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Numerically Neither Dissipative Nor Compressive Scheme for Linear Advection Equation and Its Application to the Euler System

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Abstract

The numerical dissipation is always a big issue in the numerical simulation of hyperbolic equations. The problem is that on one hand one needs it for the stability of the scheme and on the other hand one wishes to get rid of it for obtaining good quality of the solution. In this paper we are going to present a new approach for tackling this problem by developing a new type of finite volume scheme for the linear advection equation. The scheme computes approximations to both the solution and entropy, which are then used in the reconstruction of solution in each cell. Ultra-bee limitation is performed in the solution reconstruction to eliminate the spurious oscillations near discontinuities. Designed in such a way, the scheme maintains the conservation of both the solution and entropy, and in this sense the scheme is numerically neither dissipative nor compressive. We then apply this method to the linearly degenerated second characteristic field of the Euler system to improve the resolution of numerical solution there. Numerical examples of both the linear advection equation and Euler system are displayed to show the efficiency of the method.

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References

  1. Bouchut, F.: An Antidiffusive entropy scheme for monotone scalar conservation laws. J. Sci. Comput. 21, 1–30 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bouchut, F., Bourdarias, Ch., Perthame, B.: A MUSCL method satisfying all the numerical entropy inequalities. Math. Comput. 65, 1439–1461 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cui, Y.: Numerical scheme satisfying two conservation laws for KdV equation. Master’s thesis, No. 11903-02720653, Shanghai University (in Chinese)

  4. Cui, Y., Mao, D.: Numerical method satisfying the first two conservation laws for the Korteweg-de Vries equation. J. Comput. Phys. 227, 376–399 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Colella, P., Woodward, P.R.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  6. Daru, V., Tenaud, C: High order one-step monotonicity-preserving schemes for unsteady compressible flow calculations. J. Comput. Phys. 193, 563–594 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Després, B., Lagoutiére, F.: Contact discontinuity capturing schemes for linear advection and compressible gas dynamics. J. Sci. Comput. 16, 479–524 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Harten, A.: The artificial compression method for computation of shocks and contact discontinuities. 1. Single conservation laws. Comput. Pure Appl. Math. 30, 611–638 (1977)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Harten, A., Osher, S.: Uniformly high-order accurate nonoscillatory scheme I. SIAM J. Numer. Anal. 24, 279–309 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  11. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Unifomly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71, 231–303 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Huyhn, H.T.: Accurate monotone cubic interpolation. SIAM J. Numer. Anal. 30, 57 (1993)

    Article  MathSciNet  Google Scholar 

  13. Kurganov, A., Petrova, G.: Central schemes and contact discontinuities. M2AN 34, 1259–1275 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lagoutiere, F.: Non-dissipative entropic discontinuous reconstruction schemes for hyperbolic conservation laws. Pub Labo JL Lions, R06017

  15. Lax, P.D.: Shock waves and entropy. In: Zarantonello, E.A. (ed.) Contributions to Nonlinear Functional Analysis, pp. 603–634. Academic Press, New York (1971)

    Google Scholar 

  16. LeVeque, R.J.: Numerical Methods for Conservation Laws. Birkhauser, Basel (1990)

    MATH  Google Scholar 

  17. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  18. Li, H.: Entropy dissipating scheme for hyperbolic system of conservation laws in one space dimension. Doctoral thesis, No. 11903-02820022, Shanghai University (In Chinese)

  19. Li, H.: Second-order entropy dissipation scheme for scalar conservation laws in one space dimension. Master’s thesis, No. 11903-99118086, Shanghai University (In Chinese)

  20. Li, H., Mao, D.: The design of the entropy dissipator of the entropy dissipating scheme for scalar conservation law. Chin. J. Comput. Phys. 21, 319–326 (2004) (In Chinese)

    MATH  Google Scholar 

  21. Lie, K., Noelle, S.: On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws. SIAM J. Sci. Comput. 24, 1157–1174 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. Liu, Y., Mao, D.: Further development of a conservative front-tracking methods for systems of conservation laws in one space dimensions. J. Sci. Comput. 28, 85–119 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Morton, K., Mayers, D.: Numerical Solution of Partial Differential Equations. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

  24. Nakamura, T., Yabe, T.: Cubic interpolation scheme for solving hyperdimensional Vlasov-Possion equation in phase space. Comput. Phys. Commun. 120, 120 (2000)

    MathSciNet  Google Scholar 

  25. Nakamura, T., Tanaka, R., Yabe, T., Takizawa, K.: Exactly conservative semi-Lagrangian scheme for multi-dimensional hyperbolic equations with directional splitting technique. J. Comput. Phys. 174, 171–207 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  26. Roe, P.L.: Some contribution to the modelling of discontinuous flows. In: Lectures in Applied Mathematics, vol. 22, pp. 163–193. AMS, Providence (1985)

    Google Scholar 

  27. Shu, C., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J. Comput. Phys. 83, 32–78 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  28. Sod, G.A.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1–31 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  29. Suresh, A., Huyhn, H.T.: Accurate monotonicity-preserving schemes with Runge-Kutta time stepping. J. Comput. Phys. 136, 83–99 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  31. Wang, Z.: Finite difference schemes satisfying multiconservation laws for linear advection equations. Master’s thesis, No. 11903-99118086, Shanghai University (In Chinese)

  32. Wang, Z., Mao, D.: Conservative difference scheme satisfying three conservation laws for linear advection equation. J. SHU 12, 588–592 (2006) (In Chinese)

    MATH  MathSciNet  Google Scholar 

  33. Xu, Z., Shu, C.: Anti-diffusive flux corrections for high order finite difference WENO schemes. J. Comput. Phys. 205, 458–485 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  34. Yabe, T., Tanaka, R., Nakamura, T., Xiao, F.: An exactly conservative semi-Lagrangian scheme (CIP-CSL) in one dimension. Mon. Weather Rev. 129, 332 (2001)

    Article  Google Scholar 

  35. Yang, H.: An artificial compression method for ENO schemes. The slope modification method. J. Comput. Phys. 89, 125–160 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou, Zhejiang, 310018, People’s Republic of China

    Hongxia Li

  2. Department of Mathematics, Fuyang Normal College, Fuyang, Anhui, 236042, People’s Republic of China

    Zhigang Wang

  3. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    De-kang Mao

Authors
  1. Hongxia Li
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  2. Zhigang Wang
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  3. De-kang Mao
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Corresponding author

Correspondence to De-kang Mao.

Additional information

H. Li supported by Scientific Research Fund of Zhejiang Provincial Education Department No. 20070825.

D. Mao supported by Shanghai Pu Jiang Program [2006] 118.

H. Li, Z. Wang and D. Mao supported by China NSF Grant No. 10171063.

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Li, H., Wang, Z. & Mao, Dk. Numerically Neither Dissipative Nor Compressive Scheme for Linear Advection Equation and Its Application to the Euler System. J Sci Comput 36, 285–331 (2008). https://doi.org/10.1007/s10915-008-9192-x

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  • DOI: https://doi.org/10.1007/s10915-008-9192-x

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