Skip to main content Accessibility help

Quadrilateral Cell-Based Anisotropic Adaptive Solution for the Euler Equations

  • H. W. Zheng (a1), N. Qin (a1), F. C. G. A. Nicolleau (a1) and C. Shu (a2)


An anisotropic solution adaptive method based on unstructured quadrilateral meshes for inviscid compressible flows is proposed. The data structure, the directional refinement and coarsening, including the method for initializing the refined new cells, for the anisotropic adaptive method are described. It provides efficient high resolution of flow features, which are aligned with the original quadrilateral mesh structures. Five different cases are provided to show that it could be used to resolve the anisotropic flow features and be applied to model the complex geometry as well as to keep a relative high order of accuracy on an efficient anisotropic mesh.


Corresponding author


Hide All
[1]Berger, M.J. and Oliger, J.Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 1984; 53(3):484512.
[2]Hornung, R. D., and Trangenstein, J. A.Adaptive mesh refinement and multilevel iteration for flow in porous media. J. Comput. Phys. 1997; 136(2):522545.
[3]Popinet, S.Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 2003; 190(2):572600.
[4]Liang, Q., Borthwick, A.G.L., and Stelling, G.. Simulation of Dam and Dyke-break Hydrodynamics on Dynamically Adaptive Quad-tree Grids. Int. J. for Num. Meth. in Fluids 2004; 46(2):127162.
[5]Charlton, E.F., and Powell, K.G. An octree solution to conservation-laws over arbitrary regions (OSCAR). AIAA Paper 1997; 970198.
[6]Berger, M.J. and LeVeque, R.Adaptive mesh refinement for two-dimensional hyperbolic systems and the AMRCLAW software. SIAM J. Numer. Anal. 1998; 35:22982316.
[7]Zeeuw, D. De and Powell, K. G.An Adaptively Refined Cartesian Mesh Solver for the Euler Equations. J. Comput. Phys. 1993; 104(1):5668.
[8]Schmidt, G. H. and Jacobs, F. J.Adaptive local grid refinement and multi-grid in numerical reservoir simulation. J. Comput. Phys. 1988; 77(1):140165.
[9]Wang, Z.J.A Quadtree-based adaptive Cartesian/Quad grid flow solver for Navier-Stokes equations. Computers and fluids 1998; 27(44):529549.
[10]Wang, Z.J. and Srinivasan, K.An Adaptive Cartesian Grid Generation Method for ‘Dirty’ Geometry. Int. J. for Num. Meth. in Fluids 2002; 39(8):703717.
[11]Wang, Z.J., Chen, R.F., Hariharan, N., Przekwas, A.J. and Grove, D. A 2N Tree Based Automated Viscous Cartesian Grid Methodology for Feature Capturing. AIAA Paper 1999; 99-3300.
[12]Wang, Z.J. and Chen., R.F.Anisotropic Solution-Adaptive Viscous Cartesian Grid Method for Turbulent Flow Simulation. AIAA Journal 2002; 40(10):19691978.
[13]Ham, F.E., Lien, F.S., and Strong, A.B.A Cartesian Grid Method with Transient Anisotropic Adaptation. J. Comput. Phys. 2002; 179(2):469494.
[14]Keats, W.A. and Lien, F.S.Two-Dimensional Anisotropic Cartesian Mesh Adaptation for the Compressible Euler Equations. Int. J. for Num. Meth. in Fluids 2004; 46(11):10991125.
[15]Habashi, WG., Dompierre, J., Bourgault, Y., Ait-Ali-Yahia, D., Fortin, M., Vallet, MG. Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. Part II: Structured meshes. Int. J. for Num. Meth. in Fluids 2002; 39(8):657673.
[16]Habashi, WG., Dompierre, J., Bourgault, Y., Ait-Ali-Yahia, D., Fortin, M., Vallet, MG.Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD. Part III: Unstructured meshes. Int. J. for Num. Meth. in Fluids 2002; 39(8):675702.
[17]Sun, M., and Takayama, K.Conservative smoothing on an adaptive quadrilateral grid. J. Comput. Phys. 1999; 150(1):143180.
[18]Zheng, H. W., Shu, C. and Chew, Y.T.An Object-Oriented and Quadrilateral-mesh based Solution Adaptive Algorithm for Compressible Multi-fluid Flows. J. Comput. Phys. 2008; 227(14):68956921.
[19]Qin, N. and Liu, X.Flow feature aligned grid adaptation. Int. J. for Num. Meth. in Engineering 2006; 67(6):787814.
[20]Toro, EF, Spruce, M., and Speares, W.Restoration of the contact surface in the HLL Riemann solver. Shock Waves 1994; 4(1):2534.
[21]Toro, EF. Riemann Solvers and Numerical Methods for Fluid Dynamics (2nd edn) Springer: Berlin, 1999.
[22]Luo, H., Baum, J.D., and Lohner, R.A hybrid Cartesian grid and gridless method for compressible flows. J. Comput. Phys. 2006; 214(2):618632.
[23]Takayama, K. and Inoue, O.Shock wave diffraction over a 90 degree sharp corner, Posters presented at 18th ISSW. Shock Waves 1991; 1(4):301312.
[24]Takayama, K. and Jiang, Z.Shock wave reflection over wedges: A benchmark test for CFD and experiments. Shock Waves 1997; 7(4):191203.
[25]Tang, L., and Song, H.A multiresolution finite volume scheme for two-dimensional hyperbolic conservation laws. Journal of Computational and Applied Mathematics 2008; 214(2):583595.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed