A third-order weighted essentially non-oscillatory-flux limiter scheme for two-dimensional incompressible Navier-Stokes equations

Document Type : Research Paper


School of Engineering Science, College of Engineering, University of Tehran, Iran.


In this paper, the 2D incompressible Navier-Stokes (INS) equations in terms of vorticity and stream function are considered. These equations describe the physics of many phenomena of scientific and engineering. By combining monotone upwind methods and weighted essentially non-oscillatory (WENO) procedures, a new numerical algorithm is proposed to approximate the solution of INS equations. To design this algorithm, after obtaining an optimal polynomial, it is rewritten as a convex combination of second-order modified ENO polynomials. Following the methodology of the traditional WENO procedure, the new non-linear weights are calculated. The performance of the new scheme on a number of numerical examples is illustrated.


  • [1] R. Abedian, A symmetrical WENO-Z scheme for solving Hamilton-Jacobi equations, Int. J. Mod. Phys. C, 31 (2020), 2050039.
  • [2] R. Abedian, H. Adibi, and M. Dehghan, A high-order symmetrical weighted hybrid ENO-flux limiter scheme for hyperbolic conservation laws, Comput. Phys. Commun., 185 (2014), 106–127.
  • [3] R. Abedian, H. Adibi, and M. Dehghan, Symmetrical weighted essentially non-oscillatory-flux limiter schemes for Hamilton-Jacobi equations, Math. Methods Appl. Sci., 38 (2015), 4710–4728.
  • [4] R. Abedian and M. Dehghan, RBF-ENO/WENO schemes with Lax-Wendroff type time discretizations for Hamilton-Jacobi equations, Numer. Methods Partial Differ. Equ., 37 (2021), 594–613.
  • [5] R. Abedian and R. Salehi, A RBFWENO finite difference scheme for Hamilton-Jacobi equations, Comput. Math. with Appl., 79 (2020), 2002–2020.
  • [6] L. A. Barba and L. F. Rossi, Global field interpolation for particle methods, J. Comput. Phys., 229, 1292–1310.
  • [7] J. B. Bell, P. Colella, and H. M. Glaz, A second-order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys.,85 (1989), 257–283.
  • [8] D. L. Brown and M. L. Minion, Performance of under-resolved two-dimensional incompressible flow simulations, J. Comput. Phys., 122 (1995), 165–183.
  • [9] S. Bryson and D. Levy, Mapped WENO and weighted power ENO reconstructions in semi-discrete central schemes for Hamilton-Jacobi equations, Appl. Numer. Math., 56 (2006), 1211–1224.
  • [10] A. Chertock and A. Kurganov, On Splitting-Based Numerical Methods for Convection-Diffusion Equations, Quad. Mat., 24 (2009), 303–343.
  • [11] A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, Uniformly high-order essentially non-oscillatory schemes, III, J. Comput. Phys., 71 (1987), 231–303.
  • [12] A. Harten and S. Osher, Uniformly high-order accurate nonoscillatory schemes I, SIAM J. Numer. Anal., 24 (1987), 279–309.
  • [13] A. K. Henrick, T. D. Aslam, and J. M. Powers, Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points, J. Comput. Phys., 207 (2005), 542–567.
  • [14] G.-S. Jiang and D. Peng, Weighted ENO schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21 (2000), 2126–2143.
  • [15] G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), 202–228.
  • [16] P. Koumoutsakos, Inviscid axisymmetrization of an elliptical vortex, J. Comput. Phys., 138 (1997), 821–857.
  • [17] A. Kurganov, S. Noelle, and G. Petrova, Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations., SIAM J. Sci. Comput., 23 (2001), 707–740.
  • [18] X.-D. Liu, S. Osher, and T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994), 200–212.
  • [19] M. Melander, J. McWilliams, and N. Zabusky, Axisymmetrization and vorticity-gradient intensification of an isolated two-dimensional vortex through filamentation, J. Fluid Mech., 178 (1987), 137–159.
  • [20] H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys, 87 (1990), 408–463.
  • [21] S. Osher and J. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12–49.
  • [22] S. Osher and C.-W. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal., 28 (1991), 907–922.
  • [23] A. A. I. Peer, M. Z. Dauhoo, A. Gopaul, and M. Bhuruth, A weighted ENO-flux limiter scheme for hyperbolic conservation laws, Int. J. Comput. Math, 87 (2010), 3467–3488.
  • [24] A. Robinson and P. Saffman, Stability and structure of stretched vortices, Stud. Appl. Math., 70 (1984), 163–181.
  • [25] P. Saffman, M. Ablowitz, E. Hinch, J. Ockendon, and P. Olver, Vortex dynamics, Cambridge University Press, Cambridge, 1992.
  • [26] S. Serna and A. Marquina, Power ENO methods: a fifth-order accurate weighted power ENO method, J. Comput. Phys., 194 (2004), 632–658.
  • [27] S. Serna and J. Qian, Fifth order weighted power-ENO methods for Hamilton-Jacobi equations, J. Sci. Comput., 29 (2006), 57–81.
  • [28] C. W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), 439–471.
  • [29] C. W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes II, J. Comput. Phys., 83 (1989), 32–78.
  • [30] Z. Tao and J. Qiu, Dimension-by-dimension moment-based central Hermite WENO schemes for directly solving Hamilton-Jacobi equations, Adv. Comput. Math., 43 (2017), 1023–1058.
  • [31] T. Xiong, G. Russo, and J.-M. Qiu, High order multi-dimensional characteristics tracing for the incompressible Euler equation and the guiding-center Vlasov equation, J. Sci. Comput., 77 (2018), 263–282.
  • [32] F. Zheng, C.-W. Shu, and J. Qiu, Finite difference Hermite WENO schemes for the Hamilton-Jacobi equations, J. Comput. Phys., 337 (2017), 27–41.
  • [33] J. Zhu and J. Qiu, A new fifth order finite difference WENO scheme for Hamilton-Jacobi equations, Numer. Methods Partial Differ. Equ., 33 (2017), 1095–1113.