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An oscillation-free high order TVD/CBC-based upwind scheme for convection discretization

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Abstract

An oscillation-free high order scheme is presented for convection discretization by using the normalized-variable formulation in the finite volume framework. It adopts the cubic upwind interpolation scheme as the basic scheme so as to obtain high order accuracy in smooth solution domain. In order to avoid unphysical oscillations of numerical solutions, the present scheme is designed on the TVD (total variational diminishing) constraint and CBC (convection boundedness criterion) condition. Numerical results of several linear and nonlinear convection equations with smooth or discontinuous initial distributions demonstrate the present scheme possesses second-order accuracy, good robustness and high resolution.

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References

  1. Agarwal, R.K.: A third-order-accurate upwind scheme for Navier–Stokes solutions at high Reynolds numbers. In: AIAA Paper 1981-112 in 19th AIAA Aerospace Sciences Meeting, St. Louis, MO, USA (1981)

  2. Akella, S., Navon, I.M.: A comparative study of the performance of high resolution advection schemes in the context of data assimilation. Int. J. Numer. Methods Fluids 51, 719–748 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alves, M.A., Oliveira, P.J., Pinho, F.T.: A convergent and universally bounded interpolation scheme for the treatment of advection. Int. J. Numer. Methods Fluids 41, 47–75 (2003)

    Article  MATH  Google Scholar 

  4. Balsara, D.S., Shu, C.-W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160, 405–452 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Darwish, M.S.: A new high-resolution scheme based on the normalized variable formulation. Numer. Heat Transf. B 24, 337–353 (1993)

    Article  Google Scholar 

  6. Darwish, M.S., Moukalled, F.: The normalized weighting factor method: a novel technique for accelerating the convergence of high-resolution convective methods. Numer. Heat Transf. B 30, 217–237 (1996)

    Article  Google Scholar 

  7. Doswell, C.A.: A kinematic analysis of frontogenesis associated with a nondivergent vortex. J. Atmos. Sci. 41, 1242–1248 (1984)

    Article  Google Scholar 

  8. Gaskell, P.H., Lau, A.K.C.: Curvature-compensated convective trasport: SMART, a new boundedness-perserving transport algorithm. Int. J. Numer. Methods Fluids 8, 617–641 (1988)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Harten, A.: High resolution schemes for hyperbolic conservation law. J. Comput. Phys. 49, 357–4393 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  11. Harten, A.: ENO scheme with subcell resolution. J. Comput. Phys. 83, 148–184 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  12. Herrmann, M., Blanquart, G., Raman, V.: Flux corrected finite volume scheme for preserving scalar boundedness in reacting large-eddy simulations. AIAA J. 44, 2879–2886 (2006)

    Article  Google Scholar 

  13. Hirsch, C.: Numerical Computation of Internal and External Flows. Wiley, New York (1990)

    MATH  Google Scholar 

  14. Hou, P.L., Tao, W.Q., Yu, M.Z.: Refinemet of the convective boundedness criterion of Gaskell and Lau. Eng. Comput. 20, 1023–1043 (2003)

    Article  MATH  Google Scholar 

  15. Jasaka, H., Weller, H.G., Gosman, A.D.: High resolution NVD differencing scheme for aribitrarily unstructured meshes. Int. J. Numer. Methods Fluids 32, 431–449 (1999)

    Google Scholar 

  16. Kolgan, V.P.: Application of the mimmum-derivative principle in the construction of finite-difference schemes for numerical analysis of disconstinuous solutions in gas dynamics. J. Comput. Phys. 230, 2384–2390 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lax, P.D., Wendroff, B.: Systems of conservations laws. Commun. Pure Appl. Math. 13, 217–237 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  18. Leonard, B.P.: A stable and accurate modelling procedure based on quadratic interpolation. Comput. Methods Appl. Mech. Eng. 19, 59–98 (1979)

    Article  MATH  Google Scholar 

  19. Leonard, B.P.: Simple high-accuracy resolution program for convective modeling of discontinuities. Int. J. Numer. Methods Fluids 8, 1291–1318 (1988)

    Article  MATH  Google Scholar 

  20. Lin, C-H., Lin, C.A.: Simple high-order bounded convection scheme to model discontinuities. AIAA J. 35, 563–565 (1997)

    Article  Google Scholar 

  21. Lin, H., Chieng, C.C.: A characteristic-based flux limiter of an essentially 3rd-order flux-splitting method for hyperbolic conservation laws. Int. J. Numer. Methods Fluids 13, 287–301 (1991)

    Article  MATH  Google Scholar 

  22. Mingham, C.G., Causon, D.M.: A simple high-resolution advection scheme, Int. J. Numer. Methods Fluids 56, 469–484 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ng, K.C., Yusoff, M.Z., Ng, K.E.Y.: Higher-order bounded differencing schemes for compressible and incompressible flows. Int. J. Numer. Methods Fluids 53, 57–80 (2007)

    Article  MATH  Google Scholar 

  24. Roe, P.L.: Characteristic-based schemes for the Euler equations. Ann. Rev. Fluid Mech. 18, 337–365 (1986)

    Article  MathSciNet  Google Scholar 

  25. Rogerson, A., Melberg, E.: A numerical study of the convergence properties of ENO schemes. J. Sci. Comput. 18, 151–167 (1990)

    Article  Google Scholar 

  26. Shyy, W.: A study of finite difference approximations to steady-state, a convection-dominated flow problem. J. Comput. Phys. 57, 415–438 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  27. Song, B., Liu, G.R., Lam, K.Y., Amano, R.S.: On a higher-order discretization scheme. Int. J. Numer. Methods Fluids 32, 881–897 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  28. Spalding, D.B.: A novel finite difference formulation for differential expressions involving both first and second derivatives. Int. J. Numer. Methods Fluids 4, 551–559 (1972)

    Google Scholar 

  29. Sweby, P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21, 995–1011 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  30. Tamamidis, P., Assanis, D.N.: Evaluation of variaous high-order-accuracy schemes with and without flux limiters. Int. J. Numer. Methods Fluids 16, 931–948 (1993)

    Article  MATH  Google Scholar 

  31. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics, a Practical Introduction, 2nd edn. Springer, Berlin (1991)

    Google Scholar 

  32. Toro, E.F., Titarev, V.A.: ADER schemes for scalar non-linear hyperbolic conservation laws with source terms in three-space dimensions. J. Comput. Phys. 14, 361–370 (1974)

    Article  Google Scholar 

  33. Van Leer, B.: Towards the ultimate conservative difference scheme. II. Monotoinity and conservation combined in a second-order scheme. J. Comput. Phys. 202, 196–215 (2005)

    Article  MathSciNet  Google Scholar 

  34. Wei, G.W., Gu, Y.: Conjugated filter approach for solving Burger’s equation. J. Comput. Appl. Math. 149, 439–456 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  35. Wei, J.J., Bo Yu, B., Tao, W.Q.: A new high-order-accurate and bounded scheme for incompressible flow. Numer. Heat Transf. B 43, 19–41 (2003)

    Article  Google Scholar 

  36. Yu, B., Tao, W.Q., Zhang, D.S., Wang, Q.W.: Discussion on numerical stability and boundedness of convective discretized scheme. Numer. Heat Transf. B 40, 343–365 (2001)

    Article  Google Scholar 

  37. Zhu, J.: A low-diffusive and oscillation-free convective scheme. Comm. Appl. Mech. Eng. 7, 225–232 (1991)

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Inner Mongolia University, Huhhot, People’s Republic of China

    Wei Gao, Hong Li, Yang Liu & Yong-Jun Jian

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  1. Wei Gao
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  2. Hong Li
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  3. Yang Liu
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  4. Yong-Jun Jian
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Correspondence to Wei Gao.

Additional information

This work is supported by the National Natural Science Foundation of China (Grant No. 11061021,11062005) and the Program of Higher-level talents of Inner Mongolia University (SPH-IMU, Z200901004) and the Scientific Research Projection of Higher Schools of Inner Mongolia (NJ10016, NJ10006).

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Gao, W., Li, H., Liu, Y. et al. An oscillation-free high order TVD/CBC-based upwind scheme for convection discretization. Numer Algor 59, 29–50 (2012). https://doi.org/10.1007/s11075-011-9474-5

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