Skip to main content
SpringerLink
Log in

A Sequel of Inverse Lax–Wendroff High Order Wall Boundary Treatment for Conservation Laws

  • Original Paper
  • Published:
Archives of Computational Methods in Engineering Aims and scope Submit manuscript

Abstract

When solving CFD problems, the solver, or the numerical code, plays an important role. Depending on the phenomena and problem domain, designing such numerical codes can be hard work. One strategy is to start with simple problems and construct the code as building blocks. The purpose of this work is to provide a detailed review of the theory to compute analytical and exact solutions, and recent numerical methods to construct a code to solve compressible and inviscid fluid flows with high-resolution, arbitrary domains, non-linear phenomena, and on rectangular meshes. We also propose a modification to the inverse Lax–Wendroff procedure solid wall treatment and two-dimensional WENO-type extrapolation stencil selection and weights to handle more generic situations. To test our modifications, we use the finite difference method, Lax–Friedrichs splitting, WENO-Z+ scheme, and third-order strong stability preserving Runge-Kutta time discretization. Our first problem is a simple one-dimensional transient problem with periodic boundary conditions, which is useful for constructing the core solver. Then, we move to the one-dimensional Rayleigh flow, which can handle flows with heat exchange and requires more detailed boundary treatment. The next problem is the quasi-one-dimensional nozzle flow with and without shock, where the boundary treatment needs a few adjustments. The first two-dimensional problem is the Ringleb flow, and despite being smooth, it has a curved wall as the left boundary. Finally, the last problem is a two-dimensional conical flow, which presents an oblique shock and an inclined straight line wall being the cone surface. We show that the designed accuracy is being reached for smooth problems, that high-resolution is being attained for non-smooth problems, and that our modifications produce similar results while providing a more generic way to treat solid walls.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Roe P (1997) A brief introduction to high-resolution schemes. Springer, Berlin, pp 9–28

    Google Scholar 

  2. Shu C-W (1998) Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Quarteroni A (ed) Advanced numerical approximation of nonlinear hyperbolic equations. Springer, Berlin, pp 325–432

    Chapter  Google Scholar 

  3. Leveque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge texts in applied mathematics, 1st edn. Cambridge University Press, New York

    Book  Google Scholar 

  4. Hirsch C (2007) Numerical computation of internal and external flows, 2nd edn. Elsevier, Amsteradm

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Acker F, Borges RBR, Costa B (2016) An improved Weno-z scheme. J Comput Phys 313:726–753

    Article  MathSciNet  Google Scholar 

  7. Fambri F (2020) Discontinuous galerkin methods for compressible and incompressible flows on space-time adaptive meshes: toward a novel family of efficient numerical methods for fluid dynamics. Arch Comput Methods Eng 27:199–283

    Article  MathSciNet  Google Scholar 

  8. Anderson JD (2003) Modern compressible flow, 3rd edn. McGraw-Hill, New York

    Google Scholar 

  9. John JE, Keith TG (2006) Gas Dynamics, 3rd edn. Prentice Hall, New Jersey

    Google Scholar 

  10. Jiang G-S, Shu C-W (1996) Efficient implementation of weighted ENO schemes. J Comput Phys 126(1):202–228

    Article  MathSciNet  Google Scholar 

  11. Henrick AK, Aslam TD, Powers JM (2005) Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J Comput Phys 207(2):542–567

    Article  Google Scholar 

  12. Borges R, Carmona M, Costa B, Don WS (2008) An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J Comput Phys 227:3191–3211

    Article  MathSciNet  Google Scholar 

  13. Gottlieb S, Gottlieb D, Shu C-W (2006) Recovering high-order accuracy in weno computations of steady-state hyperbolic systems. J Sci Comput 28:307–318

    Article  MathSciNet  Google Scholar 

  14. Zhang R, Zhang M, Shu C-W (2011) On the order of accuracy and numerical performance of two classes of finite volume weno schemes. Commun Comput Phys 9(3):807–827

    Article  MathSciNet  Google Scholar 

  15. Helgadöttir A, Ng YT, Min C, Gibou F (2015) Imposing mixed Dirichlet-Neumann-Robin boundary conditions in a level-set framework. Comput Fluids 121:68–80

    Article  MathSciNet  Google Scholar 

  16. Wu X, Shi J-Y, Lei H, Li Y-P, Okine L (2019) Analytical solutions of transient heat conduction in multilayered slabs and application to thermal analysis of landfills. J Cent South Univ 26:3175–3187

    Article  Google Scholar 

  17. Chesshire G, Henshaw W (1990) Composite overlapping meshes for the solution of partial differential equations. J Comput Phys 90(1):1–64

    Article  MathSciNet  Google Scholar 

  18. Sebastian K, Shu C-W (2003) Multidomain weno finite difference method with interpolation at subdomain interfaces. J Sci Comput 19:405–438

    Article  MathSciNet  Google Scholar 

  19. Kreiss H-O, Petersson NA (2006) A second order accurate embedded boundary method for the wave equation with dirichlet data. SIAM J Sci Comput 27(4):1141–1167

    Article  MathSciNet  Google Scholar 

  20. Nilsson S, Petersson NA, Sjögreen B, Kreiss H-O (2007) Stable difference approximations for the elastic wave equation in second order formulation. SIAM J Numer Anal 45(5):1902–1936

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Olejniczak J, Wright MJ, Candler GV (1997) Numerical study of inviscid shock interactions on double-wedge geometries. J Fluid Mech 352:1–25

    Article  Google Scholar 

  23. Rispoli F, Saavedra R, Corsini A, Tezduyar TE (2007) Computation of inviscid compressible flows with the v-sgs stabilization and y\(\zeta \beta\) shock-capturing. Int J Numer Meth Fluids 54(6–8):695–706

    Article  MathSciNet  Google Scholar 

  24. Visbal M (2014) Viscous and inviscid interactions of an oblique shock with a flexible panel. J Fluids Struct 48:27–45

    Article  Google Scholar 

  25. Toro EF (2009) Riemann solvers and numerical methods for fluid dynamics: a practical introduction, 3rd edn. Springer, Berlin Heidelberg

    Book  Google Scholar 

  26. Filbet F, Yang C (2013) An inverse lax-wendroff method for boundary conditions applied to Boltzmann type models. J Comput Phys 245:43–61

    Article  MathSciNet  Google Scholar 

  27. Back LH, Massier PF, Gier HL (1965) Comparison of measured and predicted flows through conical supersonic nozzles, with emphasis on the transonic region. AIAA J 3(9):1606–1614

    Article  Google Scholar 

  28. Serra RCO (1972) Determination of internal gas flows by a transient numerical technique. AIAA J 10(5):603–611

    Article  Google Scholar 

  29. Shapiro AH (1953) The dynamics and thermodynamics of compressible fluid flow, vol 1. Wiley, Hoboken

    Google Scholar 

  30. Shapiro AH (1954) The dynamics and thermodynamics of compressible fluid flow. The Ronald Press Company, New York

    MATH  Google Scholar 

  31. Kitamura K, Shima E (2012) Simple and parameter-free second slope limiter for unstructured grid aerodynamic simulations. AIAA J 50(6):1415–1426

    Article  Google Scholar 

  32. Breviglieri C, Azevedo Luiz J (2017) Further development and application of high-order spectral volume methods for compressible flows. J Aerosp Technol Manag 9:301–327

    Article  Google Scholar 

  33. Sims JL (1964) Tables for supersonic flow around right circular cones at zero angle of attack. Technical report SP-3004, NASA

Download references

Funding

Funding was provided by National Science Foundation (Grant. No. DMS-1719410).

Author information

Authors and Affiliations

  1. Mathematics and Statistics Institute, Rio de Janeiro State University, Rio de Janeiro, Brazil

    Rafael B. de Rezende Borges

  2. Laboratory of Numerical Experimentation, Department of Mechanical Engineering, Federal University of Paraná, Curitiba, Brazil

    Nicholas Dicati P. da Silva

  3. Department of Mechanical Engineering, Federal University of Technology - Paraná, Pato Branco, Brazil

    Francisco A. A. Gomes

  4. Division of Applied Mathematics, Brown University, Providence, USA

    Chi-Wang Shu & Sirui Tan

Authors
  1. Rafael B. de Rezende Borges
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicholas Dicati P. da Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Francisco A. A. Gomes
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Chi-Wang Shu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Sirui Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicholas Dicati P. da Silva.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research of C.-W. Shu is supported by NSF Grant DMS-1719410.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Borges, R.B.d.R., da Silva, N.D.P., Gomes, F.A.A. et al. A Sequel of Inverse Lax–Wendroff High Order Wall Boundary Treatment for Conservation Laws. Arch Computat Methods Eng 28, 2315–2329 (2021). https://doi.org/10.1007/s11831-020-09454-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11831-020-09454-w

Search

Navigation