Journal of Scientific Computing

, Volume 77, Issue 1, pp 263–282 | Cite as

High Order Multi-dimensional Characteristics Tracing for the Incompressible Euler Equation and the Guiding-Center Vlasov Equation

  • Tao Xiong
  • Giovanni Russo
  • Jing-Mei Qiu


In this paper, we propose a high order characteristics tracing scheme for the two-dimensional nonlinear incompressible Euler system in vorticity stream function formulation and the guiding center Vlasov model. Such a scheme is incorporated into a semi-Lagrangian finite difference WENO framework for simulating the aforementioned model equations. This is an extension of our earlier work on high order characteristics tracing scheme for the 1D nonlinear Vlasov–Poisson system (Qiu and Russo in J Sci Comput 71:414–434, 2017). The effectiveness of the proposed scheme is demonstrated numerically by an extensive set of test cases.


Semi-Lagrangian Characteristics tracing Finite difference WENO Incompressible Euler equation Vorticity stream function formulation Guiding-center Vlasov equation 


  1. 1.
    Bonaventura, L., Ferretti, R., Rocchi, L.: A fully semi-Lagrangian discretization for the 2D incompressible Navier–Stokes equations in the vorticity-streamfunction formulation. Appl. Math. Comput. 323, 132–144 (2018)MathSciNetGoogle Scholar
  2. 2.
    Cai, X., Guo, W., Qiu, J.-M.: A high order semi-Lagrangian discontinuous Galerkin method for Vlasov-Poisson simulations without operator splitting. J. Comput. Phys. 354, 529–551 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Celledoni, E., Kometa, B.K., Verdier, O.: High order semi-Lagrangian methods for the incompressible Navier–Stokes equations. J. Sci. Comput. 66, 91–115 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chorin, A.J.: Numerical study of slightly viscous flow. J. Fluid Mech. 57, 785–796 (1973)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cockburn, B., Johnson, C., Shu, C.-W., Tadmor, E.: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  6. 6.
    Cottet, G.-H., Koumoutsakos, P.D.: Vortex Methods: Theory and Practice. Cambridge University Press, Cambridge (2000)CrossRefzbMATHGoogle Scholar
  7. 7.
    Crouseilles, N., Mehrenberger, M., Sonnendrücker, E.: Conservative semi-Lagrangian schemes for Vlasov equations. J. Comput. Phys. 229, 1927–1953 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Falcone, M., Ferretti, R.: Semi-Lagrangian Approximation Schemes for Linear and Hamilton–Jacobi Equations. SIAM, New York (2013)CrossRefzbMATHGoogle Scholar
  9. 9.
    Ghizzo, A., Bertrand, P., Shoucri, M., Fijalkow, E., Feix, M.: An Eulerian code for the study of the drift-kinetic Vlasov equation. J. Comput. Phys. 108, 105–121 (1993)CrossRefzbMATHGoogle Scholar
  10. 10.
    Krasny, R.: Desingularization of periodic vortex sheet roll-up. J. Comput. Phys. 65, 292–313 (1986)CrossRefzbMATHGoogle Scholar
  11. 11.
    Leonard, A.: Vortex methods for flow simulation. J. Comput. Phys. 37, 289–335 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liu, J.-G., Shu, C.-W.: A high-order discontinuous Galerkin method for 2D incompressible flows. J. Comput. Phys. 160, 577–596 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liu, J.-G., Wang, C.: High order finite difference methods for unsteady incompressible flows in multi-connected domains. Comput. Fluids 33, 223–255 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Olshanskii, M.A., Heister, T., Rebholz, L.G., Galvin, K.J.: Natural vorticity boundary conditions on solid walls. Comput. Methods Appl. Mech. Eng. 297, 18–37 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Qiu, J.-M., Russo, G.: A high order multi-dimensional characteristic tracing strategy for the Vlasov–Poisson system. J. Sci. Comput. 71, 414–434 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Qiu, J.-M., Shu, C.-W.: Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow. J. Comput. Phys. 230, 863–889 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rossi, L.F.: Merging computational elements in vortex simulations. SIAM J. Sci. Comput. 18, 1014–1027 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Russo, G.: A deterministic vortex method for the Navier–Stokes equations. J. Comput. Phys. 108, 84–94 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Russo, G., Strain, J.A.: Fast triangulated vortex methods for the 2D Euler equations. J. Comput. Phys. 111, 291–323 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shoucri, M.M.: A two-level implicit scheme for the numerical solution of the linearized vorticity equation. Int. J. Numer. Meth. Eng. 17, 1525–1538 (1981)CrossRefzbMATHGoogle Scholar
  21. 21.
    Sonnendrücker, E., Roche, J., Bertrand, P., Ghizzo, A.: The semi-Lagrangian method for the numerical resolution of the Vlasov equation. J. Comput. Phys. 149, 201–220 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Souli, M.: Vorticity boundary conditions for Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 134, 311–323 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Weinan, E., Liu, J.-G.: Vorticity boundary condition and related issues for finite difference schemes. J. Comput. Phys. 124, 368–382 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Weinan, E., Liu, J.-G.: Essentially compact schemes for unsteady viscous incompressible flows. J. Comput. Phys. 126, 122–138 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Xiong, T., Russo, G., Qiu, J.-M.: Conservative multi-dimensional semi-Lagrangian finite difference scheme: stability and applications to the kinetic and fluid simulations. arXiv:1607.07409 (2016)
  26. 26.
    Xiu, D., Karniadakis, G.E.: A semi-Lagrangian high-order method for Navier–Stokes equations. J. Comput. Phys. 172, 658–684 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zhu, H., Qiu, J., Qiu, J.-M.: An h-adaptive rkdg method for the two-dimensional incompressible euler equations and the guiding center Vlasov model. J. Sci. Comput. 73, 1316–1337 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.School of Mathematical Sciences, Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific ComputingXiamen UniversityXiamenPeople’s Republic of China
  2. 2.Department of Mathematics and Computer ScienceUniversity of CataniaCataniaItaly
  3. 3.Department of MathematicsUniversity of DelawareNewarkUSA

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