Numerical Passage from Systems of Conservation Laws to Hamilton--Jacobi Equations, and Relaxation Schemes

In this paper we study the numerical transition from a Hamilton--Jacobi (H--J) equation to its associated system of conservation laws in arbitrary space dimensions. We first study how, in a very generic setting, one can recover the viscosity solutions of the H--J equation using the numerical solutions to the system of conservation laws. We then introduce a simple, second-order relaxation scheme to solve the underlying weakly hyperbolic system of conservation laws.

[1]  S. Kruzhkov.  The method of finite differences for a first-order non-linear equation with many independent variables , 1966 .

[2]  S. Kružkov,et al.  GENERALIZED SOLUTIONS OF NONLINEAR FIRST ORDER EQUATIONS WITH SEVERAL INDEPENDENT VARIABLES. II , 1967 .

[3]  S. Kružkov.  GENERALIZED SOLUTIONS OF THE HAMILTON-JACOBI EQUATIONS OF EIKONAL TYPE. I. FORMULATION OF THE PROBLEMS; EXISTENCE, UNIQUENESS AND STABILITY THEOREMS; SOME PROPERTIES OF THE SOLUTIONS , 1975 .

[4]  P. Lions.  Generalized Solutions of Hamilton-Jacobi Equations , 1982 .

[5]  P. Lions,et al.  Viscosity solutions of Hamilton-Jacobi equations , 1983 .

[6]  M. Slemrod,et al.  DYNAMICS OF FIRST ORDER PHASE TRANSITIONS , 1984 .

[7]  P. Sweby.  High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws , 1984 .

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

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

[10]  Woodrow Whitlow,et al.  Entropy Condition Satisfying Approximations for the Full Potential Equation of Transonic Flow , 1985 .

[11]  P. Lax.  Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves , 1987 .

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

[13]  S. Osher,et al.  Algorithms Based on Hamilton-Jacobi Formulations , 1988 .

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

[15]  S. Osher,et al.  The nonconvex multi-dimensional Riemann problem for Hamilton-Jacobi equations , 1991 .

[16]  V. Caselles.  Scalar conservation laws and Hamilton-Jacobi equations in one-space variable , 1992 .

[17]  C. D. Levermore,et al.  Hyperbolic conservation laws with stiff relaxation terms and entropy , 1994 .

[18]  Shi Jin,et al.  Numerical Integrations of Systems of Conservation Laws of Mixed Type , 1995, SIAM J. Appl. Math..

[19]  P. Souganidis,et al.  Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton-Jacobi equations , 1995 .

[20]  M. Falcone,et al.  Numerical schemes for conservation laws via Hamilton-Jacobi equations , 1995 .

[21]  Z. Xin,et al.  The relaxation schemes for systems of conservation laws in arbitrary space dimensions , 1995 .

[22]  R. Natalini.  Convergence to equilibrium for the relaxation approximations of conservation laws , 1996 .

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

[24]  Bernardo Cockburn,et al.  A Model Numerical Scheme for the Propagation of phase Transitions in Solids , 1996, SIAM J. Sci. Comput..

[25]  L. Corrias.  Fast Legendre--Fenchel Transform and Applications to Hamilton--Jacobi Equations and Conservation Laws , 1996 .

[26]  Shi Jin,et al.  The Effects of Numerical Viscosities , 1996 .

[27]  B. V. Leer,et al.  Towards the Ultimate Conservative Difference Scheme , 1997 .

[28]  A. Tzavaras,et al.  CONTRACTIVE RELAXATION SYSTEMS AND THE SCALARMULTIDIMENSIONAL CONSERVATION , 1997 .

[29]  Shi Jin,et al.  Relaxation approximations to front propagation , 1997 .

[30]  Eitan Tadmor,et al.  Nonoscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws , 1998, SIAM J. Sci. Comput..

[31]  R. Natalini.  A Discrete Kinetic Approximation of Entropy Solutions to Multidimensional Scalar Conservation Laws , 1998 .

[32]  ASYMPTOTIC LIMIT OF INITIAL BOUNDARY VALUE PROBLEMS FOR CONSERVATION LAWS WITH RELAXATIONAL EXTENSIONS , 1998 .

[33]  Z. Xin,et al.  Relaxation schemes for curvature-dependent front propagation , 1999 .