### A CELL-CENTERED ALE METHOD WITH HLLC-2D RIEMANN SOLVER IN 2D CYLINDRICAL GEOMETRY

Jian Ren1, Zhijun Shen1,2, Wei Yan1, Guangwei Yuan1

1. 1. Institute of Applied Physics and Computational Mathematics, Beijing 100088, China;
2. Center for Applied Physics and Technology, HEDPS, Peking University, Beijing 100081, China
• Received:2019-07-19 Revised:2020-02-18 Online:2021-09-15 Published:2021-10-15
• Supported by:
Project supported by the National Natural Science Foundation of China (U1630249, 11971071, 11971069, 11871113), the Science Challenge Project (JCKY2016212A502) and the Foundation of Laboratory of Computation Physics. The authors appreciate the reviewers’ help and valuable suggestions during the revision of this paper.

Jian Ren, Zhijun Shen, Wei Yan, Guangwei Yuan. A CELL-CENTERED ALE METHOD WITH HLLC-2D RIEMANN SOLVER IN 2D CYLINDRICAL GEOMETRY[J]. Journal of Computational Mathematics, 2021, 39(5): 666-692.

This paper presents a second-order direct arbitrary Lagrangian Eulerian (ALE) method for compressible flow in two-dimensional cylindrical geometry. This algorithm has half-face fluxes and a nodal velocity solver, which can ensure the compatibility between edge fluxes and the nodal flow intrinsically. In two-dimensional cylindrical geometry, the control volume scheme and the area-weighted scheme are used respectively, which are distinguished by the discretizations for the source term in the momentum equation. The two-dimensional second-order extensions of these schemes are constructed by employing the monotone upwind scheme of conservation law (MUSCL) on unstructured meshes. Numerical results are provided to assess the robustness and accuracy of these new schemes.

CLC Number:

 [1] B.N. Azarenok, Realization of a second-order Godunov's scheme, Comput. Meth. Appl. Mech. Eng. 189(2000), 1031-1052.[2] B.N. Azarenok, Adaptive mesh redistribution method based on Godunov schemes, Comm. Math. Sci., 1(2003), 152-179.[3] D. Burton, T. Carney, N. Morgan, S. Sambasivan, and M. Shashkov, A cell centered Lagrangian Godunov-like method of solid dynamics, Comput. & Fluids, 83(2013), 33-47.[4] E.J. Caramana, D.E. Burton, M.J. Shashkov, P.P. Whalen, The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, J. Comput. Phys., 146(1998), 227-262.[5] G. Carré, S. Del Pino, B. Després, E. Labourasse, A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension,J. Comput. Phys., 228(2009), 5160-5183.[6] G.X. Chen, H.Z. Tang, P.W. Zhang, Second-order accurate Godunov scheme for multicomponent Flows on Moving Triangular Meshes, J. Sci. Comput., 34(2008), 64-86.[7] J. Cheng and C.W. Shu, A cell-centered Lagrangian scheme with the preservation of symmetry and conservation properties for compressible fluid flows in two-dimensional cylindrical geometry, J. Comput. Phys., 229(2010), 7191-7206.[8] J. Cheng and C.W. Shu, Second order symmetry-preserving conservative Lagrangian scheme for compressible Euler equations in two-dimensional cylindrical coordinates, J. Comput. Phys., 272(2014), 245-265.[9] B. Després and C. Mazeran, Lagrangian gas dynamics in two dimensions and lagrangian systems, Arch. Rational Mech. Anal., 178(2005), 327-372.[10] J.K. Dukowicz, M.C. Cline, and F.S. Addessio, A general topology Godunov method. J. Comput. Phys., 82(1989), 29-63.[11] J.K. Dukowicz and B. Meltz, Vorticity errors in multidimensional Lagrangian codes, J. Comput. Phys., 99(1992), 115-134.[12] S.K. Godunov, A.V. Zabrodin, and M. Ya. Ivanov, et al., Numerical Solution of Multidimensional Problems of Gas Dynamics, (in Russian), Moscow, Nauka Press, 1976.[13] J.O. Langseth and R.J. LeVeque, A wave propagation method for 3D hyperbolic conservation laws. J. Comput. Phys., 165(2000), 126-166.[14] R. Li, T. Tang, P.W. Zhang, Moving mesh methods in multiple dimensions based on harmonic maps. J. Comput. Phys., 170(2001), 562-588.[15] H. Luo, J.D. Baum, R. Lohner, On the computation of multi-material flows using ALE formulation, J. Comput. Phys., 194(2004), 304-328.[16] P.H. Maire, R. Abgrall, J. Breil and J. Ovadia, A cell-centered Lagrangian scheme for compressible flow problems, SIAM J. Sci. Comput., 29:4(2007), 1781-1824.[17] P.H. Maire, A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured mesh, J. Comput. Phys., 228:7(2009), 2391-2425.[18] P.H. Maire, A high-order cell-centered Lagrangian scheme for compressible fluid flows in twodimensional cylindrical geometry, J. Comput. Phys.,228:18(2009), 6882-6915.[19] R. Loubére, P. H. Maire, M. Shashkov. ReALE:A Reconnection Arbitrary-Lagrangian-Eulerian method in cylindrical geometry. Comput. & Fluids, 46(2011), 59-69.[20] M. Friess, J. Breil, P.H. Maire, M. Shashkov. A multi-material CCALE-MOF approach in cylindrical Geometry, Commun. Comput. Phys., 15(2014), 330-364.[21] L. Margolin, M. Shashkov, M. Taylor, Symmetry-preserving discretizations for Lagrangian gas dynamics, in:P. NeittaanmSki, T. Tiihonen, P. Tarvainen (Eds.), Proceedings of the Third European Conference Numerical Mathematics and Advanced Applications, World Scientific, (2000), 725-732.[22] N. Morgan, M. Kenamond, D. Burton, T. Carney, and D. Ingraham, An approach for treating contact surfaces in Lagrangian cell-centered hydrodynamics, J. Comput. Phys., 250(2013), 527-554.[23] W.F. Noh, Errors for calculations of strong shocks using artificial viscosity and an artificial heat flux, J. Comput. Phys., 72(1987), 78-120.[24] L. Sedov, Similarity and Dimensional Methods in Mechanics, Academic Press, 1959.[25] Z.J. Shen, W. Yan, and G.W. Yuan, A robust and contact resolving Riemann solver on unstructured mesh, Part I, Euler method, J. Comput. Phys., 268(2014), 432-455.[26] Z.J. Shen, W. Yan, and G.W. Yuan, A robust and contact resolving Riemann solver on unstructured mesh, Part Ⅱ, ALE method, J. Comput. Phys., 268(2014), 456-484.[27] Z.J. Shen, X. Li, J. Ren. Comparisons of some difference forms for compressible flow in cylindrical geometry on ALE framework, Appl. Math. Mech., 37:11(2016), 1571-1586.[28] E.F. Toro, M. Spruce and W. Speares, Restoration of the contact surface in the HLL-Riemann solver, Shock Wave, 4(1994), 25-34.[29] E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag, Berlin, 2008.[30] T.J. Barth, D.C. Jespersen, The design and application of upwind schemes on unstructured meshes, 27th Aerospace Science Meeting, AIAA Paper 89-0366, Reno, Nevada, 1989[31] R. Abgrall, How to prevent oscillations in multicomponent flow calculations:a quasi conservative approach, J. Comput. Phys., 125(1996), 150-160.[32] N R. Morgan, K.N. Lipnikov, D.E. Burton, M.A. Kenamond, A Lagrangian staggered grid Godunov-like approach for hydrodynamics, J. Comput. Phys., 259(2014), 568-597.
 [1] Tianliang Hou, Chunmei Liu, Chunlei Dai, Luoping Chen, Yin Yang. TWO-GRID ALGORITHM OF H1-GALERKIN MIXED FINITE ELEMENT METHODS FOR SEMILINEAR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS [J]. Journal of Computational Mathematics, 2022, 40(5): 667-685. [2] Huan Liu, Xiangcheng Zheng, Hongfei Fu. ANALYSIS OF A MULTI-TERM VARIABLE-ORDER TIME-FRACTIONAL DIFFUSION EQUATION AND ITS GALERKIN FINITE ELEMENT APPROXIMATION [J]. Journal of Computational Mathematics, 2022, 40(5): 814-834. [3] Rong Zhang, Hongqi Yang. A DISCRETIZING LEVENBERG-MARQUARDT SCHEME FOR SOLVING NONLIEAR ILL-POSED INTEGRAL EQUATIONS [J]. Journal of Computational Mathematics, 2022, 40(5): 686-710. [4] Abdelhamid Zaghdani, Sayed Sayari, Miled EL Hajji. A NEW HYBRIDIZED MIXED WEAK GALERKIN METHOD FOR SECOND-ORDER ELLIPTIC PROBLEMS* [J]. Journal of Computational Mathematics, 2022, 40(4): 499-516. [5] Didi Lv, Xiaoqun Zhang. A GREEDY ALGORITHM FOR SPARSE PRECISION MATRIX APPROXIMATION [J]. Journal of Computational Mathematics, 2021, 39(5): 693-707. [6] Yuping Zeng, Feng Wang, Zhifeng Weng, Hanzhang Hu. A POSTERIORI ERROR ESTIMATES FOR A MODIFIED WEAK GALERKIN FINITE ELEMENT APPROXIMATION OF SECOND ORDER ELLIPTIC PROBLEMS WITH DG NORM [J]. Journal of Computational Mathematics, 2021, 39(5): 755-776. [7] Tamal Pramanick. ERROR ESTIMATES FOR TWO-SCALE COMPOSITE FINITE ELEMENT APPROXIMATIONS OF NONLINEAR PARABOLIC EQUATIONS [J]. Journal of Computational Mathematics, 2021, 39(4): 493-517. [8] Yong Liu, Chi-Wang Shu, Mengping Zhang. SUB-OPTIMAL CONVERGENCE OF DISCONTINUOUS GALERKIN METHODS WITH CENTRAL FLUXES FOR LINEAR HYPERBOLIC EQUATIONS WITH EVEN DEGREE POLYNOMIAL APPROXIMATIONS [J]. Journal of Computational Mathematics, 2021, 39(4): 518-537. [9] Qingguo Hong, Jinchao Xu. UNIFORM STABILITY AND ERROR ANALYSIS FOR SOME DISCONTINUOUS GALERKIN METHODS [J]. Journal of Computational Mathematics, 2021, 39(2): 283-310. [10] Xuhong Yu, Lusha Jin, Zhongqing Wang. EFFICIENT AND ACCURATE CHEBYSHEV DUAL-PETROV-GALERKIN METHODS FOR ODD-ORDER DIFFERENTIAL EQUATIONS [J]. Journal of Computational Mathematics, 2021, 39(1): 43-62. [11] Qilong Zhai, Xiaozhe Hu, Ran Zhang. THE SHIFTED-INVERSE POWER WEAK GALERKIN METHOD FOR EIGENVALUE PROBLEMS [J]. Journal of Computational Mathematics, 2020, 38(4): 606-623. [12] Qianting Ma. IMAGE DENOISING VIA TIME-DELAY REGULARIZATION COUPLED NONLINEAR DIFFUSION EQUATIONS [J]. Journal of Computational Mathematics, 2020, 38(3): 417-436. [13] Leiwu Zhang. A STOCHASTIC MOVING BALLS APPROXIMATION METHOD OVER A SMOOTH INEQUALITY CONSTRAINT [J]. Journal of Computational Mathematics, 2020, 38(3): 528-546. [14] Baiju Zhang, Yan Yang, Minfu Feng. A C0-WEAK GALERKIN FINITE ELEMENT METHOD FOR THE TWO-DIMENSIONAL NAVIER-STOKES EQUATIONS IN STREAM-FUNCTION FORMULATION [J]. Journal of Computational Mathematics, 2020, 38(2): 310-336. [15] Yijun Zhong, Chongjun Li. PIECEWISE SPARSE RECOVERY VIA PIECEWISE INVERSE SCALE SPACE ALGORITHM WITH DELETION RULE [J]. Journal of Computational Mathematics, 2020, 38(2): 375-394.
Viewed
Full text

Abstract