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A splitting method for the augmented Burgers equation

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Abstract

In this paper we consider a splitting method for the augmented Burgers equation and prove that it is of first order. We also analyze the large-time behavior of the approximated solution by obtaining the first term in the asymptotic expansion. We prove that, when time increases, these solutions behave as the self-similar solutions of the viscous Burgers equation.

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References

  1. Aguirre, J., Escobedo, M., Zuazua, E.: Self-similar solutions of a convection diffusion equation and related semilinear elliptic problems. Commun. Partial Differ. Equ. 15(2), 139–157 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alonso, J.J., Colonno, M.R.: Multidisciplinary optimization with applications to sonic-boom minimization. Annu. Rev. Fluid Mech. 44, 505–526 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Biler, P., Funaki, T., Woyczyński, W.A.: Fractal Burgers equations. J. Differ. Equ. 148(1), 9–46 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Biler, P., Karch, G., Woyczyński, W.A.: Asymptotics for conservation laws involving Lévy diffusion generators. Stud. Math. 148(2), 171–192 (2001)

    Article  MATH  Google Scholar 

  5. Biler, P., Karch, G., Woyczyński, W.A.: Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws. Ann. l’Inst. Henri Poincare (C) Non Linear Anal. 18(5), 613–637 (2001)

    Article  MATH  Google Scholar 

  6. Bouharguane, A., Carles, R.: Splitting methods for the nonlocal fowler equation. Math. Comput. 83, 1121–1141 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Castro, C., Palacios, F., Zuazua, E.: Optimal control and vanishing viscosity for the Burgers equation. In: Costanda, C., Pérez, M.E. (eds.) Integral Methods in Science and Engineering, Chap. 7, vol. 2, pp. 65–90. Birkhäuser Verlag, Boston (2010)

    Google Scholar 

  8. Cazenave, T., Haraux, A.: An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, vol. 13. Oxford Univeristy Press, New York (1998)

    MATH  Google Scholar 

  9. Cleveland, R.O.: Propagation of sonic booms through a real, stratified atmosphere. Ph.D. thesis, University of Texas at Austin (1995)

  10. Crandall, M.G., Majda, A.: The method of fractional steps for conservation laws. Numer. Math. 34, 285–314 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften, vol. 325. Springer, Berlin (2010)

    Book  Google Scholar 

  12. Escobedo, M., Zuazua, E.: Large time behavior for convection–diffusion equations in \(\mathbb{R}^n\). J. Funct. Anal. 100(1), 119–161 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Holden, H., Karlsen, K.H., Lie, K.A., Risebro, N.H.: Splitting methods for partial differential equations with rough solutions. EMS Series of Lectures in Mathematics. European Mathematical Society (2010)

  14. Holden, H., Lubich, C., Risebro, N.H.: Operator splitting for partial differential equations with burgers nonlinearity. Math. Comput. 82(281), 173–185 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ignat, L.I., Ignat, T.I., Stancu-Dumitru, D.: A compactness tool for the analysis of nonlocal evolution equations. SIAM J. Math. Anal. 47(2), 1330–1354 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ignat, L.I., Pozo, A.: A semi-discrete large-time behavior preserving scheme for the augmented Burgers equation. doi:10.1051/m2an/2017029 (2017)

  17. Ignat, L.I., Pozo, A., Zuazua, E.: Large-time asymptotics, vanishing viscosity and numerics for 1-d scalar conservation laws. Math. Comput. 84, 1633–1662 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ignat, L.I., Rossi, J.D.: A nonlocal convection–diffusion equation. J. Funct. Anal. 251, 399–437 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ignat, L.I., Rossi, J.D.: Decay estimates for nonlocal problems via energy methods. J. Math. Pures Appl. 92(2), 163–187 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ignat, L.I., Stan, D.: Asymptotic behaviour for fractional diffusion–convection equations. https://arxiv.org/abs/1703.02908 (2017)

  21. Iserles, A.: A First Course in the Numerical Analysis of Differential Equations. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  22. Laurençot, P.: Asymptotic self-similarity for a simplified model for radiating gases. Asymptot. Anal. 42, 251–262 (2005)

    MathSciNet  MATH  Google Scholar 

  23. Rallabhandi, S.K.: Advanced sonic boom prediction using augmented Burger’s equation. J. Aircr. 48(4), 1245–1253 (2011)

    Article  Google Scholar 

  24. Rallabhandi, S.K.: Sonic boom adjoint methodology and its applications. In: 29th AIAA Applied Aerodynamics Conference. American Institute of Aeronautics and Astronautics (2011)

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Acknowledgements

The authors thank the anonymous referees for the valuable comments that improved the first version of this paper. This work began during the visit of A.P. to IMAR and was finished during the visit of L.I. to BCAM. The authors thank these centers for the hospitality during their visits.

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Authors and Affiliations

  1. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21 Calea Grivitei Street, 010702, Bucharest, Romania

    Liviu I. Ignat

  2. Innovalia Association, Carretera de Asua 6, 48930, Las Arenas, Getxo, Spain

    Alejandro Pozo

  3. BCAM - Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009, Bilbao, Spain

    Alejandro Pozo

Authors
  1. Liviu I. Ignat
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  2. Alejandro Pozo
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Corresponding author

Correspondence to Alejandro Pozo.

Additional information

Communicated by Christian Lubich.

L. I. Ignat was partially supported by Grant PN-II-RU-TE-2014-4-0007 of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, Grant MTM2014-52347, MICINN, Spain and FA9550-15-1-0027 of AFOSR. A. Pozo was granted by the Basque Government, reference PRE_2013_2_150, and partially supported by ERCEA under Grant 246775 NUMERIWAVES, by the Basque Government through the BERC 2014-2017 program and by Spanish MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323.

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Ignat, L.I., Pozo, A. A splitting method for the augmented Burgers equation. Bit Numer Math 58, 73–102 (2018). https://doi.org/10.1007/s10543-017-0673-x

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  • DOI: https://doi.org/10.1007/s10543-017-0673-x

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