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A Finite Difference Mapped WENO Scheme with Unequal-Size Stencils for Hyperbolic Conservation Laws

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Abstract

In this paper, a fifth-order finite difference mapped weighted essentially non-oscillatory scheme with unequal-size stencils, termed as MUS-WENO, is designed for hyperbolic conservation laws. The new mapping function and nonlinear weights are proposed to reduce the difference between the linear weights and nonlinear weights. It could get smaller numerical errors and obtain fifth-order accuracy, give optimal fifth-order convergence with a tiny \(\varepsilon \) even near critical points in smooth regions, and simultaneously suppress spurious oscillations near strong discontinuities. Comparing with the classical fifth-order finite difference WENO schemes and fifth-order finite difference mapped WENO (MWENO) schemes, this mapped WENO scheme uses three unequal-size stencils and the linear weights which can be any positive numbers on condition that their summation is one. Another advantage is that we can reconstruct polynomial over the whole big stencil, while many classical high-order WENO reconstructions only reconstruct the values at the boundary points or discrete quadrature points. Extensive examples including some steady-state problems and some low density, low pressure, or low energy extreme problems are used to testify the good representations of this new MUS-WENO scheme.

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Data availability

All datasets generated during the current study are available from the corresponding author upon reasonable request.

References

  1. Balsara, D.S., Garain, S., Shu, C.-W.: An efficient class of WENO schemes with adaptive order. J. Comput. Phys. 326, 780–804 (2016)

    MathSciNet  MATH  Google Scholar 

  2. Balsara, D.S., Rumpf, T., Dumbser, M., Munz, C.D.: Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics. J. Comput. Phys. 228, 2480–2516 (2009)

    MathSciNet  MATH  Google Scholar 

  3. Balsara, D.S., Shu, C.-W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160, 405–452 (2000)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. Bryson, S., Levy, D.: Mapped WENO and weighted power ENO reconstructions in semi-discrete central schemes for Hamilton–Jacobi equations. Appl. Numer. Math. 56, 1211–1224 (2006)

    MathSciNet  MATH  Google Scholar 

  6. Capdeville, G.: A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes. J. Comput. Phys. 227, 2977–3014 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Casper, J.: Finite-volume implementation of high-order essentially nonoscillatory schemes in two dimensions. AIAA J. 30, 2829–2835 (1992)

    MATH  Google Scholar 

  8. Casper, J., Atkins, H.L.: A finite-volume high-order ENO scheme for two-dimensional hyperbolic systems. J. Comput. Phys. 106, 62–76 (1993)

    MathSciNet  MATH  Google Scholar 

  9. Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230, 1766–1792 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Colella, P., Woodward, P.: The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys. 54, 174–201 (1984)

    MATH  Google Scholar 

  11. Cravero, I., Semplice, M.: On the accuracy of WENO and CWENO reconstructions of third order on nonuniform meshes. J. Sci. Comput. 67, 1219–1246 (2016)

    MathSciNet  MATH  Google Scholar 

  12. Dumbser, M., Käser, M.: Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J. Comput. Phys. 221, 693–723 (2007)

    MathSciNet  MATH  Google Scholar 

  13. Feng, H., Hu, F.X., Wang, R.: A new mapped weighted essentially non-oscillatory scheme. J. Sci. Comput. 51, 449–473 (2012)

    MathSciNet  MATH  Google Scholar 

  14. Feng, H., Huang, C., Wang, R.: An improved mapped weighted essentially non-oscillatory scheme. Appl. Math. Comput. 232, 453–468 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Fu, L., Hu, X.Y., Adams, N.A.: A family of high order targeted ENO schemes for compressible-fluid simulations. J. Comput. Phys. 305, 333–359 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Gao, Z., Don, W.S.: Mapped hybrid central-WENO finite difference scheme for detonation waves simulations. J. Sci. Comput. 55, 351–371 (2013)

    MathSciNet  MATH  Google Scholar 

  17. Gardner, C., Dwyer, S.: Numerical simulation of the XZ Tauri supersonic astrophysical jet. Acta Math. Sci. 29, 1677–1683 (2009)

    MATH  Google Scholar 

  18. Godunov, S.K.: A finite-difference scheme for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47, 271–290 (1959)

    MathSciNet  MATH  Google Scholar 

  19. Ha, Y., Gardner, C.: Positive scheme numerical simulation of high Mach number astrophysical jets. J. Sci. Comput. 34, 247–259 (2008)

    MathSciNet  MATH  Google Scholar 

  20. Ha, Y., Gardner, C., Gelb, A., Shu, C.-W.: Numerical simulation of high Mach number astrophysical jets with radiative cooling. J. Sci. Comput. 24, 597–612 (2005)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  22. Harten, A.: Preliminary results on the extension of ENO schemes to two-dimensional problems. In: Proceedings, International Conference on Nonlinear Hyperbolic Problems. Springer, Berlin (1987)

  23. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.: Uniformly high order accurate essentially non-oscillatory schemes III. J. Comput. Phys. 71, 231–323 (1987)

    MathSciNet  MATH  Google Scholar 

  24. Harten, A., Osher, S.: Uniformly high-order accurate non-oscillatory schemes. IMRC Technical Summary Rept. 2823, University of Wisconsin, Madison (1985)

  25. Henrick, A.K., Aslam, T.D., Powers, J.M.: Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207, 542–567 (2005)

    MATH  Google Scholar 

  26. Hu, G., Li, R., Tang, T.: A robust WENO type finite volume solver for steady Euler equations on unstructured grids. Commun. Comput. Phys. 9, 627–648 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Hu, C., Shu, C.-W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150, 97–127 (1999)

    MathSciNet  MATH  Google Scholar 

  28. Hong, Z., Ye, Z., Meng, X.: A mapping-function-free WENO-M scheme with low computational cost. J. Comput. Phys. 405, 109145 (2020)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  30. Kolb, O.: On the full and global accuracy of a compact third order WENO scheme. SIAM J. Numer. Anal. 52(5), 2335–2355 (2014)

    MathSciNet  MATH  Google Scholar 

  31. Korobeinikov, V.P.: Problems of Point-Blast Theory. American Institute of Physics, College Park (1991)

    Google Scholar 

  32. Kurganov, A., Noelle, S., Petrova, G.: Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations. SIAM J. Sci. Comput. 23, 707–740 (2001)

    MathSciNet  MATH  Google Scholar 

  33. Leonard, B.P.: The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection. Comput. Meth. Appl. Mech. Eng. 88, 17–74 (1991)

    MATH  Google Scholar 

  34. Levy, D., Puppo, G., Russo, G.: Central WENO schemes for hyperbolic systems of conservation laws, M2AN. Math. Model. Numer. Anal. 33, 547–571 (1999)

    MathSciNet  MATH  Google Scholar 

  35. Levy, D., Puppo, G., Russo, G.: Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput. 22(2), 656–672 (2000)

    MathSciNet  MATH  Google Scholar 

  36. Linde, T., Roe, P.L.: Robust Euler codes. In: 13th Computational Fluid Dynamics Conference, AIAA Paper-97-2098 (1997)

  37. Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)

    MathSciNet  MATH  Google Scholar 

  38. Liu, Y., Zhang, Y.T.: A robust reconstruction for unstructured WENO schemes. J. Sci. Comput. 54, 603–621 (2013)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  40. Qiu, J., Shu, C.-W.: On the construction, comparison, and local characteristic decomposition for high order central WENO schemes. J. Comput. Phys. 183, 187–209 (2002)

    MathSciNet  MATH  Google Scholar 

  41. Sedov, L.I.: Similarity and Dimensional Methods in Mechanics. Academic Press, New York (1959)

    MATH  Google Scholar 

  42. Semplice, M., Coco, A., Russo, G.: Adaptive mesh refinement for hyperbolic systems based on third-order compact WENO reconstruction. J. Sci. Comput. 66, 692–724 (2016)

    MathSciNet  MATH  Google Scholar 

  43. Serna, S., Marquina, A.: Power-ENO methods: a fifth-order accurate weighted power ENO method. J. Comput. Phys. 194, 632–658 (2004)

    MathSciNet  MATH  Google Scholar 

  44. Serna, S., Qian, J.: Fifth order weighted power-ENO methods for Hamilton–Jacobi equations. J. Sci. Comput. 29, 57–81 (2006)

    MathSciNet  MATH  Google Scholar 

  45. Shen, Y.Q., Yang, G.W.: Hybrid finite compact-WENO schemes for shock calculation. Int. J. Numer. Methods Fluids 53, 531–560 (2007)

    MathSciNet  MATH  Google Scholar 

  46. Shen, Y.Q., Zha, G.C.: A robust seventh-order WENO scheme and its applications. 46th AIAA Aerospace Sciences Meeting and Exhibit, AIAA 2008-757 (2008)

  47. Shi, J., Hu, C.Q., Shu, C.-W.: A technique of treating negative weights in WENO schemes. J. Comput. Phys. 175, 108–127 (2002)

    MATH  Google Scholar 

  48. Shu, C.-W.: High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51, 82–126 (2009)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  51. Van Leer, B.: Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136 (1979)

    MATH  Google Scholar 

  52. Vevek, U.S., Zang, B., New, T.H.: A new mapped WENO method for hyperbolic problems. In: ICCFD10, Barcelona, Spain (2018)

  53. Wang, Z.J., Chen, R.F.: Optimized weighted essentially non-oscillatory schemes for linear waves with discontinuity. J. Comput. Phys. 174, 381–404 (2001)

    MATH  Google Scholar 

  54. Wang, R., Feng, H., Huang, C.: A new mapped weighted essentially non-oscillatory method using rational mapping function. J. Sci. Comput. 67, 540–580 (2016)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  56. Zhang, X., Shu, C.-W.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229, 3091–3120 (2010)

    MathSciNet  MATH  Google Scholar 

  57. Zhang, X., Shu, C.-W.: On positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229, 8918–8934 (2010)

    MathSciNet  MATH  Google Scholar 

  58. Zhang, X., Shu, C.-W.: Maximum-principle-satisfying and positivity-preserving high order schemes for conservation laws: survey and new developments. Proc. R. Soc. A 467, 2752–2776 (2011)

    MathSciNet  MATH  Google Scholar 

  59. Zhang, X., Shu, C.-W.: Positivity-preserving high order finite difference WENO schemes for compressible Euler equations. J. Comput. Phys. 231, 2245–2258 (2012)

    MathSciNet  MATH  Google Scholar 

  60. Zhang, Y.T., Shu, C.-W.: High order WENO schemes for Hamilton–Jacobi equations on triangular meshes. SIAM J. Sci. Comput. 24, 1005–1030 (2003)

    MathSciNet  MATH  Google Scholar 

  61. Zhang, Y.T., Shu, C.-W.: Third order WENO scheme on three dimensional tetrahedral meshes. Commun. Comput. Phys. 5, 836–848 (2009)

    MathSciNet  MATH  Google Scholar 

  62. Zhong, X., Shu, C.-W.: A simple weighted essentially nonoscillatory limiter for Runge–Kutta discontinuous Galerkin methods. J. Comput. Phys. 232, 397–415 (2013)

    MathSciNet  Google Scholar 

  63. Zhu, J., Qiu, J.: A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 318, 110–121 (2016)

    MathSciNet  MATH  Google Scholar 

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Funding

Research of J. Zhu was supported by NSFC Grant 11872210 and MCMS-I-0120G01. Research of J. Qiu was supported by NSFC Grant 12071392.

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Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, Jiangsu, People’s Republic of China

    Jun Zhu

  2. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Jianxian Qiu

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  1. Jun Zhu
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Zhu, J., Qiu, J. A Finite Difference Mapped WENO Scheme with Unequal-Size Stencils for Hyperbolic Conservation Laws. J Sci Comput 93, 72 (2022). https://doi.org/10.1007/s10915-022-02034-z

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