Essentially Non-Oscillatory Adaptive Tree Methods

Abstract We develop high order essentially non-oscillatory (ENO) schemes on non-uniform meshes based on generalized binary trees. The idea is to adopt an appropriate data structure which allows to communicate information easily between unstructured data structure and virtual uniform meshes. While the generalized binary trees as an unstructured data structure can store solution information efficiently if combined with a good adaptive strategy, virtual uniform meshes allow us to take advantage of many well-developed ENO numerical methods based on uniform meshes. Therefore, the ENO adaptive tree methods proposed here can leverage the merits from both tree structures and uniform meshes. Numerical examples demonstrate that the new method is efficient and accurate.

[1]  Martin J. Dürst,et al.  The design and analysis of spatial data structures. Applications of spatial data structures: computer graphics, image processing, and GIS , 1991 .

[2]  Wang Hai-bing,et al.  High-order essentially non-oscillatory schemes for Hamilton-Jacobi equations , 2006 .

[3]  Mario Ohlberger,et al.  A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions , 2000, Math. Comput..

[4]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[5]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[6]  Rémi Abgrall,et al.  On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation , 1994 .

[7]  Doug Moore.  The cost of balancing generalized quadtrees , 1995, SMA '95.

[8]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[9]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[10]  Jean-François Remacle,et al.  An Adaptive Discontinuous Galerkin Technique with an Orthogonal Basis Applied to Compressible Flow Problems , 2003, SIAM Rev..

[11]  Ronald N. Perry,et al.  Simple and Efficient Traversal Methods for Quadtrees and Octrees , 2002, J. Graphics, GPU, & Game Tools.

[12]  S. Osher,et al.  High-order essentially nonsocillatory schemes for Hamilton-Jacobi equations , 1990 .

[13]  Harold L. Atkins,et al.  A Finite-Volume High-Order ENO Scheme for Two-Dimensional Hyperbolic Systems , 1993 .

[14]  Yong-Tao Zhang,et al.  Resolution of high order WENO schemes for complicated flow structures , 2003 .

[15]  Stanley Osher,et al.  Simplex free adaptive tree fast sweeping and evolution methods for solving level set equations in arbitrary dimension , 2006, J. Comput. Phys..

[16]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[17]  M. Berger,et al.  Adaptive mesh refinement for hyperbolic partial differential equations , 1982 .

[18]  Jianliang Qian,et al.  Fifth-Order Weighted Power-ENO Schemes for Hamilton-Jacobi Equations , 2006, J. Sci. Comput..

[19]  Jianliang Qian,et al.  An adaptive finite-difference method for traveltimes and amplitudes , 2002 .

[20]  Laurent Gosse,et al.  Two A Posteriori Error Estimates for One-Dimensional Scalar Conservation Laws , 2000, SIAM J. Numer. Anal..

[21]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[22]  Barry Merriman,et al.  Understanding the Shu–Osher Conservative Finite Difference Form , 2003, J. Sci. Comput..

[23]  Jianliang Qian,et al.  Adaptive Finite Difference Method For Traveltime And Amplitude , 1999 .

[24]  Bernardo Cockburn,et al.  A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations. Part I: The steady state case , 2001, Math. Comput..

[25]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[26]  J. Strain.  Tree Methods for Moving Interfaces , 1999 .

[27]  R. LeVeque,et al.  Adaptive Mesh Refinement Using Wave-Propagation Algorithms for Hyperbolic Systems , 1998 .

[28]  Bernardo Cockburn,et al.  An adaptive method with rigorous error control for the Hamilton--Jacobi equations. Part I: The one-dimensional steady state case , 2005 .

[29]  David H. Sharp,et al.  The dynamics of bubble growth for Rayleigh-Taylor unstable interfaces , 1987 .

[30]  Donald E. Knuth,et al.  The art of computer programming: V.1.: Fundamental algorithms , 1997 .

[31]  Danping Peng,et al.  Weighted ENO Schemes for Hamilton-Jacobi Equations , 1999, SIAM J. Sci. Comput..

[32]  Chohong Min.  Local level set method in high dimension and codimension , 2004 .