Boundary Closures for Sixth-Order Energy-Stable Weighted Essentially Non-Oscillatory Finite-Difference Schemes

  • Mark H. CarpenterEmail author
  • Travis C. Fisher
  • Nail K. Yamaleev
Part of the Fields Institute Communications book series (FIC, volume 66)


A general strategy was presented in 2009 by Yamaleev and Carpenter (J. Comput. Phys. 228(11):4248–4272, 2009; J. Comput. Phys. 228(8):3025–3047, 2009), for constructing Energy Stable Weighted Essentially Non-Oscillatory (ESWENO) finite-difference schemes on periodic domains. Fisher et al. (J. Comput. Phys. 230(10):3727–3752, 2011) provided boundary closures for the fourth-order ESWENO scheme that maintain, the WENO stencil biasing properties and satisfy the summation-by-parts (SBP) operator convention, thereby ensuring stability in an L 2 norm. Herein, the general capability of finite-domain schemes is extended by providing closures for the sixth-order case. Third-order and fifth-order boundary closures are developed that are stable in diagonal and block norms, respectively, and achieve fourth- and sixth-order global accuracy for hyperbolic systems. A novel set of nonuniform flux interpolation points is necessary near the boundaries to simultaneously achieve (1) accuracy, (2) the SBP convention, and (3) WENO stencil biasing mechanics. Complete implementation details for the diagonal-norm sixth-order operator are provided as well as examples that demonstrate shock capturing and multi-domain capabilities.


WENO Scheme Boundary Closure Free Stream Mach Number Design Order Linear Advection Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    R. Borges, M. Carmona, B. Costa, and W. S. Don. An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. Journal of Computational Physics, 227(6):3191–3211, 2008. MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    M. H. Carpenter, D. Gottlieb, and S. Abarbanel. Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: Methodology and application to high-order compact schemes. Journal of Computational Physics, 111(2):220–236, 1994. MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    M. H. Carpenter and C. A. Kennedy. Fourth-order 2n-storage Runge-Kutta schemes. Technical Report TM 109112, NASA, 1994. Google Scholar
  4. 4.
    M. H. Carpenter, J. Nordström, and D. Gottlieb. A stable and conservative interface treatment of arbitrary spatial accuracy. Journal of Computational Physics, 148(2):341–365, 1999. MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    M. H. Carpenter, J. Nordström, and D. Gottlieb. Revisiting and extending interface penalties for multi-domain summation-by-parts operators. Technical Report TM 214892, NASA, 2007. Google Scholar
  6. 6.
    M. H. Carpenter, J. Nordström, and D. Gottlieb. Revisiting and extending interface penalties for multi-domain summation-by-parts operators. Journal of Scientific Computing, 45(1–3):118–150, 2010. MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    J. Casper and M. H. Carpenter. Computational considerations for the simulation of shock-induced sound. SIAM Journal on Scientific Computing, 19(3):813–828, 1998. MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    B. Cockburn, C. Johnson, C.-W. Shu, and E. Tadmor. Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Springer, Berlin, 1998. zbMATHGoogle Scholar
  9. 9.
    G. Erlebacher, M. Hussaini, and C.-W. Shu. Interaction of a shock with a longitudinal vortex. Journal of Fluid Mechanics, 337:129–153, 1997. MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    T. C. Fisher, M. H. Carpenter, N. K. Yamaleev, and S. H. Frankel. Boundary closures for fourth-order energy stable weighted essentially non-oscillatory finite-difference schemes. Journal of Computational Physics, 230(10):3727–3752, 2011. MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    B. Gustaffson. High Order Finite Difference Methods for Time Dependent PDE. Springer, Berlin, 2008. Google Scholar
  12. 12.
    B. Gustafsson. The convergence rate for difference approximations to mixed initial boundary value problems. Mathematics of Computation, 29:396–406, 1975. MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    G. Jiang and C.-W. Shu. Efficient implementation of weighted ENO schemes. Journal of Computational Physics, 126(1):202–228, 1996. MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    A. Kitson, R. I. McLachlan, and N. Robidoux. Skew-adjoint finite difference methods on nonuniform grids. New Zealand Journal of Mathematics, 32(2):139–159, 2003. MathSciNetzbMATHGoogle Scholar
  15. 15.
    H.-O. Kreiss and G. Scherer. Finite element and finite difference methods for hyperbolic partial differential equations. In Mathematical Aspects of Finite Elements in Partial Differential Equations, pages 195–212. Academic Press, San Diego, 1974. Google Scholar
  16. 16.
    P. Lax and M. Mock. The computation of discontinuous solutions of linear hyperbolic equations. Communications in Pure and Applied Mathematics, 31:423–430, 1978. MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    M. Martin, E. Taylor, M. Wu, and V. Weris. A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence. Journal of Computational Physics, 220(1):270–289, 2006. zbMATHCrossRefGoogle Scholar
  18. 18.
    K. Mattsson. Summation by parts operators for finite difference approximations of second-derivatives with variable coefficients. Journal of Scientific Computing, 29(1):1–33, 2011. Google Scholar
  19. 19.
    R. I. McLachlan and N. Robidoux. EQUADIFF 99. In Antisymmetry, Pseudospectral Methods, and Conservative PDEs, pages 994–999. World Scientific, Singapore, 2000. Google Scholar
  20. 20.
    J. Nordström. Conservative finite difference formulations, variable coefficients, energy estimates and artificial dissipation. Journal of Scientific Computing, 29(3):375–404, 2006. MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    J. Nordström and M. H. Carpenter. Boundary and interface conditions for high-order finite-difference methods applied to the Euler and Navier-Stokes equations. Journal of Computational Physics, 148(2):621–645, 1999. MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    J. Nordström, J. Gong, E. van der Weide, and M. Svärd. A stable and conservative high order multi-block method for the compressible Navier-stokes equation. Journal of Computational Physics, 228(24):9020–9035, 2009. MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    C.-W. Shu. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In Advanced Numerical Approximation of Nonlinear Hyperbolic Equations: Lecture Notes in Mathematics, volume 1697, pages 325–432. Springer, Berlin, 1998. CrossRefGoogle Scholar
  24. 24.
    B. Strand. Summation by parts for finite difference approximations for d/dx. Journal of Computational Physics, 110(1):47–67, 1994. MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    M. Svärd and J. Nordström. On the order of accuracy for difference approximations of initial-boundary value problems. Journal of Computational Physics, 218(1):333–352, 2006. MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    N. K. Yamaleev and M. H. Carpenter. A systematic methodology for constructing high-order energy stable WENO schemes. Journal of Computational Physics, 228(11):4248–4272, 2009. MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    N. K. Yamaleev and M. H. Carpenter. Third-order energy stable WENO scheme. Journal of Computational Physics, 228(8):3025–3047, 2009. MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    S. Zhang, S. Jiang, Y. Zhang, and C. Shu. The mechanism of sound generation in the interaction between a shock wave and two counter-rotating vortices. Physics of Fluids, 21:076101, 2009. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Mark H. Carpenter
    • 1
    Email author
  • Travis C. Fisher
    • 1
  • Nail K. Yamaleev
    • 2
  1. 1.NASA Langley Research CenterHamptonUSA
  2. 2.Department of MathematicsNorth Carolina A&T State UniversityGreensboroUSA

Personalised recommendations