Arch. Elektron. Rechnen) 9, 9–24 (1972)

    MathSciNet  MATH  Google Scholar 

  23. Ketcheson, D.I.: Highly efficient strong stability preserving Runge-Kutta methods with low-storage implementations. SIAM J. Sci. Comput. 30, 2113–2136 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ketcheson, D.I., Gottlieb, S., Macdonald, C.B.: Strong stability preserving two-step Runge-Kutta methods. SIAM J. Numer. Anal. 2618–2639 (2012)

  25. Ketcheson, D.I., Macdonald, C.B., Gottlieb, S.: Optimal implicit strong stability preserving Runge-Kutta methods. Appl. Numer. Math. 52, 373 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ketcheson, D.I., Parsani, M., Ahmadia, A.J.: RK-Opt: software for the design of Runge-Kutta methods, version 0.2. https://github.com/ketch/RK-opt. Accessed 15 Feb 2018

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, J., Du, Z.: A two-stage fourth order time-accurate discretization for Lax-Wendroff type flow solvers I. hyperbolic conservation laws. SIAM J. Sci. Comput. 38, 3046–3069 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Mitsui, T.: Runge-Kutta type integration formulas including the evaluation of the second derivative. i. Publ. Res. Inst. Math. Sci. 18, 325–364 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  32. Nguyen-Ba, T., Nguyen-Thu, H., Giordano, T., Vaillancourt, R.: One-step strong-stability-preserving Hermite–Birkhoff–Taylor methods. Sci. J. Riga Tech. Univ. 45, 95–104 (2010)

    MATH  Google Scholar 

  33. Obreschkoff, N.: Neue quadraturformeln. Abh. Preuss. Akad. Wiss. Math.-Nat. Kl. 1940(4), 20 (1940)

    MathSciNet  MATH  Google Scholar 

  34. Ono, H., Yoshida, T.: Two-stage explicit Runge-Kutta type methods using derivatives. Jpn. J. Ind. Appl. Math. 21, 361–374 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Pan, L., Xu, K., Li, Q., Li, J.: An efficient and accurate two-stage fourth-order gas-kinetic scheme for the Euler and Navier-Stokes equations. J. Comput. Phys. 326, 197–221 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Seal, D.C., Guclu, Y., Christlieb, A.J.: High-order multiderivative time integrators for hyperbolic conservation laws. J. Sci. Comput. 60, 101–140 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Shintani, H.: On one-step methods utilizing the second derivative. Hiroshima Math. J. 1, 349–372 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  41. Shintani, H.: On explicit one-step methods utilizing the second derivative. Hiroshima Math. J. 2, 353–368 (1972)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  44. 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)

    Article  MathSciNet  MATH  Google Scholar 

  45. Stancu, D.D., Stroud, A.H.: Quadrature formulas with simple Gaussian nodes and multiple fixed nodes. Math. Comput. 17, 384–394 (1963)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

  48. Toro, E., Titarev, V.A.: Solution of the generalized Riemann problem for advection-reaction equations. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 458, 271–281 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  49. Toro, E.F., Titarev, V.A.: Derivative Riemann solvers for systems of conservation laws and ADER methods. J. Comput. Phys. 212, 150–165 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  50. Tsai, A.Y.J., Chan, R.P.K., Wang, S.: Two-derivative Runge-Kutta methods for PDEs using a novel discretization approach. Numer. Algorithms 65, 687–703 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  51. Turán, P.: On the theory of the mechanical quadrature. Acta Universitatis Szegediensis Acta Scientiarum Mathematicarum 12, 30–37 (1950)

    MathSciNet  MATH  Google Scholar 

  52. Verner, J.: Explicit Runge-Kutta methods with estimates of the local truncation error. SIAM J. Numer. Anal. 15, 772–790 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  53. Xing, Y., Zhang, X., Shu, C.-W.: Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Adv. Water Resour. 33, 1476–1493 (2010)

    Article  Google Scholar 

  54. Zhang, X., Shu, C.-W.: On Maximum-principle-satisfying High Order Schemes for Scalar Conservation Laws. J. Comput. Phys. 229, 3091–3120 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The work of D. C. Seal was supported in part by the Naval Academy Research Council. The work of S. Gottlieb and Z. J. Grant was supported by the AFOSR Grant #FA9550-15-1-0235. A part of this research is sponsored by the Office of Advanced Scientific Computing Research; US Department of Energy, and was performed at the Oak Ridge National Laboratory, which is managed by UT-Battelle, LLC under Contract no. De-AC05-00OR22725. This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.

Author information

Authors and Affiliations

  1. Department of Computational and Applied Mathematics, Oak Ridge National Laboratory, Oak Ridge, USA

    Zachary Grant

  2. Department of Mathematics, University of Massachusetts Dartmouth, Dartmouth, USA

    Sigal Gottlieb

  3. Department of Mathematics, U.S. Naval Academy, Annapolis, USA

    David C. Seal

Authors
  1. Zachary Grant
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  2. Sigal Gottlieb
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  3. David C. Seal
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Corresponding author

Correspondence to Zachary Grant.

Appendices

Appendix 1 Order Conditions

Any method of the form (9) must satisfy the order conditions for all \(p \le P\) to be of order P.

\(p = 1,\)

$$\begin{aligned} b^{\text {T}} e =1; \end{aligned}$$

\(p = 2,\)

$$\begin{aligned} b^{\text {T}} c+{\hat{b}}^{\text {T}}e =\frac{1}{2}; \end{aligned}$$

\(p = 3,\)

$$\begin{aligned}&b^{\text {T}} c^2 + 2{\hat{b}}^{\text {T}} c=\frac{1}{3}, \\&b^{\text {T}}Ac+b^{\text {T}}{\hat{c}}+{\hat{b}}^{\text {T}}c=\frac{1}{6}; \end{aligned}$$

\(p = 4,\)

$$\begin{aligned}&b^{\text {T}}c^3+3{\hat{b}}^{\text {T}}c^2=\frac{1}{4}, \\&b^{\text {T}}\left( c \odot Ac\right) +b^{\text {T}}\left( c \odot {\hat{c}}\right) +{\hat{b}}^{\text {T}}c^2+{\hat{b}}^{\text{T}}Ac+{\hat{b}}^{\text {T}}{\hat{c}}=\frac{1}{8},\\&b^{\text {T}}Ac^2+2b^{\text {T}}{\hat{A}}c+{\hat{b}}^{\text {T}}c^2=\frac{1}{12}, \\&b^{\text {T}}A^2c+ b^{\text {T}}A{\hat{c}}+ b^{\text {T}}{\hat{A}}c+{\hat{b}}^{\text {T}}Ac+{\hat{b}}^{\text{T}}{\hat{c}}=\frac{1}{24}; \end{aligned}$$

\(p = 5,\)

$$\begin{aligned}&b^{\text {T}}c^4 + 4{\hat{b}}^{\text {T}}c^3 =\frac{1}{5} , \\&b^{\text {T}}\left( c^2 \odot Ac\right) + b^{\text {T}}\left( c^2 \odot {\hat{c}}\right) +{\hat{b}}^{\text {T}}c^3+2{\hat{b}}^{\text{T}}\left( c \odot Ac\right) +2{\hat{b}}^{\text {T}}\left( c \odot {\hat{c}}\right) =\frac{1}{10},\\&b^{\text {T}}\left( c \odot Ac^2\right) +2b^{\text {T}}\left( c \odot {\hat{A}}c\right) +{\hat{b}}^{\text {T}}c^3+{\hat{b}}^{\text{T}} Ac^2+2{\hat{b}}^{\text {T}}{\hat{A}}c=\frac{1}{15} ,\\&b^{\text {T}}\left( c \odot A^2c \right) +b^{\text {T}}\left( c \odot A{\hat{c}}\right) +b^{\text {T}}\left( c \odot {\hat{A}}c\right) +{\hat{b}}^{\text {T}}\left( c \odot Ac\right) +{\hat{b}}^{\text {T}}\left( c \odot {\hat{c}}\right) +{\hat{b}}^{\text {T}}A^2c+{\hat{b}}^{\text{T}}A{\hat{c}}+{\hat{b}}^{\text {T}}{\hat{A}}c=\frac{1}{30},\\&b^{\text {T}}\left( Ac \odot Ac \right) +2b^{\text {T}}\left( {\hat{c}} \odot Ac\right) +b^{\text {T}}{\hat{c}}^2+ 2{\hat{b}}^{\text {T}}\left( c \odot Ac\right) +2{\hat{b}}^{\text {T}}\left( c \odot {\hat{c}}\right) =\frac{1}{20},\\&b^{\text {T}}Ac^3+3b^{\text {T}}{\hat{A}}c^2+{\hat{b}}^{\text {T}}c^3=\frac{1}{20}, \\&b^{\text {T}}A\left( c \odot Ac\right) +b^{\text {T}}A\left( c \odot {\hat{c}}\right) +b^{\text {T}}{\hat{A}}c^2+b^{\text {T}}{\hat{A}}Ac+b^{\text {T}}{\hat{A}}{\hat{c}} + {\hat{b}}^{\text {T}}\left( c \odot Ac\right) +{\hat{b}}^{\text {T}}\left( c \odot {\hat{c}}\right) =\frac{1}{40},\\&b^{\text {T}}A^2c^2+2b^{\text {T}}A{\hat{A}}c+b^{\text {T}}{\hat{A}}c^2+{\hat{b}}^{\text {T}} Ac^2+2{\hat{b}}^{\text{T}}{\hat{A}}c=\frac{1}{60}, \\&b^{\text {T}}A^3 c+b^{\text {T}}A^2 {\hat{c}}+b^{\text {T}} A {\hat{A}}c+b^{\text {T}}{\hat{A}}Ac+b^{\text {T}}{\hat{A}}{\hat{c}} +{\hat{b}}^{\text {T}}A^2c +{\hat{b}}^{\text{T}}A{\hat{c}}+{\hat{b}}^{\text {T}}{\hat{A}}c=\frac{1}{120}; \end{aligned}$$

\(p = 6,\)

$$\begin{aligned}&b^{\text {T}} c^5 + 5{\hat{b}}^{\text {T}} c^4 = \frac{1}{6},\\&b^{\text {T}} \left( c^3 \odot Ac\right) + 3{\hat{b}}^{\text {T}} \left( c^2 \odot Ac \right) + {\hat{b}}^{\text {T}} c^4 + b^{\text {T}} \left( c^3 \odot {\hat{c}}\right) + 3{\hat{b}}^{\text {T}} \left( c^2 \odot {\hat{c}}\right) = \frac{1}{12},\\&b^{\text {T}} \left( c^2 \odot A c^2\right) + 2{\hat{b}}^{\text {T}}\left( c \odot A c^2\right) + 2b^{\text {T}} \left( c^2\odot {\hat{A}}c\right) + {\hat{b}}^{\text {T}} c^4 + 4{\hat{b}}^{\text{T}}\left( c \odot {\hat{A}}c\right) = \frac{1}{18},\\&b^{\text {T}}\left( c \odot A c^3\right) + 3b^{\text {T}}\left( c \odot {\hat{A}} c^2\right) + {\hat{b}}^{\text {T}}A c^3 + 3{\hat{b}}^{\text{T}}{\hat{A}} c^2 + {\hat{b}}^{\text {T}} c^4 = \frac{1}{24}, \\&b^{\text {T}}A c^4 + 4b^{\text {T}}{\hat{A}} c^3 + {\hat{b}}^{\text {T}} c^4 = \frac{1}{30},\\&b^{\text {T}}\left( c^2 \odot A^2c\right) + 2{\hat{b}}^{\text {T}}\left( c \odot A^2c\right) + b^{\text {T}} \left( c^2 \odot A{\hat{c}}\right) + b^{\text {T}} \left( c^2 \odot {\hat{A}}c\right) + {\hat{b}}^{\text {T}} \left( c^2 \odot Ac\right) + 2{\hat{b}}^{\text {T}}\left( c \odot A{\hat{c}}\right)\\&+ 2{\hat{b}}^{\text {T}}\left( c \odot {\hat{A}}c\right) + {\hat{b}}^{\text {T}} \left( c^2 \odot {\hat{c}}\right) = \frac{1}{36}, \\&b^{\text {T}}\left( c \odot A^2 c^2\right) + {\hat{b}}^{\text {T}}A^2 c^2 + {\hat{b}}^{\text{T}}\left( c \odot A c^2\right) + b^{\text {T}}\left( c \odot {\hat{A}} c^2\right) + 2b^{\text {T}}\left( c \odot A{\hat{A}}c\right) + {\hat{b}}^{\text {T}}{\hat{A}} c^2 + 2{\hat{b}}^{\text {T}}A{\hat{A}}c \\&+ 2{\hat{b}}^{\text {T}}\left( c \odot {\hat{A}}c\right) =\frac{1}{72},\\&b^{\text {T}}A^2 c^3 + {\hat{b}}^{\text {T}}A c^3 + b^{\text {T}}{\hat{A}} c^3 + 3b^{\text {T}}A{\hat{A}} c^2 + 3{\hat{b}}^{\text {T}}{\hat{A}} c^2 = \frac{1}{120},\\&b^{\text {T}}\left( c \odot Ac \odot Ac\right) + {\hat{b}}^{\text {T}}\left( Ac \odot Ac\right) + b^{\text {T}}\left( c \odot {\hat{A}}Ac\right) + b^{\text {T}}\left( c\odot A\left( c\odot {\hat{c}}\right) \right) + {\hat{b}}^{\text {T}} \left( c^2 \odot Ac\right) + b^{\text {T}}\left( c \odot {\hat{A}} c^2\right) \\& +{\hat{b}}^{\text {T}}{\hat{A}}Ac + {\hat{b}}^{\text {T}}A\left( c \odot {\hat{c}}\right) + b^{\text {T}}\left( c \odot {\hat{A}}{\hat{c}}\right) + {\hat{b}}^{\text {T}}{\hat{A}} c^2 + {\hat{b}}^{\text {T}} c^2{\hat{c}} + {\hat{b}}^{\text {T}}{\hat{A}}{\hat{c}} = \frac{1}{48}, \\&b^{\text {T}}A \left( c^2 \odot Ac\right) + b^{\text {T}}A \left( c^2 \odot {\hat{c}}\right) + {\hat{b}}^{\text {T}} \left( c^2 \odot Ac\right) + 2b^{\text {T}}\left( {\hat{A}}c \odot Ac\right) + b^{\text {T}}{\hat{A}} c^3 + 2b^{\text {T}}\left( {\hat{A}}c \odot {\hat{c}}\right) \\& + {\hat{b}}^{\text {T}} \left( c^2 \odot {\hat{c}}\right) = \frac{1}{60},\\&b^{\text {T}}\left( Ac \odot A c^2\right) + {\hat{b}}^{\text {T}}\left( c \odot A c^2\right) + b^{\text {T}}{\hat{A}}A c^2 + b^{\text {T}}{\hat{A}} c^3 + 2b^{\text {T}}\left( Ac \odot {\hat{A}}c\right) + 2b^{\text {T}}{\hat{A}}^2c + 2{\hat{b}}^{\text {T}}\left( c \odot {\hat{A}}c\right) = \frac{1}{90}, \\&b^{\text {T}}\left( c \odot A^3c\right) + {\hat{b}}^{\text {T}}A^3c + {\hat{b}}^{\text{T}}\left( c \odot A^2c \right) + b^{\text {T}}\left( c \odot {\hat{A}}Ac\right) + b^{\text {T}}\left( c \odot A{\hat{A}}c\right) + b^{\text {T}}\left( c \odot A^2{\hat{c}}\right) + {\hat{b}}^{\text {T}}{\hat{A}}Ac\\& + {\hat{b}}^{\text {T}}A{\hat{A}}c + {\hat{b}}^{\text {T}}A^2{\hat{c}} + b^{\text {T}}\left( c \odot {\hat{A}}{\hat{c}}\right) + {\hat{b}}^{\text {T}}CA{\hat{c}} + {\hat{b}}^{\text {T}}C{\hat{A}}c + {\hat{b}}^{\text {T}}{\hat{A}}{\hat{c}} = \frac{1}{144},\\&b^{\text {T}}\left( Ac \odot A^2c\right) + b^{\text {T}}\left( Ac \odot A{\hat{c}}\right) + b^{\text {T}}\left( Ac \odot {\hat{A}}c\right) + b^{\text {T}}\left( {\hat{A}}c \odot Ac\right) + b^{\text {T}}{\hat{A}}A^2c + {\hat{b}}^{\text {T}}\left( c \odot A^2c\right) \\& + b^{\text {T}}\left( {\hat{A}}c \odot {\hat{c}}\right) + b^{\text {T}}{\hat{A}}A{\hat{c}} + {\hat{b}}^{\text {T}}\left( c \odot A{\hat{c}}\right) + b^{\text {T}}{\hat{A}}^2c + {\hat{b}}^{\text {T}}\left( c \odot {\hat{A}}c\right) = \frac{1}{180},\\&b^{\text {T}}(A^2\left( c \odot Ac\right) + b^{\text {T}}A^2\left( c \odot {\hat{c}}\right) + b^{\text {T}}A{\hat{A}}c^2 + b^{\text {T}}A{\hat{A}}Ac + b^{\text {T}}{\hat{A}}\left( c \odot Ac\right) + {\hat{b}}^{\text {T}}A\left( c \odot Ac\right) + b^{\text {T}}A{\hat{A}}{\hat{c}} \\& + b^{\text {T}}{\hat{A}}\left( c \odot {\hat{c}}\right) + {\hat{b}}^{\text {T}}A\left( c \odot {\hat{c}}\right) + {\hat{b}}^{\text {T}}{\hat{A}}c^2 + {\hat{b}}^{\text {T}}{\hat{A}}Ac + {\hat{b}}^{\text {T}}{\hat{A}}{\hat{c}} = \frac{1}{240},\\&b^{\text {T}}A^3 c^2 + {\hat{b}}^{\text {T}}A^2 c^2 + b^{\text {T}}{\hat{A}}A c^2 + b^{\text {T}}A{\hat{A}} c^2 + 2b^{\text {T}}A^2{\hat{A}}c + {\hat{b}}^{\text {T}}{\hat{A}} c^2 + 2{\hat{b}}^{\text {T}}A{\hat{A}}c + 2b^{\text {T}}{\hat{A}}^2c = \frac{1}{360},\\&b^{\text {T}}\left( c \odot Ac \odot Ac\right) + {\hat{b}}^{\text {T}}\left( Ac \odot Ac\right) + 2b^{\text {T}}\left( c \odot {\hat{c}} \odot Ac\right) + 2{\hat{b}}^{\text {T}}\left( c^2 \odot Ac\right) + 2{\hat{b}}^{\text {T}}\left( {\hat{c}} \odot Ac\right) + 2{\hat{b}}^{\text {T}}\left( c^2 \odot {\hat{c}}\ \right) \\& + b^{\text {T}}\left( c \odot {\hat{c}}^2\right) + {\hat{b}}^{\text {T}} {\hat{c}}^2 = \frac{1}{24}, \end{aligned}$$
$$\begin{aligned}&b^{\text {T}}\left( Ac \odot A^2c\right) + b^{\text {T}}\left( {\hat{c}} \odot A^2c\right) + {\hat{b}}^{\text {T}}\left( c \odot A^2c\right) + {\hat{b}}^{\text {T}}\left( Ac \odot Ac\right) + b^{\text {T}}\left( Ac \odot {\hat{A}}c\right) + b^{\text {T}}\left( Ac \odot A{\hat{c}}\right) \\& + 2 {\hat{b}}^{\text {T}}\left( {\hat{c}} \odot Ac\right) + b^{\text {T}}\left( {\hat{c}} \odot {\hat{A}}c\right) + b^{\text {T}}\left( {\hat{c}} \odot A{\hat{c}}\right) + {\hat{b}}^{\text {T}}\left( c \odot {\hat{A}}c\right) + {\hat{b}}^{\text {T}}\left( c \odot A{\hat{c}}\right) + {\hat{b}}^{\text {T}} {\hat{c}}^2 = \frac{1}{72},\\&b^{\text {T}}\left( Ac \odot Ac^2\right) + b^{\text {T}}\left( {\hat{c}} \odot Ac^2\right) + {\hat{b}}^{\text {T}}\left( c \odot Ac^2\right) + {\hat{b}}^{\text {T}}\left( Ac \odot c^2\right) + 2b^{\text {T}}\left( Ac \odot {\hat{A}}c\right) + {\hat{b}}^{\text {T}}\left( {\hat{c}} \odot c^2\right) \\& + 2b^{\text {T}}\left( {\hat{c}} \odot {\hat{A}}c\right) + 2{\hat{b}}^{\text {T}}\left( c \odot {\hat{A}}c\right) = \frac{1}{36},\\&b^{\text {T}}A\left( Ac \odot Ac\right) + 2b^{\text {T}}A\left( {\hat{c}} \odot Ac\right) + 2b^{\text {T}}{\hat{A}}\left( c \odot Ac\right) + {\hat{b}}^{\text {T}}\left( Ac \odot Ac\right) + 2{\hat{b}}^{\text {T}}\left( {\hat{c}} \odot Ac\right) + 2b^{\text {T}}{\hat{A}}\left( c \odot {\hat{c}}\right) \\& + b^{\text {T}}A{\hat{c}}^2 + {\hat{b}}^{\text {T}} {\hat{c}}^2 = \frac{1}{120},\\&b^{\text {T}}A^4c + b^{\text {T}}A^3{\hat{c}} + b^{\text {T}}A^2{\hat{A}}c + b^{\text {T}}A{\hat{A}}Ac + b^{\text {T}}{\hat{A}}A^2c + {\hat{b}}^{\text {T}}A^3c + b^{\text {T}}A{\hat{A}}{\hat{c}} + b^{\text {T}}{\hat{A}}A{\hat{c}} + {\hat{b}}^{\text {T}}A^2{\hat{c}} \\& + b^{\text {T}}{\hat{A}}^2c + {\hat{b}}^{\text {T}}A{\hat{A}}c + {\hat{b}}^{\text {T}}{\hat{A}}Ac + {\hat{b}}^{\text {T}}{\hat{A}}{\hat{c}} = \frac{1}{720}. \end{aligned}$$

Appendix 2 Coefficients of Selected Methods

All the time-stepping methods in this work can be downloaded as Matlab files from [14]. In this appendix we present selected methods.

SSP-TS M2(4,5,1) This method has \(\mathcal{{C}}_{\text {TS}}= 2.186\,48,\)

$$\begin{aligned} \begin{array}{lll} a_{21}=4.280\,141\,748\,183\,123{\text {E}}-01 ,\; \; & {\hat{a}}_{21}=9.159\,806\,692\,270\,039{\text {E}}-02, \; \; & b_{1}=3.456\,442\,194\,983\,256{\text {E}}-01, \\ a_{31}=3.174\,364\,422\,211\,321{\text {E}}-01, & {\hat{a}}_{31}=2.068\,159\,838\,961\,376{\text {E}}-02, & b_{2}=1.551\,487\,425\,849\,178{\text {E}}-01, \\ a_{32}=1.032\,647\,478\,325\,804{\text {E}}-01 , & {\hat{a}}_{32}=2.361\,437\,143\,530\,821{\text {E}}-02 , & b_{3}=3.458\,932\,447\,335\,502{\text {E}}-01, \\ a_{41}=3.280\,547\,501\,426\,051{\text {E}}-01, & {\hat{a}}_{41}=1.869\,435\,227\,642\,530{\text {E}}-02 , & b_{4}=1.533\,137\,931\,832\,064{\text {E}}-01, \\ a_{42}=9.334\,228\,125\,655\,676{\text {E}}-02 , & {\hat{a}}_{42}=2.134\,532\,206\,271\,365{\text {E}}-02 , & {\hat{b}}_{1}=3.226\,836\,941\,745\,746{\text {E}}-02, \\ a_{43}=4.134\,096\,583\,922\,347E-01 , & {\hat{a}}_{43}=9.453\,767\,556\,809\,974{\text {E}}-02, &{\hat{b}}_2=1.785\,928\,934\,720\,153{\text {E}}-02, \\ & & {\hat{b}}_3=7.490\,191\,551\,289\,183{\text {E}}-02, \\ & & {\hat{b}}_4=3.505\,948\,481\,328\,697{\text {E}}-02. \\ \end{array} \end{aligned}$$

SSP-TS M3(8,6,1) This method has \(\mathcal{{C}}_{\text {TS}}= 1.736\,9,\)

$$\begin{aligned} \begin{array}{lll} a_{21}=3.498\,630\,949\,258\,150{\text {E}}-01,\;\; & a_{65}=3.779\,563\,241\,192\,044{\text {E}}-01, \;\;& {\hat{a}}_{21}=6.120\,209\,259\,553\,491{\text {E}}-02, \\ a_{31}=2.253\,295\,269\,463\,227{\text {E}}-01 , & a_{71}=2.148\,681\,581\,922\,796{\text {E}}-01 , & {\hat{a}}_{31}=1.921\,063\,160\,949\,869{\text {E}}-02, \\ a_{32}=1.807\,161\,013\,759\,724{\text {E}}-01 , & a_{72}=1.533\,420\,472\,452\,636{\text {E}}-01 , &{\hat{a}}_{41}=4.358\,605\,297\,856\,505{\text {E}}-03, \\ a_{41}=2.071\,695\,605\,568\,409{\text {E}}-01 , & a_{73}=1.813\,808\,417\,863\,181{\text {E}}-02 , & {\hat{a}}_{51}=2.136\,333\,816\,692\,593{\text {E}}-03, \\ a_{42}=4.100\,178\,308\,548\,576{\text {E}}-02 , & a_{74}=7.994\,387\,176\,143\,736{\text {E}}-02 , & {\hat{a}}_{61}=1.402\,456\,855\,983\,780{\text {E}}-03, \\ a_{43}=1.306\,253\,212\,278\,126{\text {E}}-01 , & a_{75}=1.630\,752\,796\,649\,391{\text {E}}-01 , &{\hat{a}}_{71}=1.631\,142\,330\,728\,269{\text {E}}-02, \\ a_{51}=1.667\,117\,585\,911\,237{\text {E}}-01 , & a_{76}=2.484\,093\,806\,816\,690{\text {E}}-01 , &{\hat{a}}_{81}=1.548\,804\,492\,637\,956{\text {E}}-02, \\ a_{52}=2.009\,667\,996\,165\,993{\text {E}}-02 , & a_{81}=2.036\,762\,412\,289\,922{\text {E}}-01 , &b_1\;\;=1.927\,179\,349\,665\,056{\text {E}}-01, \\ a_{53}=6.402\,490\,521\,280\,881{\text {E}}-02 , & a_{82}=1.456\,707\,401\,767\,411{\text {E}}-01 , &b_2\;\;=7.457\,643\,792\,836\,192{\text {E}}-02, \\ a_{54}=2.821\,909\,187\,189\,924{\text {E}}-01 , & a_{83}=2.379\,744\,031\,395\,224{\text {E}}-02 , &b_3 \;\;=1.097\,549\,250\,079\,706{\text {E}}-01, \\ a_{61}=1.493\,141\,923\,275\,556{\text {E}}-01 , & a_{84}=1.048\,777\,345\,557\,326{\text {E}}-01 , &b_4\;\;=1.166\,274\,027\,628\,658{\text {E}}-01, \\ a_{62}=1.319\,303\,489\,675\,465{\text {E}}-02 , & a_{85}=2.139\,668\,745\,571\,685{\text {E}}-01 , &b_5 \;\;=1.862\,061\,970\,475\,841{\text {E}}-01, \\ a_{63}=4.203\,095\,914\,776\,495{\text {E}}-02 , & a_{86}=6.560\,681\,670\,556\,633{\text {E}}-02 , & b_6\;\;=1.088\,089\,628\,270\,683{\text {E}}-01, \\ a_{64}=1.852\,522\,020\,371\,737{\text {E}}-01 , & a_{87}=1.520\,556\,075\,200\,664{\text {E}}-01 , & b_7\;\;=4.414\,821\,350\,738\,243{\text {E}}-02, \\ &&b_8\;\;=1.671\,599\,259\,522\,612{\text {E}}-01, \\ &&{\hat{b}}_1\;\;=1.1565\,185\,169\,801\,32{\text {E}}-02.\\ \end{array} \end{aligned}$$

Appendix 3: Fifth-Order WENO Method of Jiang and Shu

To solve the PDE

$$\begin{aligned} u_t + f(u)_x = 0, \end{aligned}$$

we approximate the spatial derivative to obtain \(F(u) \approx -f(u)_x\), and then use a time-stepping method to solve the resulting system of ODEs. In this section we describe the fifth-order WENO scheme presented in [20].

First, we split the flux into the positive and negative parts

$$\begin{aligned} f(u) = f^{+}(u) + f^{-}(u). \end{aligned}$$

This can be accomplished in a variety of ways, e.g., the Lax-Friedrichs flux splitting

$$\begin{aligned} f^{+}(u) = \frac{1}{2} \left( f(u) + m u \right), \; \; \; \; f^{-}(u) = \frac{1}{2} \left( f(u) - m u \right) , \end{aligned}$$

where \(m = \max |f'(u)|\). In this way we ensure that \(\frac{{\text{d}} f^{+}}{{\text{d}}u} \ge 0\) and \(\frac{{\text{d}} f^{-}}{{\text{d}}u} \le 0\).

To calculate the numerical fluxes \({\hat{f}}^{+}_{j+ \frac{1}{2}}\) and \({\hat{f}}^{-}_{j+ \frac{1}{2}}\), we begin by calculating the smoothness measurements to determine if a shock lies within the stencil. For our fifth-order scheme, these are

$$\begin{aligned} IS_0^+& = \frac{13}{12} \left( f^+_{j-2} -2 f^+_{j-1} + f^+_{j} \right) ^2 + \frac{1}{4} \left( f^+_{j-2} -4 f^+_{j-1} + 3 f^+_{j} \right) ^2, \\ IS_1^+& = \frac{13}{12} \left( f^+_{j-1} -2 f^+_{j} + f^+_{j+1} \right) ^2 + \frac{1}{4} \left( f^+_{j-1} - f^+_{j+1} \right) ^2, \\ IS_2^+& = \frac{13}{12} \left( f^+_{j} -2 f^+_{j+1} + f^+_{j+2} \right) ^2 + \frac{1}{4} \left( 3 f^+_{j} -4 f^+_{j+1} + f^+_{j+2} \right) ^2, \end{aligned}$$

and

$$\begin{aligned} IS_0^-& = \frac{13}{12} \left( f^-_{j+1} -2 f^-_{j+2} + f^-_{j+3} \right) ^2 + \frac{1}{4} \left( 3 f^-_{j+1} -4 f^-_{j+2} + f^-_{j+3} \right) ^2,\\ IS_1^-& = \frac{13}{12} \left( f^-_{j} -2 f^-_{j+1} + f^-_{j+2} \right) ^2 + \frac{1}{4} \left( f^-_{j} - f^-_{j+2} \right) ^2, \\ IS_2^-& = \frac{13}{12} \left( f^-_{j-1} -2 f^-_{j} + f^-_{j+1} \right) ^2 + \frac{1}{4} \left( f^-_{j-1} -4 f^-_{j} + 3 f^-_{j+1} \right) ^2 \; . \end{aligned}$$

Next, we use the smoothness measurements to calculate the stencil weights

$$\begin{aligned} \alpha ^\pm _0 = \frac{1}{10} \left( \frac{1}{\epsilon + IS_0^\pm }\right) ^2, \; \; \; \; \; \alpha ^\pm _1 = \frac{6}{10} \left( \frac{1}{\epsilon + IS_1^\pm }\right) ^2, \; \; \; \; \; \alpha ^\pm _2 = \frac{3}{10} \left( \frac{1}{\epsilon + IS_2^\pm }\right) ^2, \end{aligned}$$

and

$$\begin{aligned} \omega ^\pm _0 = \frac{\alpha ^\pm _0}{\alpha ^\pm _0 + \alpha ^\pm _1 + \alpha ^\pm _2}, \; \; \; \; \; \omega ^\pm _1 = \frac{\alpha ^\pm _1}{\alpha ^\pm _0 + \alpha ^\pm _1 + \alpha ^\pm _2}, \; \; \; \; \; \omega ^\pm _2 = \frac{\alpha ^\pm _2}{\alpha ^\pm _0 + \alpha ^\pm _1 + \alpha ^\pm _2}. \end{aligned}$$

Finally, the numerical fluxes are

$$\begin{aligned} {\hat{f}}^{+}_{j+ \frac{1}{2}}& = \omega ^+_0 \left( \frac{2}{6} f^+_{j-2} -\frac{7}{6} f^+_{j-1} + \frac{11}{6} f^+_{j} \right) + \omega ^+_1 \left( - \frac{1}{6} f^+_{j-1} + \frac{5}{6} f^+_{j} + \frac{2}{6} f^+_{j+1} \right) \\&\quad + \omega ^+_2 \left( \frac{2}{6} f^+_{j} + \frac{5}{6} f^+_{j+1} - \frac{1}{6} f^+_{j+2} \right), \end{aligned}$$

and

$$\begin{aligned} {\hat{f}}^{-}_{j+ \frac{1}{2}}& = \omega ^-_2 \left( - \frac{1}{6} f^-_{j-1} + \frac{5}{6} f^-_{j} + \frac{2}{6} f^-_{j+1} \right) + \omega ^-_1 \left( \frac{2}{6} f^-_{j} + \frac{5}{6} f^-_{j+1} - \frac{1}{6} f^-_{j+2} \right) \\&\quad + \omega ^-_0 \left( \frac{11}{6} f^-_{j+1} - \frac{7}{6} f^-_{j+2} + \frac{2}{6} f^-_{j+3} \right) . \end{aligned}$$

Finally, we compute

$$\begin{aligned} {\text {WENO}}^+ f^{+}(u)& = - \frac{1}{\Delta x} \left( {\hat{f}}^{+}_{j+ \frac{1}{2}} - {\hat{f}}^{+}_{j- \frac{1}{2}} \right), \end{aligned}$$
(41)
$$\begin{aligned} {\text {WENO}}^- f^{-}(u)& = - \frac{1}{\Delta x} \left( {\hat{f}}^{-}_{j+ \frac{1}{2}} - {\hat{f}}^{-}_{j- \frac{1}{2}} \right), \end{aligned}$$
(42)

and put it all together

$$\begin{aligned} F(u) = {\text {WENO}}^+ f^{+}(u) + {\text {WENO}}^- f^{-}(u) \approx -f(u)_x . \end{aligned}$$

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Grant, Z., Gottlieb, S. & Seal, D.C. A Strong Stability Preserving Analysis for Explicit Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions. Commun. Appl. Math. Comput. 1, 21–59 (2019). https://doi.org/10.1007/s42967-019-0001-3

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