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Semi-Lagrangian Runge-Kutta Exponential Integrators for Convection Dominated Problems

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Abstract

In this paper we consider the case of nonlinear convection-diffusion problems with a dominating convection term and we propose exponential integrators based on the composition of exact pure convection flows. These methods can be applied to the numerical integration of the considered PDEs in a semi-Lagrangian fashion. Semi-Lagrangian methods perform well on convection dominated problems (Pironneau in Numer. Math. 38:309–332, 1982; Hockney and Eastwood in Computer simulations using particles. McGraw-Hill, New York, 1981; Rees and Morton in SIAM J. Sci. Stat. Comput. 12(3):547–572, 1991; Baines in Moving finite elements. Monographs on numerical analysis. Clarendon Press, Oxford, 1994).

In these methods linear convective terms can be integrated exactly by first computing the characteristics corresponding to the gridpoints of the adopted discretization, and then producing the numerical approximation via an interpolation procedure.

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References

  1. Ascher, U.M., Ruuth, S.J., Spiteri, R.: Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25, 151–167 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ascher, U.M., Ruuth, S.J., Wetton, B.T.R.: Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32, 797–823 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baines, M.J.: Moving Finite Elements. Monographs on Numerical Analysis. Clarendon Press, Oxford (1994)

    MATH  Google Scholar 

  4. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988)

    MATH  Google Scholar 

  5. Crouch, P.E., Grossman, R.: Numerical integration of ordinary differential equations on manifolds. J. Nonlinear Sci. 3, 1–33 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  6. Celledoni, E., Marthinsen, A., Owren, B.: Commutator-free Lie group methods. FCGS 19, 341–352 (2003)

    Google Scholar 

  7. Celledoni, E.: Eulerian and semi-Lagrangian commutator-free exponential integrators. CRM Proc. 39, 19 (2004)

    Google Scholar 

  8. Celledoni, E., Kometa, B.K.: Order conditions for the semi-Lagrangian exponential integrators. Preprint numerics nr. 4, Department of Mathematical Sciences, NTNU, Trondheim, Norway (2009)

  9. Celledoni, E., Moret, I.: A Krylov projection method for systems of ODEs. Appl. Numer. Math. 24, 365–378 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Celledoni, E., Cohen, D., Owren, B.: Symmetric exponential integrators for the cubic Schroedinger equation. J. FoCM 8(3), 303–317 (2008)

    MATH  MathSciNet  Google Scholar 

  11. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II, 2nd edn. Springer Series in Computational Mathematics. Springer, Berlin (1996)

    MATH  Google Scholar 

  12. Grava, T., Klein, C.: Numerical solution of the small dispersion limit of Korteweg-de Vries and Whitham equations. Commun. Pure Appl. Math. 60(11), 1623–1664 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Giraldo, F.X., Perot, J.B., Fischer, P.F.: A spectral element semi-Lagrangian (SESL) method for the spherical shallow water equations. J. Comput. Phys. 190, 623–650 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hockney, R.W., Eastwood, J.W.: Computer Simulations Using Particles. McGraw-Hill, New York (1981)

    Google Scholar 

  15. Kennedy, C.A., Carpenter, M.H.: Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math. 44, 139–181 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Moret, I., Novati, P.: RD-rational approximations of the matrix exponential BIT. Numer. Math. 44, 595–615 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Owren, B.: Order conditions for commutator-free Lie group methods. J. Phys. A 39, 5585–5599 (2006)

    MATH  MathSciNet  Google Scholar 

  18. Pirroneau, O.: On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numer. Math. 38, 309–332 (1982)

    Article  Google Scholar 

  19. Pietra, P., Pohl, C.: Weak limits of the quantum hydrodynamic model. VLSI Des. 9, 427–434 (1999)

    Article  Google Scholar 

  20. Rees, M.D., Morton, K.W.: Moving point, particle and free Lagrange methods for convection-diffusion equations. SIAM J. Sci. Stat. Comput. 12(3), 547–572 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  21. Tritton, D.J.: Physical Fluid Dynamics. Van Nostrand Reinhold, London (1977)

    MATH  Google Scholar 

  22. Xiu, D., Karniadakis, G.E.: A semi-Lagrangian high-order method for Navier-Stokes equation. J. Comput. Phys. 172, 658–684 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. Zheng, Z., Petzold, L.: Runge-Kutta-Chebyshev projection method. J. Comput. Phys. 219(2), 976–991 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Institutt for Matematiske Fag, NTNU, 7049, Trondheim, Norway

    Elena Celledoni & Bawfeh Kingsley Kometa

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  1. Elena Celledoni
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  2. Bawfeh Kingsley Kometa
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Correspondence to Elena Celledoni.

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Celledoni, E., Kometa, B.K. Semi-Lagrangian Runge-Kutta Exponential Integrators for Convection Dominated Problems. J Sci Comput 41, 139–164 (2009). https://doi.org/10.1007/s10915-009-9291-3

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  • DOI: https://doi.org/10.1007/s10915-009-9291-3

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