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Shock Capturing Artificial Dissipation for High-Order Finite Difference Schemes

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Abstract

We study 2nd-, 4th-, 6th- and 8th-order accurate finite difference schemes approximating systems of conservation laws. Our goal is to utilize the high order of accuracy of the schemes for approximating complicated flow structures and add suitable diffusion operators to capture shocks.

We choose appropriate viscosity terms and prove non-linear entropy stability. In the scalar case, entropy stability enables us to prove convergence to the unique entropy solution. Moreover, a limiter function that localizes the effect of the dissipation around discontinuities is derived. The resulting scheme is entropy stable for systems, and also converges to the entropy solution in the scalar case.

We present a number of numerical experiments in order to demonstrate the robustness and accuracy of our scheme. The set of examples consists of a moving shock solution to the Burgers’ equation, a solution to the Euler equations that consists of a rarefaction and two contact discontinuities and a shock/entropy wave solution to the Euler equations (Shu’s test problem). Furthermore, we use the limited scheme to compute the solution to the linear advection equation and demonstrate that the limiter quickly vanishes for smooth flows and design/high-order of accuracy is retained. The numerical results in all experiments were very good. We observe a remarkable gain in accuracy when the order of the scheme is increased.

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References

  1. Adams, N.A., Shariff, K.: A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems. JCP 127, 27–51 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  2. Carpenter, M.H., Gottlieb, D., Abarbanel, S.: Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: Methodology and application to high-order compact schemes. J. Comput. Phys. 111(2), 220–236 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Coquel, F., LeFloch, P.: Convergence of finite difference schemes for conservation laws in several space dimensions: A general theory. SIAM J. Numer. Anal. 30, 675–700 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin (2000)

    MATH  Google Scholar 

  5. Engquist, B., Sjögreen, B.: The convergence rate of finite difference schemes in the presence of shocks. SIAM J. Numer. Anal. 35(6), 2464–2485 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gustafsson, B.: The convergence rate for difference approximations to mixed initial boundary value problems. Math. Comput. 29(130), 396–406 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gustafsson, B.: The convergence rate for difference approximations to general mixed initial boundary value problems. SIAM J. Numer. Anal. 18(2), 179–190 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  8. Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357–393 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  9. Harten, A., Hyman, J.M.: Self adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comput. Phys. 50, 235–269 (1983)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Kreiss, H.-O., Scherer, G.: On the existence of energy estimates for difference approximations for hyperbolic systems. Technical Report, Dept. of Scientific Computing, Uppsala University (1977)

  13. Larsson, J., Gustafsson, B.: Stability criteria for hybrid difference methods. J. Comput. Phys. 227(5), 2886–2898 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  15. Lu, Y.: Hyperbolic Conservation Laws and the Compensated Compactness Method. Surveys in Pure and Applied Mathematics. Chapman and Hall, CRC, Boca Raton (2003)

    MATH  Google Scholar 

  16. Martin, M.P., Taylor, E.M., Wu, M., Weirs, V.G.: A bandwidth-optimized weno scheme for the effective direct numerical simulation of compressible turbulence. J. Comput. Phys. 220, 270–289 (2006)

    Article  MATH  Google Scholar 

  17. Mattsson, K., Svärd, M., Nordström, J.: Stable and accurate artificial dissipation. J. Sci. Comput. 21(1), 57–79 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Mattsson, K., Svärd, M., Carpenter, M.H., Nordström, J.: High-order accurate computations for unsteady aerodynamics. Comput. Fluids 36(3), 636–649 (2007)

    Article  Google Scholar 

  19. Osher, S.: Riemann solvers, the entropy condition and difference approximations. SIAM J. Numer. Anal. 21, 955–984 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  20. Pirozzoli, S.: Conservative hybrid compact-WENO schemes for shock-turbulence interaction. J. Comput. Phys. 178, 81–117 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Roe, P.L.: Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43, 357–372 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  22. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Technical Report ICASE Report No. 97–65, Institute for Computer Applications in Science and Engineering (1997)

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

    Article  MATH  MathSciNet  Google Scholar 

  24. Sod, G.: A survey of several finite difference methods for systems of conservation laws. J. Comput. Phys. 17, 1–31 (1978)

    Article  MathSciNet  Google Scholar 

  25. Strang, G.: Accurate partial difference methods II. Non-linear problems. Numer. Math. 6, 37–46 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  26. Svärd, M., Nordström, J.: On the order of accuracy for difference approximations of initial-boundary value problems. J. Comput. Phys. 218(1), 333–352 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Svärd, M., Nordstrom, J.: A stable high-order finite difference scheme for the compressible Navier-Stokes equations, wall boundary conditions. J. Comput. Phys. 227, 4805–4824 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. Svärd, M., Mattsson, K., Nordström, J.: Steady state computations using summation-by-parts operators. J. Sci. Comput. 24(1), 79–95 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  29. Svärd, M., Carpenter, M.H., Nordström, J.: A stable high-order finite difference scheme for the compressible Navier-Stokes equations, far-field boundary conditions. J. Comput. Phys. 225(1), 1020–1038 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  30. Tadmor, E.: The numerical viscosity and the entropy condition for conservative difference schemes. Math. Comput. 43, 369–381 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  31. Tadmor, E.: The numerical viscosity of entropy stable schemes for systems of conservation laws, I. Math. Comput. 49, 91–103 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  32. Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numerica 12, 451–512 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  33. Taylor, E.M., Wu, M., Martin, M.P.: Optimization of nonlinear error for weighted essentially non-oscillatory methods in direct numerical simulations of compressible turbulence. J. Comput. Phys. 223, 384–397 (2007)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053 Blindern, 0316, Oslo, Norway

    Magnus Svärd & Siddhartha Mishra

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  1. Magnus Svärd
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  2. Siddhartha Mishra
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Correspondence to Magnus Svärd.

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Svärd, M., Mishra, S. Shock Capturing Artificial Dissipation for High-Order Finite Difference Schemes. J Sci Comput 39, 454–484 (2009). https://doi.org/10.1007/s10915-009-9285-1

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  • DOI: https://doi.org/10.1007/s10915-009-9285-1

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