Communications on Applied Mathematics and Computation >

2023 , Vol. 5 >Issue 1: 315 - 342

DOI: https://doi.org/10.1007/s42967-021-00163-0

ORIGINAL PAPERS

A Fourth-Order Unstructured NURBS-Enhanced Finite Volume WENO Scheme for Steady Euler Equations in Curved Geometries

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  • 1 Research Center for Mathematics, Beijing Normal University at Zhuhai, Zhuhai 519087, Guangdong, China;
    2 Division of Science and Technology, BNU-HKBU United International College, Zhuhai 519087, Guangdong, China;
    3 School of Mathematical Sciences, Ocean University of China, Qingdao 266100, Shandong, China;
    4 Department of Mathematics, University of Macau, Macao S. A. R., China;
    5 Zhuhai UM Science and Technology Research Institute, Zhuhai 519000, Guangdong, China;
    6 Guangdong-Hong Kong-Macao Joint Laboratory for Data-Driven Fluid Dynamics and Engineering Applications, University of Macau, Macao S. A. R., China

Received date: 2021-03-11

  Revised date: 2021-07-22

  Online published: 2023-03-08

Supported by

The research of Xucheng Meng is partially supported by the Scientific Research Fund of Beijing Normal University (Grant No. 28704-111032105), and the Start-up Research Fund from BNU-HKBU United International College (Grant No. R72021112). The research of Guanghui Hu was partially supported by the FDCT of the Macao S. A. R. (0082/2020/A2), the National Natural Science Foundation of China (Grant Nos. 11922120, 11871489), the Multi-Year Research Grant (2019-00154-FST) of University of Macau, and a Grant from Department of Science and Technology of Guangdong Province (2020B1212030001).

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Abstract

In Li and Ren (Int. J. Numer. Methods Fluids 70: 742–763, 2012), a high-order k-exact WENO finite volume scheme based on secondary reconstructions was proposed to solve the two-dimensional time-dependent Euler equations in a polygonal domain, in which the high-order numerical accuracy and the oscillations-free property can be achieved. In this paper, the method is extended to solve steady state problems imposed in a curved physical domain. The numerical framework consists of a Newton type finite volume method to linearize the nonlinear governing equations, and a geometrical multigrid method to solve the derived linear system. To achieve high-order non-oscillatory numerical solutions, the classical k-exact reconstruction with k = 3 and the efficient secondary reconstructions are used to perform the WENO reconstruction for the conservative variables. The non-uniform rational B-splines (NURBS) curve is used to provide an exact or a high-order representation of the curved wall boundary. Furthermore, an enlarged reconstruction patch is constructed for every element of mesh to significantly improve the convergence to steady state. A variety of numerical examples are presented to show the effectiveness and robustness of the proposed method.

Key words： Steady Euler equations; Curved boundary; NURBS-enhanced finite volume method; WENO reconstruction; Secondary reconstruction

Cite this article

Xucheng Meng, Yaguang Gu, Guanghui Hu . A Fourth-Order Unstructured NURBS-Enhanced Finite Volume WENO Scheme for Steady Euler Equations in Curved Geometries[J]. Communications on Applied Mathematics and Computation, 2023 , 5(1) : 315 -342 . DOI: 10.1007/s42967-021-00163-0

References

1. Aftosmis, M., Gaitonde, D., Tavares, T.S.: On the accuracy, stability and monotonicity of various reconstruction algorithms for unstructured meshes. AIAA 94-0415 (1994)
2. Barth, T.J.: Recent developments in high order k-exact reconstruction on unstructured meshes. AIAA 93-0668 (1993)
3. Barth, T.J., Frederickson, P.O.: Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction. In: 28th Aerospace Sciences Meeting, A90-26902. Reno, NV, USA (1990)
4. Bassi, F., Rebay, S.: High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys. 138(2), 251–285 (1997)
5. Carpentieri, G., Koren, B., van Tooren, M.J.L.: Adjoint-based aerodynamic shape optimization on unstructured meshes. J. Comput. Phys. 224(1), 267–287 (2007)
6. Chiocchia, G.: Exact solutions to transonic and supersonic flows. In: AGARD. Technical report AR-211, AGARD (1985)
7. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, vol.40. SIAM, USA (2002)
8. Cockburn, B., Lin, S.-Y., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. J. Comput. Phys. 84(1), 90–113 (1989)
9. Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II. General framework. Math. Comput. 52(186), 411–435 (1989)
10. Dumbser, M., Käser, M.: Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J. Comput. Phys. 221(2), 693–723 (2007)
11. Duvigneau, R.: Isogeometric analysis for compressible flows using a discontinuous Galerkin method. Comput. Methods Appl. Mech. Eng. 333, 443–461 (2018)
12. Engquist, B., Froese, B.D., Tsai, Y.-H.R.: Fast sweeping methods for hyperbolic systems of conservation laws at steady state II. J. Comput. Phys. 286, 70–86 (2015)
13. Goldenthal, R., Bercovier, M.: Spline curve approximation and design by optimal control over the knots. In: Hahmann, S., Brunnett, G., Farin, G., Goldman, R. (eds) Computing, pp. 53–64. Springer, Berlin (2004)
14. Goodman, J.B., LeVeque, R.J.: On the accuracy of stable schemes for 2D scalar conservation laws. Math. Comput. 45, 15–21 (1985)
15. Haider, F., Croisille, J.-P., Courbet, B.: Stability analysis of the cell centered finite-volume Muscl method on unstructured grids. Numer. Math. 113(4), 555–600 (2009)
16. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71(2), 231–303 (1987)
17. Hu, G.H.: An adaptive finite volume method for 2D steady Euler equations with WENO reconstruction. J. Comput. Phys. 252, 591–605 (2013)
18. Hu, C.Q., Shu, C.-W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150, 97–127 (1999)
19. Hu, G.H., Li, R., Tang, T.: A robust high-order residual distribution type scheme for steady Euler equations on unstructured grids. J. Comput. Phys. 229(5), 1681–1697 (2010)
20. Hu, G.H., Li, R., Tang, T.: A robust WENO type finite volume solver for steady Euler equations on unstructured grids. Commun. Comput. Phys. 9(3), 627–648 (2011)
21. Hu, G.H., Meng, X.C., Yi, N.Y.: Adjoint-based an adaptive finite volume method for steady Euler equations with non-oscillatory k-exact reconstruction. Comput. Fluids 139, 174–183 (2016)
22. Hu, G.H., Yi, N.Y.: An adaptive finite volume solver for steady Euler equations with non-oscillatory k-exact reconstruction. J. Comput. Phys. 312, 235–251 (2016)
23. Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39/40/41), 4135-4195 (2005)
24. Huynh, H.T., Wang, Z.J., Vincent, P.E.: High-order methods for computational fluid dynamics: a brief review of compact differential formulations on unstructured grids. Comput. Fluids 98, 209-220 (2014)
25. Jameson, A.: Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence in transonic and hypersonic flows. In: 11th Computational Fluid Dynamics Conference, AIAA 93–3359, Reston, VA, USA (1993)
26. Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)
27. John, V., Matthies, G.: Higher-order finite element discretizations in a benchmark problem for incompressible flows. Int. J. Numer. Methods Fluids 37(8), 885–903 (2001)
28. Kim, K.H., Kim, C.: Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows: part II: multi-dimensional limiting process. J. Comput. Phys. 208(2), 570–615 (2005)
29. Krivodonova, L., Berger, M.: High-order accurate implementation of solid wall boundary conditions in curved geometries. J. Comput. Phys. 211(2), 492–512 (2006)
30. Li, R., Wang, X., Zhao, W.B.: A multigrid block LU-SGS algorithm for Euler equations on unstructured grids. Numer. Math. Theory Method 1(1), 92–112 (2008)
31. Li, W.A., Ren, Y.-X.: High-order k-exact WENO finite volume schemes for solving gas dynamic Euler equations on unstructured grids. Int. J. Numer. Methods Fluids 70(6), 742–763 (2012)
32. Li, W.A., Ren, Y.-X.: The multi-dimensional limiters for solving hyperbolic conservation laws on unstructured grids II: extension to high order finite volume schemes. J. Comput. Phys. 231(11), 4053–4077 (2012)
33. Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)
34. Liu, Y., Vinokur, M., Wang, Z.J.: Spectral difference method for unstructured grids I: basic formulation. J. Comput. Phys. 216(2), 780–801 (2006)
35. Liu, Y.L., Zhang, W.W.: Accuracy preserving limiter for the high-order finite volume method on unstructured grids. Comput. Fluids 149, 88–99 (2017)
36. Meng, X.C., Hu, G.H.: A NURBS-enhanced finite volume solver for steady Euler equations. J. Comput. Phys. 359, 77–92 (2018)
37. Michalak, C., Ollivier-Gooch, C.: Accuracy preserving limiter for the high-order accurate solution of the Euler equations. J. Comput. Phys. 228(23), 8693–8711 (2009)
38. Ollivier-Gooch, C., Van Altena, M.: A high-order-accurate unstructured mesh finite-volume scheme for the advection-diffusion equation. J. Comput. Phys. 181(2), 729–752 (2002)
39. Piegl, L., Tiller, W.: The NURBS Book (Monographs in Visual Communication), 2nd edn. Springer, New York (1997)
40. Sevilla, R., Fernández-Méndez, S., Huerta, A.: NURBS-enhanced finite element method for Euler equations. Int. J. Numer. Methods Fluids 57(9), 1051–1069 (2008)
41. Sevilla, R., Fernández-Méndez, S., Huerta, A.: NURBS-enhanced finite element method (NEFEM). Int. J. Numer. Meth. Eng. 76(1), 56–83 (2008)
42. Shi, L., Wang, Z.J.: Adjoint-based error estimation and mesh adaptation for the correction procedure via reconstruction method. J. Comput. Phys. 295, 261–284 (2015)
43. Shu, C.-W.: High-order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD. Int. J. Comut. Fluid Dyn. 17(2), 107–118 (2003)
44. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes. Acta Numer. 29, 701–762 (2020)
45. Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4(1), 25–34 (1994)
46. Venkatakrishnan, V.: Convergence to steady state solutions of the Euler equations on unstructured grids with limiters. J. Comput. Phys. 118(1), 120–130 (1995)
47. Wang, Z.J.: Evaluation of high-order spectral volume method for benchmark computational aeroacoustic problems. AIAA J. 43(2), 337–348 (2005)
48. Wang, Z.J., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., Deconinck, H., Hartmann, R., Hillewaert, K., Huynh, H.T., Kroll, N., May, G., Persson, P.-O., van Leer, B., Visbal, M.: High-order CFD methods: current status and perspective. Int. J. Numer. Methods Fluids 72(8), 811–845 (2013)
49. Wang, Z.J., Liu, Y.: Extension of the spectral volume method to high-order boundary representation. J. Comput. Phys. 211(1), 154–178 (2006)
50. Wang, K., Yu, S.J., Wang, Z., Feng, R.Z., Liu, T.G.: Adjoint-based airfoil optimization with adaptive isogeometric discontinuous Galerkin method. Comput. Methods Appl. Mech. Eng. 344, 602–625 (2019)
51. Zhang, S.H., Jiang, S.F., Shu, C.-W.: Improvement of convergence to steady state solutions of Euler equations with the WENO schemes. J. Sci. Comput. 47(2), 216–238 (2011)
52. Zhu, J., Shu, C.-W.: Numerical study on the convergence to steady state solutions of a new class of high order WENO schemes. J. Comput. Phys. 349, 80–96 (2017)
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