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Lax-Wendroff Approximate Taylor Methods with Fast and Optimized Weighted Essentially Non-oscillatory Reconstructions

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Abstract

The goal of this work is to introduce new families of shock-capturing high-order numerical methods for systems of conservation laws that combine Fast WENO (FWENO) and Optimal WENO (OWENO) reconstructions with Approximate Taylor methods for the time discretization. FWENO reconstructions are based on smoothness indicators that require a lower number of calculations than the standard ones. OWENO reconstructions are based on a definition of the nonlinear weights that allows one to unconditionally attain the optimal order of accuracy regardless of the order of critical points. Approximate Taylor methods update the numerical solutions by using a Taylor expansion in time in which, instead of using the Cauchy–Kovalevskaya procedure, the time derivatives are computed by combining spatial and temporal numerical differentiation with Taylor expansions in a recursive way. These new methods are compared between them and against methods based on standard WENO implementations and/or SSP-RK time discretization. A number of test cases are considered ranging from scalar linear 1d problems to nonlinear systems of conservation laws in 2d.

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References

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Gottlieb, S., Shu, C.W.: Total variation diminishing Runge-Kutta schemes. Math. Comput. 67(221), 73–85 (1998)

    Article  MathSciNet  Google Scholar 

  4. Gottlieb, S., Ketcheson, D., Shu, C.W.: Strong stability preserving Runge-Kutta and multistep time discretizations. Word Scientific, 1 edition (2011)

  5. Arándiga, F., Baeza, A., Belda, A.M., Mulet, P.: Analysis of WENO schemes for full and global accuracy. SIAM J. Numer. Anal. 49, 893–915 (2011)

    Article  MathSciNet  Google Scholar 

  6. Arándiga, F., Martí, M.C., Mulet, P.: Weights design for maximal order WENO scheme. J. Sci. Comput. 60, 641–659 (2014)

    Article  MathSciNet  Google Scholar 

  7. Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230(5), 1766–1792 (2011)

    Article  MathSciNet  Google Scholar 

  8. Henrick, Andrew K., Aslam, Tariq D., Powers, Joseph M.: Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207(2), 542–567 (2005)

    Article  Google Scholar 

  9. Yamaleev, N.K., Carpenter, M.H.: A systematic methodology to for constructing high-order energy stable weno schemes. J. Comput. Phys. 11, 4248–4272 (2009)

    Article  MathSciNet  Google Scholar 

  10. Baeza, A., Bürger, R., Mulet, P., Zorío, D.: An efficient third-order WENO scheme with unconditionally optimal accuracy. SIAM J. Sci. Comput. (To appear) (2020)

  11. Baeza, A., Bürger, R., Mulet, P., Zorío, D.: WENO reconstructions of unconditionally optimal high order. SIAM J. Numer. Anal. 57, 2760–2784 (2019)

    Article  MathSciNet  Google Scholar 

  12. Baeza, A., Bürger, R., Mulet, P., Zorío, D.: On the efficient computation of smoothness indicators for a class of WENO reconstructions. J. Sci. Comput. 80, 1240–1263 (2019)

    Article  MathSciNet  Google Scholar 

  13. Millington, R.C., Toro, E.F., Nejad, L.A.M.: Towards very high order godunov schemes. In: Toro, E.F. (ed.) Godunov Methods. Theory and Applications, pp. 905–938. Kluwer/Plenum Academic Publishers, Amsterdam (2001)

    Google Scholar 

  14. Titarev, V.A., Toro, E.F.: ADER: arbitrary high order godunov approach. J. Sci. Comput. 17, 609–618 (2002)

    Article  MathSciNet  Google Scholar 

  15. Munz, C.D., Schwartzkopff, T., Toro, E.F.: A high-order approach for linear hyperbolic systems in 2d. J. Sci. Comput. 17, 231–240 (2002)

    Article  MathSciNet  Google Scholar 

  16. Dumbser, M., Enaux, C., Toro, E.F.: Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J. Comput. Phys. 227(2), 3971–4001 (2008)

    Article  MathSciNet  Google Scholar 

  17. Dumbser, M., Balsara, D., Toro, E.F., Munz, C.D.: A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. J. Comput. Phys. 227, 8209–8253 (2008)

    Article  MathSciNet  Google Scholar 

  18. Zorío, D., Baeza, A., Mulet, P.: An approximate Lax-Wendroff-type procedure for high order accurate schemes for hyperbolic conservation laws. J. Sci. Comput. 71, 246–273 (2017)

    Article  MathSciNet  Google Scholar 

  19. LeVeque, R.J.: Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems (Classics in Applied Mathematics). Society for Industrial and Applied Mathematics, Philadelpia, PA. USA., 1 edition, (2007)

  20. Carrillo, H., Parés, C.: Compact approximate Taylor methods for systems of conservation laws. J. Sci. Comput. 80, 1832–1866 (2019)

    Article  MathSciNet  Google Scholar 

  21. Shu, C.-W., Osher, S.: Primitive, conservative and adaptive schemes for hyperbolic conservation laws. J. Comput. Phys. 83, 32–78 (1989)

    Article  MathSciNet  Google Scholar 

  22. Henrick, Andrew K., Aslam, Tariq D., Powers, Joseph M.: Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points. J. Comput. Phys. 207(2), 542–567 (2005)

    Article  Google Scholar 

  23. Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory weno-z schemes for hyperbolic conservation laws. J. Comput. Phys. 230(5), 1766–1792 (2011)

    Article  MathSciNet  Google Scholar 

  24. Sod, G.A.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27(1), 1–31 (1978)

    Article  MathSciNet  Google Scholar 

  25. Einfeldt, B., Roe, P.L., Munz, C.D., Sjogreen, B.: On Godunov-type methods near low densities. J. Comput. Phys. 92, 273–295 (1991)

    Article  MathSciNet  Google Scholar 

  26. Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 1, 115–173 (1984)

    Article  MathSciNet  Google Scholar 

  27. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edn. Springer, Berlin (2009)

    Book  Google Scholar 

  28. Donat, R., Marquina, A.: Capturing shock reflections: an improved flux formula. J. Comput. Phys. 125(1), 42–58 (1996)

    Article  MathSciNet  Google Scholar 

  29. Lax, P., Xu-Dong, L.: Solution of two-dimensional riemann problems of gas dynamics by positive schemes. SIAM J. Sci. Comput. 19(2), 319–340 (1998)

    Article  MathSciNet  Google Scholar 

  30. Kurganov, A., Tadmor, E.: Solution of two-dimensional riemann problems for a gas dynamics without Riemann problem solvers. Numer. Methods Part. Differ. Equ. 18, 584–608 (2002)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This research has received funding from the European Union’s Horizon 2020 research and innovation program, under the Marie Sklodowska-Curie grant agreement No 642768. It has been also partially supported by the Spanish Government and FEDER through the Research project RTI2018-096064-B-C21. D. Zorío is also supported by Fondecyt Project 3170077.

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Authors and Affiliations

  1. ITESRC, Km 120, Agujita, Coahuila, Mexico

    H. Carrillo

  2. Departamento de Matemática Aplicada, Universidad de Málaga, Avda. Cervantes, 2., 29071, Málaga, Spain

    C. Parés

  3. CI2MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    D. Zorío

  4. Universidad Internacional de Valencia, Valencia, Spain

    D. Zorío

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  1. H. Carrillo
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Correspondence to H. Carrillo.

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Carrillo, H., Parés, C. & Zorío, D. Lax-Wendroff Approximate Taylor Methods with Fast and Optimized Weighted Essentially Non-oscillatory Reconstructions. J Sci Comput 86, 15 (2021). https://doi.org/10.1007/s10915-020-01380-0

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  • DOI: https://doi.org/10.1007/s10915-020-01380-0

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