Relaxation schemes for curvature-dependent front propagation

Jin, Shi; Katsoulakis, Markos A.; Xin, Zhouping

doi:10.1002/(sici)1097-0312(199912)52:12<1587::aid-cpa4>3.0.co;2-a

Cited by 9 publications

(3 citation statements)

References 37 publications

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“…Thus this relaxation system yields asymptotically the level set equation of Osher-Sethian [26] for front propagating in the normal direction with speed V = 1 − ǫκ, where κ is the mean curvature. More detailed analysis, as well as its impact on numerical approximation to such a motion, is carried out in [15].…”

Section: Discussionmentioning

confidence: 99%

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

Jin¹,

Xin²

1998

SIAM J. Numer. Anal.

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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.

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Section: Discussionmentioning

confidence: 99%

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

Jin¹,

Xin²

1998

SIAM J. Numer. Anal.

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“…Several other relaxation approximation have been introduced recently. For example we mentioned here the work of Coquel and Perthame [10] for real gas computation, the relaxation schemes of Jin et al [19] for curvaturedependent front propagation, the relaxation approximation in the rapid granular flow [22], the relaxation approximation and relaxation schemes for diffusion and convection-diffusion problems [21,25,27,28]. Moreover, there are strong and interesting links between the relaxation approximation and the kinetic approach to nonlinear transport equations, based upon analogies with the passage from the Boltzmann equation to fluid mechanics (see for example [1,2,7]).…”

Section: Introductionmentioning

confidence: 99%

High-Order Relaxation Schemes for Nonlinear Degenerate Diffusion Problems

Cavalli

Naldi

Puppo

et al. 2007

SIAM J. Numer. Anal.

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Several relaxation approximations to partial differential equations have been recently proposed. Examples include conservation laws, HamiltonJacobi equations, convection-diffusion problems, gas dynamics problems. The present paper focuses onto diffusive relaxation schemes for the numerical approximation of nonlinear parabolic equations. These schemes are based on a suitable semilinear hyperbolic system with relaxation terms. High order methods are obtained by coupling ENO and WENO schemes for space discretization with IMEX schemes for time integration. Error estimates and convergence analysis are developed for semidiscrete schemes with numerical analysis for fully discrete relaxed schemes. Various numerical results in one and two dimensions illustrate the high accuracy and good properties of the proposed numerical schemes, also in the degenerate case. These schemes can be easily implemented on parallel computers and applied to more general systems of nonlinear parabolic equations in twoand three-dimensional cases.

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“…Luo [8] studied the stability of rarefaction wave in the scalar, multidimensional setting of the Jin-Xin model and Luo and Xin [9] showed nonlinear stability of the traveling wave solution in the scalar, multidimensional case. The relaxation approximation was later generalized to Hamilton-Jacobi equation [6] and to curvature dependent front propagation [4].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic towards rarefaction wave of the Jin–Xin relaxation model for the $p$ system

Wang¹

2001

Methods and Applications of Analysis

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We study the asymptotic equivalence of the Jin-Xin relaxation model to its formal limit of genuinely nonlinear 2 by 2 conservation laws (isentropic Euler equation in Lagrangian coordinate). We consider the case where the initial data are allowed to have jump discontinuities such that the corresponding solutions to the Euler equation contain centered rarefaction waves. In particular, Riemann data connected by rarefaction curves are included. We show that, as long as the initial data is a small perturbation of a non-vacuum constant state, the solution for the relaxation system exists globally in time and converges, as e -> 0, to the solution of the corresponding Euler equation uniformly except for an initial layer whose width is essentially of order O(e).

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Relaxation schemes for curvature-dependent front propagation

Cited by 9 publications

References 37 publications

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

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

High-Order Relaxation Schemes for Nonlinear Degenerate Diffusion Problems

Asymptotic towards rarefaction wave of the Jin–Xin relaxation model for the $p$ system

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