One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

doi:10.1007/s42967-021-00151-4

Abstract

Abstract: We address the issue of point value reconstructions from cell averages in the context of third-order finite volume schemes, focusing in particular on the cells close to the boundaries of the domain. In fact, most techniques in the literature rely on the creation of ghost cells outside the boundary and on some form of extrapolation from the inside that, taking into account the boundary conditions, fills the ghost cells with appropriate values, so that a standard reconstruction can be applied also in the boundary cells. In Naumann et al. (Appl. Math. Comput. 325: 252–270. https:// doi. org/ 10. 1016/j. amc. 2017. 12. 041, 2018), motivated by the difficulty of choosing appropriate boundary conditions at the internal nodes of a network, a different technique was explored that avoids the use of ghost cells, but instead employs for the boundary cells a different stencil, biased towards the interior of the domain. In this paper, extending that approach, which does not make use of ghost cells, we propose a more accurate reconstruction for the one-dimensional case and a two-dimensional one for Cartesian grids. In several numerical tests, we compare the novel reconstruction with the standard approach using ghost cells.

Key words: High-order finite volume schemes, Boundary conditions without ghost cells, Hyperbolic systems, CWENOZ reconstruction, Adaptive order reconstructions

CLC Number:

65M08
76M12

M. Semplice, E. Travaglia, G. Puppo. One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells[J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 143-169.

TrendMD

References

1. Baeza, A., Bürger, R., Mulet, P., Zorío, D.: Central WENO schemes through a global average weight. J. Sci. Comput. 78(1), 499–530 (2019). https:// doi. org/ 10. 1007/ s10915-018-0773-z
2. Baeza, A., Mulet, P., Zorío, D.: High order weighted extrapolation for boundary conditions for finite difference methods on complex domains with Cartesian meshes. J. Sci. Comput. 69(1), 170–200 (2016). https:// doi. org/ 10. 1007/ s10915-016-0188-7
3. Baeza, A., Mulet, P., Zorío, D.: Weighted extrapolation techniques for finite difference methods on complex domains with Cartesian meshes. In: Trends in Differential Equations and Applications, pp. 243–259, Springer, Cham (2016). https:// doi. org/ 10. 1007/ 978-3-319-32013-7_ 14
4. Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W.D., Karpeyev, D., Kaushik, D., Knepley, M.G., May, D.A., McInnes, L.C., Mills, R.T., Munson, T., Rupp, K., Sanan, P., Smith, B.F., Zampini, S., Zhang, H., Zhang, H.: PETSc users manual. Tech. Rep. ANL-95/11-Revision 3.11, Argonne National Laboratory (2019)
5. Balay, S., Gropp, W.D., McInnes, L.C., Smith, B.F.: Efficient management of parallelism in object oriented numerical software libraries. In: Arge, E., Bruaset, A.M., Langtangen, H.P. (eds) Modern Software Tools in Scientific Computing, pp. 163–202. Birkhäuser Press, Boston (1997)
6. Balsara, D.S., Garain, S., Florinski, V., Boscheri, W.: An efficient class of WENO schemes with adaptive order for unstructured meshes. J. Comput. Phys. 404, 109062 (2020). https:// doi. org/ 10. 1016/j. jcp.2019. 109062
7. Borsche, R., Kall, J.: ADER schemes and high order coupling on networks of hyperbolic conservation laws. J. Comput. Phys. 273, 658–670 (2014). https:// doi. org/ 10. 1016/j. jcp. 2014. 05. 042
8. Cada, M., Torrilhon, M.: Compact third-order limiter functions for finite volume methods. J. Comput. Phys. 228(11), 4118–4145 (2009). https:// doi. org/ 10. 1016/j. jcp. 2009. 02. 020
9. 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(6), 1241–1252 (1995). https:// doi. org/ 10. 1137/ 09160 72
10. Castro-Dìaz, M.J., Semplice, M.: Third-and fourth-order well-balanced schemes for the shallow water equations based on the CWENO reconstruction. Int. J. Numer. Meth. Fluid 89(8), 304–325 (2019). https:// doi. org/ 10. 1002/ fld. 4700
11. Contarino, C., Toro, E., Montecinos, G., Borsche, R., Kall, J.: Junction-generalized Riemann problem for stiff hyperbolic balance laws in networks: an implicit solver and ADER schemes. J. Comput. Phys. 315, 409–433 (2016). https:// doi. org/ 10. 1016/j. jcp. 2016. 03. 049
12. Cravero, I., Puppo, G., Semplice, M., Visconti, G.: Cool WENO schemes. Comp. Fluids 169, 71–86 (2018). https:// doi. org/ 10. 1016/j. compfluid. 2017. 07. 022
13. Cravero, I., Puppo, G., Semplice, M., Visconti, G.: CWENO: uniformly accurate reconstructions for balance laws. Math. Comp. 87(312), 1689–1719 (2018). https:// doi. org/ 10. 1090/ mcom/ 3273
14. 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). https:// doi. org/ 10. 1007/ s10915-015-0123-3
15. Cravero, I., Semplice, M., Visconti, G.: Optimal definition of the nonlinear weights in multidimensional central WENOZ reconstructions. SIAM J. Numer. Anal. 57(5), 2328–2358 (2019). https:// doi. org/ 10. 1007/ s10915-015-0123-3
16. Don, W.S., Borges, R.: Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes. J. Comput. Phys. 250, 347–372 (2013). https:// doi. org/ 10. 1016/j. jcp. 2013. 05. 018
17. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)
18. 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). https:// doi. org/10. 1016/j. jcp. 2005. 01. 023
19. Hu, C., Shu, C.-W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150(1), 97–127 (1999). https:// doi. org/ 10. 1006/ jcph. 1998. 6165
20. Hui, W., Li, P., Li, Z.: A unified coordinate system for solving the two-dimensional Euler equations. J. Comput. Phys. 153(2), 596–637 (1999). https:// doi. org/ 10. 1006/ jcph. 1999. 6295
21. Jiang, G.S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)
22. Kolb, O.: On the full and global accuracy of a compact third order WENO scheme. SIAM J. Numer. Anal. 52(5), 2335–2355 (2014). https:// doi. org/ 10. 1137/ 13094 7568
23. Levy, D., Puppo, G., Russo, G.: Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput. 22(2), 656–672 (2000). https:// doi. org/ 10. 1137/ S1064 82759 93594 61
24. 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. Computat. Appl. Math. 299, 140–158 (2016). https:// doi. org/ 10. 1016/j. cam. 2015. 11. 038
25. Liska, R.: http://www-troja.fjfi.cvut.cz/liska/CompareEuler/animations/ (2020)
26. Liska, R., Wendroff, B.: Comparison of several difference schemes for the Euler equations in 1D and 2D. In: Hou, T.Y., Tadmor, E. (eds) Hyperbolic Problems: Theory, Numerics, Applications, pp. 831–840. Springer-Verlag, Berlin, Heidelberg (2003). https:// doi. org/ 10. 1007/ 978-3-642-55711-8_78. The 9th International Conference on Hyperbolic Problems, Calf Inst Tech, Pasadena, Ca, Mar 25–29, 2002–2003
27. 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). https:// doi. org/ 10. 1016/j. jcp. 2016. 04. 059
28. Lu, J., Shu, C.-W., Tan, S., Zhang, M.: An inverse Lax-Wendroff procedure for hyperbolic conservation laws with changing wind direction on the boundary. J. Comput. Phys. 426, 109940 (2021). https:// doi. org/ 10. 1016/j. jcp. 2020. 109940
29. Naumann, A., Kolb, O., Semplice, M.: On a third order CWENO boundary treatment with application to networks of hyperbolic conservation laws. Appl. Math. Comput. 325, 252–270 (2018). https:// doi. org/ 10. 1016/j. amc. 2017. 12. 041
30. Pirozzoli, S.: On the spectral properties of shock capturing schemes. J. Comput. Phys. 219, 489-497 (2006)
31. Schulz-Rinne, C.W.: Classification of the Riemann problem for two-dimensional gas dynamics. SIAM J. Math. Anal. 24, 76–88 (1993). https:// doi. org/ 10. 1137/ 05240 06
32. Semplice, M., Coco, A., Russo, G.: Adaptive mesh refinement for hyperbolic systems based on thirdorder compact WENO reconstruction. J. Sci. Comput. 66, 692–724 (2016). https:// doi. org/ 10. 1007/ s10915-015-0038-z
33. Semplice, M., Visconti, G.: Efficient implementation of adaptive order reconstructions. J. Sci. Comput. 83, 1 (2020). https:// doi. org/ 10. 1007/ s10915-020-01156-6
34. Semplice, M., Visconti, G.: claw1dArena v1.2 (2021). https:// doi. org/ 10. 5281/ zenodo. 26417 24
35. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: NASA/CR-97-206253 ICASE Report No.97-65 (1997)
36. Shu, C.-W.: High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51(1), 82–126 (2009). https:// doi. org/ 10. 1137/ 07067 9065
37. Shu, C.-W.: High order WENO and DG methods for time-dependent convection-dominated PDEs: a brief survey of several recent developments. J. Comput. Phys. 316, 598–613 (2016). https:// doi. org/ 10. 1016/j. jcp. 2016. 04. 030
38. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes. Acta Numer. 29, 701–762 (2020). https:// doi. org/ 10. 1017/ S0962 49292 00000 57
39. Shu, C.-W., Tan, S.: Inverse Lax-Wendroff procedure for numerical boundary treatment of hyperbolic equations. Handb. Numer. Anal. 18, 23–52 (2017). https:// doi. org/ 10. 1016/ bs. hna. 2016. 10. 001
40. Tan, S., Shu, C.-W.: Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws. J. Comput. Phys. 229(21), 8144–8166 (2010). https:// doi. org/ 10. 1016/j. jcp. 2010. 07. 014
41. 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(6), 2510–2527 (2012). https:// doi. org/10. 1016/j. jcp. 2011. 11. 037
42. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edn. Springer, Berlin (2009)
43. Zhao, W., Huang, J., Ruuth, S.: Boundary treatment of high order Runge-Kutta methods for hyperbolic conservation laws. J. Comput. Phys. 421, 109697 (2020). https:// doi. org/ 10. 1016/j. jcp. 2020. 109697
44. Zhu, J., Qiu, J.: New finite volume weighted essentially nonoscillatory schemes on triangular meshes. SIAM J. Sci. Comput. 40(2), A903–A928 (2018). https:// doi. org/ 10. 1137/ 17M11 12790

[1]	Sigal Gottlieb, Jan S. Hesthaven, Jianxian Qiu, Chi-Wang Shu, Qiang Zhang, Yong-Tao Zhang. Preface to the Focused Issue on WENO Schemes [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 1-2.
[2]	Bao-Shan Wang, Wai Sun Don, Alexander Kurganov, Yongle Liu. Fifth-Order A-WENO Schemes Based on the Adaptive Diffusion Central-Upwind Rankine-Hugoniot Fluxes [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 295-314.
[3]	G. Puppo, M. Semplice, G. Visconti. Quinpi: Integrating Conservation Laws with CWENO Implicit Methods [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 343-369.
[4]	R. Abgrall. A Combination of Residual Distribution and the Active Flux Formulations or a New Class of Schemes That Can Combine Several Writings of the Same Hyperbolic Problem: Application to the 1D Euler Equations [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 370-402.
[5]	Dinshaw S. Balsara, Saurav Samantaray, Sethupathy Subramanian. Efficient WENO-Based Prolongation Strategies for Divergence-Preserving Vector Fields [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 428-484.
[6]	Andreas Dedner, Robert Klöfkorn. Extendible and Efcient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Methods [J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 657-696.
[7]	Alexander Kurganov, Zhuolin Qu, Olga S. Rozanova, Tong Wu. Adaptive Moving Mesh Central-Upwind Schemes for Hyperbolic System of PDEs: Applications to Compressible Euler Equations and Granular Hydrodynamics [J]. Communications on Applied Mathematics and Computation, 2021, 3(3): 445-480.
[8]	Jun-Feng Yin, Yi-Shu Du. A Class of Preconditioners Based on Positive-Definite Operator Splitting Iteration Methods for Variable-Coefficient Space-Fractional Diffusion Equations [J]. Communications on Applied Mathematics and Computation, 2021, 3(1): 157-176.
[9]	Rémi Abgrall. The Notion of Conservation for Residual Distribution Schemes (or Fluctuation Splitting Schemes), with Some Applications [J]. Communications on Applied Mathematics and Computation, 2020, 2(3): 341-368.
[10]	Christiane Helzel. A Third-Order Accurate Wave Propagation Algorithm for Hyperbolic Partial Diferential Equations [J]. Communications on Applied Mathematics and Computation, 2020, 2(3): 403-427.
[11]	Steven Jöns, Claus, Dieter Munz. An Approximate Riemann Solver for Advection–Difusion Based on the Generalized Riemann Problem [J]. Communications on Applied Mathematics and Computation, 2020, 2(3): 515-539.