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Stability Analysis of the Inverse Lax–Wendroff Boundary Treatment for High Order Central Difference Schemes for Diffusion Equations

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Abstract

In this paper, high order central finite difference schemes in a finite interval are analyzed for the diffusion equation. Boundary conditions of the initial-boundary value problem are treated by the simplified inverse Lax–Wendroff procedure. For the fully discrete case, a third order explicit Runge–Kutta method is used as an example for the analysis. Stability is analyzed by both the Gustafsson, Kreiss and Sundström theory and the eigenvalue visualization method on both semi-discrete and fully discrete schemes. The two different analysis techniques yield consistent results. Numerical tests are performed to demonstrate and validate the analysis results.

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References

  1. Berger, M.J., Helzel, C., LeVeque, R.J.: h-box methods for the approximation of hyperbolic conservation laws on irregular grids. SIAM J. Numer. Anal. 41, 893–918 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Carpenter, M.H., Gottlieb, D., Abarbanel, S., Don, W.-S.: The theoretical accuracy of Runge–Kutta time discretizations for the initial boundary value problem: a study of the boundary error. SIAM J. Sci. Comput. 16, 1241–1252 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Goldberg, M., Tadmor, E.: Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. I. Math. Comput. 32, 1097–1107 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  4. Goldberg, M., Tadmor, E.: Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II. Math. Comput. 36, 603–626 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gustafsson, B., Kreiss, H.-O., Sundström, A.: Stability theory of difference approximations for mixed initial boundary value problem. II. Math. Comput. 26, 649–686 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  6. Henshaw, W.D.: A high-order accurate parallel solver for Maxwell’s equations on overlapping grids. SIAM J. Sci. Comput. 28, 1730–1765 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Henshaw, W.D., Kreiss, H.-O., Reyna, L.G.M.: A fourth-order accurate difference approximation for the incompressible Navier–Stokes equations. Comput. & Fluids 23, 575–593 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Huang, L., Shu, C.-W., Zhang, M.: Numerical boundary conditions for the fast sweeping high order WENO methods for solving the Eikonal equation. J. Comput. Math. 26, 336–346 (2008)

    MathSciNet  MATH  Google Scholar 

  9. Kreiss, H.-O., Petersson, N.A.: A second order accurate embedded boundary method for the wave equation with Dirichlet data. SIAM J. Sci. Comput. 27, 1141–1167 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kreiss, H.-O., Petersson, N.A., Yström, J.: Difference approximations for the second order wave equation. SIAM J. Numer. Anal. 40, 1940–1967 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kreiss, H.-O., Petersson, N.A., Yström, J.: Difference approximations of the Neumann problem for the second order wave equation. SIAM J. Numer. Anal. 42, 1292–1323 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, T., Shu, C.-W., Zhang, M.: Stability analysis of the inverse Lax–Wendroff boundary treatment for high order upwind-biased finite difference schemes. J. Comput. Appl. Math. 299, 140–158 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Liu, Y., Shu, C.-W., Zhang, M.: High order finite difference WENO schemes for nonlinear degenerate parabolic equations. SIAM J. Sci. Comput. 33, 939–965 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lu, J., Fang, J., Tan, S., Shu, C.-W., Zhang, M.: Inverse Lax–Wendroff procedure for numerical boundary conditions of convection–diffusion equations. J. Comput. Phys. 317, 276–300 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Osher, S.: Stability of parabolic difference approximations to certain mixed initial boundary value problems. Math. Comput. 26, 13–39 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  16. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MATH  Google Scholar 

  17. Sjögreen, B., Petersson, N.A.: A Cartesian embedded boundary method for hyperbolic conservation laws. Commun. Comput. Phys. 2, 1199–1219 (2007)

    MathSciNet  MATH  Google Scholar 

  18. Sousa, E.: Stability analysis of difference methods for parabolic initial value problems. J. Sci. Comput. 26, 45–66 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Strikwerda, J.C.: Initial boundary value problems for the method of lines. J. Comput. Phys. 34, 94–107 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tadmor, E.: From semi-discrete to fully discrete: stability of Runge-Kutta schemes by the energy method. II. ”Collected lectures on the preservation of stability under discretization”, In: Estep, D., Tavener, S. (eds.) Proceedings in Applied Mathematics 109, SIAM, 25–49 (2002)

  21. Tan, S., Shu, C.-W.: Inverse Lax–Wendroff procedure for numerical boundary conditions of conservation laws. J. Comput. Phys. 229, 8144–8166 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tan, S., Shu, C.-W.: A high order moving boundary treatment for compressible inviscid flows. J. Comput. Phys. 230, 6023–6036 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tan, S., Wang, C., Shu, C.-W., Ning, J.: Efficient implementation of high order inverse Lax–Wendroff boundary treatment for conservation laws. J. Comput. Phys. 231, 2510–2527 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Varah, J.M.: Stability of difference approximations to the mixed initial boundary value problems for parabolic systems. SIAM J. Numer. Anal. 8, 598–615 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  25. Vilar, F., Shu, C.-W.: Development and stability analysis of the inverse Lax–Wendroff boundary treatment for central compact schemes. Math. Model. Numer. Anal. 49, 39–67 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

We thank one of the referees for giving the remark about the alternative approach for the stability analysis at the end of the concluding remarks section.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

    Tingting Li & Mengping Zhang

  2. Division of Applied Mathematics, Brown University, Providence, 02912, RI, USA

    Chi-Wang Shu

Authors
  1. Tingting Li
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  2. Chi-Wang Shu
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  3. Mengping Zhang
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Corresponding author

Correspondence to Mengping Zhang.

Additional information

C.-W. Shu: Research supported by AFOSR Grant F49550-12-1-0399 and NSF Grant DMS-1418750.

M. Zhang: Research supported by NSFC Grant 11471305.

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Li, T., Shu, CW. & Zhang, M. Stability Analysis of the Inverse Lax–Wendroff Boundary Treatment for High Order Central Difference Schemes for Diffusion Equations. J Sci Comput 70, 576–607 (2017). https://doi.org/10.1007/s10915-016-0258-x

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  • DOI: https://doi.org/10.1007/s10915-016-0258-x

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