High-Resolution Flux-Based Level Set Method

A new high-resolution flux-based finite volume method for general advection equations in nondivergent form including a level set equation for moving interfaces is introduced. The method is applicable to the case of nondivergence free velocity and to general unstructured grids in higher dimensions. We show that the method is consistent and that the numerical solution fulfills the discrete minimum/maximum principle. Numerical experiments show its second order accuracy for smooth solutions as well as for solutions with discontinuous derivatives and on general unstructured meshes. Numerical examples for passive transport and shrinking of dynamic interfaces, including examples with topological changes, are presented using locally adapted two-dimensional and three-dimensional grids.

[1]  Ian M. Mitchell,et al.  A hybrid particle level set method for improved interface capturing , 2002 .

[2]  Rémi Abgrall,et al.  High Order Numerical Discretization for Hamilton–Jacobi Equations on Triangular Meshes , 2000, J. Sci. Comput..

[3]  Kokichi Sugihara,et al.  Approximation of the Boat-Sail Voronoi Diagram and Its Application , 2004, ICCSA.

[4]  Panagiotis E. Souganidis,et al.  Finite volume schemes for Hamilton–Jacobi equations , 1999, Numerische Mathematik.

[5]  Chi-Wang Shu,et al.  Central WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes , 2006, SIAM J. Sci. Comput..

[6]  H. Rentz-Reichert,et al.  UG – A flexible software toolbox for solving partial differential equations , 1997 .

[7]  Chi-Tien Lin,et al.  High-Resolution Nonoscillatory Central Schemes for Hamilton-Jacobi Equations , 1999, SIAM J. Sci. Comput..

[8]  T. F. Russell,et al.  NUMERICAL METHODS FOR CONVECTION-DOMINATED DIFFUSION PROBLEMS BASED ON COMBINING THE METHOD OF CHARACTERISTICS WITH FINITE ELEMENT OR FINITE DIFFERENCE PROCEDURES* , 1982 .

[9]  G. Sapiro,et al.  Geometric partial differential equations and image analysis [Book Reviews] , 2001, IEEE Transactions on Medical Imaging.

[10]  Riccardo Sacco,et al.  A semi-Lagrangian discontinuous Galerkin method for scalar advection by incompressible flows , 2006, J. Comput. Phys..

[11]  P. Lions,et al.  Two approximations of solutions of Hamilton-Jacobi equations , 1984 .

[12]  Ian M. Mitchell,et al.  A Toolbox of Level Set Methods , 2005 .

[13]  Thomas Sonar,et al.  On the design of an upwind scheme for compressible flow on general triangulations , 1993, Numerical Algorithms.

[14]  V. Cristini,et al.  Adaptive unstructured volume remeshing - II: Application to two- and three-dimensional level-set simulations of multiphase flow , 2005 .

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

[16]  S. Osher,et al.  Geometric Level Set Methods in Imaging, Vision, and Graphics , 2011, Springer New York.

[17]  M. Sussman,et al.  A Coupled Level Set and Volume-of-Fluid Method for Computing 3D and Axisymmetric Incompressible Two-Phase Flows , 2000 .

[18]  Peter Frolkovič.  Flux-based methods of characteristics for coupled transport equations in porous media , 2004 .

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

[20]  J. Strain.  Semi-Lagrangian Methods for Level Set Equations , 1999 .

[21]  Jean-François Remacle,et al.  A quadrature-free discontinuous Galerkin method for the level set equation , 2006, J. Comput. Phys..

[22]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[23]  A. Reusken,et al.  A finite element based level set method for two-phase incompressible flows , 2006 .

[24]  K. Mikula,et al.  Flux-based level set method: A finite volume method for evolving interfaces , 2007 .

[25]  Xiaofeng Yang,et al.  An adaptive coupled level-set/volume-of-fluid interface capturing method for unstructured triangular grids , 2006, J. Comput. Phys..

[26]  S. Osher,et al.  A level set approach for computing solutions to incompressible two-phase flow , 1994 .

[27]  Björn Engquist,et al.  A finite element based level-set method for multiphase flow applications , 2000 .

[28]  Kokichi Sugihara,et al.  Stable marker-particle method for the Voronoi diagram in a flow field , 2007 .

[29]  Chi-Wang Shu,et al.  High-Order WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes , 2003, SIAM J. Sci. Comput..

[30]  S. Osher,et al.  Spatially adaptive techniques for level set methods and incompressible flow , 2006 .

[31]  G. Wittum,et al.  Two-Phase Flows on Interface Refined Grids Modeled with VOF, Staggered Finite Volumes, and Spline Interpolants , 2001 .

[32]  M. Falcone,et al.  Convergence Analysis for a Class of High-Order Semi-Lagrangian Advection Schemes , 1998 .

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

[34]  M. Falcone,et al.  Level Sets of Viscosity Solutions: some Applications to Fronts and Rendez-vous Problems , 1994, SIAM J. Appl. Math..

[35]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods , 1999 .

[36]  Peter Bastian,et al.  On a monotonicity preserving Eulerian–Lagrangian localized adjoint method for advection–diffusion equations , 2005 .

[37]  T. F. Russell,et al.  An overview of research on Eulerian-Lagrangian localized adjoint methods (ELLAM). , 2002 .

[38]  P. Koumoutsakos,et al.  A Lagrangian particle level set method. , 2005 .

[39]  J. Sethian,et al.  Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains , 1998 .

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

[41]  R. Abgrall.  Numerical discretization of the first‐order Hamilton‐Jacobi equation on triangular meshes , 1996 .

[42]  T. Barth,et al.  Finite Volume Methods: Foundation and Analysis , 2004 .

[43]  G. Kreiss,et al.  A conservative level set method for two phase flow II , 2005, Journal of Computational Physics.

[44]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[45]  Frank Losasso,et al.  A fast and accurate semi-Lagrangian particle level set method , 2005 .

[46]  Nicola Parolini,et al.  Mass preserving finite element implementations of the level set method , 2006 .

[47]  Mark Sussman,et al.  An Efficient, Interface-Preserving Level Set Redistancing Algorithm and Its Application to Interfacial Incompressible Fluid Flow , 1999, SIAM J. Sci. Comput..