Fast Semi-Lagrangian Schemes for the Eikonal Equation and Applications

Cristiani, Emiliano; Falcone, Maurizio

doi:10.1137/050637625

Cited by 58 publications

(67 citation statements)

References 24 publications

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“…This FMM scheme has been proved to be convergent, using a relation between the FMM solution and the numerical solution to finite difference schemes for the Level Sets formulation, for which it is known that these schemes are convergent (see Sethian, Vladimirsky [24] and Cristiani, Falcone [11]). More recently, the method has been extended to more general Hamilton-Jacobi equation by Sethian and Vladimirsky [24,25] and it has been also adapted to the case of time-dependent non-negative velocities c(x, t) ≥ 0 by Vladimirsky [28].…”

Section: (12)mentioning

confidence: 99%

Convergence of a Generalized Fast-Marching Method for an Eikonal Equation with a Velocity-Changing Sign

Carlini¹,

Falcone

Forcadel³

et al. 2008

SIAM J. Numer. Anal.

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We present a new Fast Marching algorithm for an eikonal equation with a velocity changing sign. This first order equation models a front propagation in the normal direction. The algorithm is an extension of the Fast Marching Method in two respects. The first is that the new scheme can deal with a time-dependent velocity and the second is that there is no restriction on its change in sign. We analyze the properties of the algorithm and we prove its convergence in the class of discontinuous viscosity solutions. Finally, we present some numerical simulations of fronts propagating in R 2 .

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Section: (12)mentioning

confidence: 99%

Convergence of a Generalized Fast-Marching Method for an Eikonal Equation with a Velocity-Changing Sign

Carlini¹,

Falcone

Forcadel³

et al. 2008

SIAM J. Numer. Anal.

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“…In section 2 we recall the FM method for the eikonal equation based on the semi-Lagrangian discretization introduced in [7]. We show why the FM technique does not work for more general equations.…”

Section: Introductionmentioning

confidence: 99%

“…The semi-Lagrangian scheme will be detailed in section 2.1 and will be used also for the new method proposed in this paper. In [7] it is also shown that the finite difference discretization (2) must be implemented carefully. If the numerical scheme is solved explicitly after evaluating the min's and max's using all the available values on the grid then the resulting second order equation in T i,j can have imaginary solutions.…”

Section: Introductionmentioning

confidence: 99%

“…The author and Falcone [8] introduced the Characteristic FM method for the eikonal equation which, similarly to Kim [16], accepts more than one node at the same time and it is faster than the FM method in most cases. Again the author and Falcone [7] (see also [5]) introduced the FM method based on the semi-Lagrangian discretization in the framework of control theory and minimum time problem. The semi-Lagrangian scheme is proved to be more accurate than the finite difference scheme classically used in FM method although they are both first order schemes.…”

Section: Introductionmentioning

confidence: 99%

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A Fast Marching Method for Hamilton-Jacobi Equations Modeling Monotone Front Propagations

Cristiani

2008

J Sci Comput

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In this paper we present a generalization of the Fast Marching method introduced by J. A. Sethian in 1996 to solve numerically the eikonal equation. The new method, named Buffered Fast Marching (BFM), is based on a semi-Lagrangian discretization and is suitable for Hamilton-Jacobi equations modeling monotonically advancing fronts, including Hamilton-Jacobi-Bellman and Hamilton-JacobiIsaacs equations which arise in the framework of optimal control problems and differential games. We also show the convergence of the algorithm to the viscosity solution. Finally we present several numerical tests proving that the BFM method is accurate and faster than the classical iterative algorithm in which every node of the grid is computed at every iteration.

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“…ease of implementation, fast solvers, or the convergence proof). Semi-Lagrangian schemes [FF02,CF07], are accurate, but they involve solving the characteristic ordinary differential equations, and are generally more complicated to implement. Central schemes [LT00] achieve second order accuracy, at the expense of a slightly more complicated, non-explicit formulation.…”

Section: Introductionmentioning

confidence: 99%

Filtered schemes for Hamilton–Jacobi equations: A simple construction of convergent accurate difference schemes

Oberman¹,

Salvador²

2015

Journal of Computational Physics

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Abstract. We build a simple and general class of finite difference schemes for first order Hamilton-Jacobi (HJ) Partial Differential Equations. These filtered schemes are convergent to the unique viscosity solution of the equation. The schemes are accurate: we implement second, third and fourth order accurate schemes in one dimension and second order accurate schemes in two dimensions, indicating how to build higher order ones. They are also explicit, which means they can be solved using the fast sweeping method. The accuracy of the method is validated with computational results for the eikonal equation and other HJ equations in one and two dimensions, using filtered schemes made from standard centered differences, higher order upwinding and ENO interpolation.

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Fast Semi-Lagrangian Schemes for the Eikonal Equation and Applications

Cited by 58 publications

References 24 publications

Convergence of a Generalized Fast-Marching Method for an Eikonal Equation with a Velocity-Changing Sign

Convergence of a Generalized Fast-Marching Method for an Eikonal Equation with a Velocity-Changing Sign

A Fast Marching Method for Hamilton-Jacobi Equations Modeling Monotone Front Propagations

Filtered schemes for Hamilton–Jacobi equations: A simple construction of convergent accurate difference schemes

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