A Level Set Framework for Capturing Multi-Valued Solutions of Nonlinear First-Order Equations

We introduce a level set method for the computation of multi-valued solutions of a general class of nonlinear first-order equations in arbitrary space dimensions. The idea is to realize the solution as well as its gradient as the common zero level set of several level set functions in the jet space. A very generic level set equation for the underlying PDEs is thus derived. Specific forms of the level set equation for both first-order transport equations and first-order Hamilton-Jacobi equations are presented. Using a local level set approach, the multi-valued solutions can be realized numerically as the projection of single-valued solutions of a linear equation in the augmented phase space. The level set approach we use automatically handles these solutions as they appear

[1]  S. Zagatti.  On viscosity solutions of Hamilton-Jacobi equations , 2008 .

[2]  Stanley Osher,et al.  A level set method for the computation of multi-valued solutions to quasi-linear hyperbolic PDE's and Hamilton-Jacobi equations , 2003 .

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

[4]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .

[5]  Björn Engquist,et al.  High-Frequency Wave Propagation by the Segment Projection Method , 2002 .

[6]  Yann Brenier,et al.  Averaged Multivalued Solutions for Scalar Conservation Laws , 1984 .

[7]  Dr. M. G. Worster.  Methods of Mathematical Physics , 1947, Nature.

[8]  Laurent Gosse,et al.  TWO MOMENT SYSTEMS FOR COMPUTING MULTIPHASE SEMICLASSICAL LIMITS OF THE SCHRÖDINGER EQUATION , 2003 .

[9]  Y. Brenier,et al.  A kinetic formulation for multi-branch entropy solutions of scalar conservation laws , 1998 .

[10]  Christof Sparber,et al.  Wigner functions versus WKB‐methods in multivalued geometrical optics , 2001 .

[11]  J. Benamou.  Direct computation of multivalued phase space solutions for Hamilton-Jacobi equations , 1999 .

[12]  S. Osher,et al.  Reflection in a Level Set Framework for Geometric Optics , 2004 .

[13]  S. Osher.  A level set formulation for the solution of the Dirichlet problem for Hamilton-Jacobi equations , 1993 .

[14]  Yoshikazu Giga,et al.  A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations , 2003, Math. Comput..

[15]  S. Osher,et al.  A level set-based Eulerian approach for anisotropic wave propagation , 2003 .

[16]  Stanley Osher,et al.  Numerical solution of the high frequency asymptotic expansion for the scalar wave equation , 1995 .

[17]  S. Osher,et al.  Motion of curves in three spatial dimensions using a level set approach , 2001 .

[18]  J. Sethian,et al.  FRONTS PROPAGATING WITH CURVATURE DEPENDENT SPEED: ALGORITHMS BASED ON HAMILTON-JACOB1 FORMULATIONS , 2003 .

[19]  S. Osher,et al.  COMPUTATIONAL HIGH-FREQUENCY WAVE PROPAGATION USING THE LEVEL SET METHOD, WITH APPLICATIONS TO THE SEMI-CLASSICAL LIMIT OF SCHRÖDINGER EQUATIONS∗ , 2003 .

[20]  Lawrence C. Evans.  A Geometric Interpretation of the Heat Equation with Multivalued Initial Data , 1996 .

[21]  L. Gosse.  Using K-Branch Entropy Solutions for Multivalued Geometric Optics Computations , 2002 .

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

[23]  Jean-David Benamou.  Regular ArticleBig Ray Tracing: Multivalued Travel Time Field Computation Using Viscosity Solutions of the Eikonal Equation , 1996 .

[24]  P. Lions,et al.  Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations. , 1984 .

[25]  P. Markowich,et al.  Quantum hydrodynamics, Wigner transforms, the classical limit , 1997 .

[26]  P. Markowich,et al.  Homogenization limits and Wigner transforms , 1997 .

[27]  Olof Runborg,et al.  Some new results in multiphase geometrical optics , 2000 .

[28]  Chohong Min,et al.  Simplicial isosurfacing in arbitrary dimension and codimension , 2003 .

[29]  B. Engquist,et al.  Computational high frequency wave propagation , 2003, Acta Numerica.

[30]  Jean-David Benamou,et al.  Big Ray Tracing , 1996 .

[31]  Björn Engquist,et al.  Acta Numerica 2003: Computational high frequency wave propagation , 2003 .

[32]  B. Engquist,et al.  Multi-phase computations in geometrical optics , 1996 .

[33]  Stanley Osher,et al.  Geometric Optics in a Phase-Space-Based Level Set and Eulerian Framework , 2002 .

[34]  S. Gray,et al.  Kirchhoff migration using eikonal equation traveltimes , 1994 .

[35]  Hailiang Liu,et al.  GEOMETRICAL SOLUTIONS OF HAMILTON-JACOBI EQUATIONS , 2004 .

[36]  Shi Jin,et al.  Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner , 2003 .