A discontinuous Galerkin scheme for front propagation with obstacles

Bokanowski, Olivier; Cheng, Yingda; Shu, Chi-Wang

doi:10.1007/s00211-013-0555-3

Cited by 12 publications

(17 citation statements)

References 23 publications

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“…The iterations are stopped when the difference between too successive time step is small enough or a fixed number of iterations is passed, i.e., in this example, Example 6 Advection with an obstacle. Here we consider an obstacle problem, which is taken from [3]:…”

Section: Example 1 Eikonal Equation We Consider the Case Ofmentioning

confidence: 99%

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An Efficient Filtered Scheme for Some First Order Time-Dependent Hamilton--Jacobi Equations

Bokanowski¹,

Falcone²,

Sahu³

2016

SIAM J. Sci. Comput.

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We introduce a new class of "filtered" schemes for some first order non-linear Hamilton-Jacobi-Bellman equations. The work follows recent ideas of Froese and Oberman (SIAM J. Numer. Anal., Vol 51, pp.423-444, 2013). The proposed schemes are not monotone but still satisfy some -monotone property. Convergence results and precise error estimates are given, of the order of √ ∆x where ∆x is the mesh size. The framework allows to construct finite difference discretizations that are easy to implement, high-order in the domains where the solution is smooth, and provably convergent, together with error estimates. Numerical tests on several examples are given to validate the approach, also showing how the filtered technique can be applied to stabilize an otherwise unstable high-order scheme.

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Section: Example 1 Eikonal Equation We Consider the Case Ofmentioning

confidence: 99%

“…(Example 6) Plots at T=0(initial data), T=0.3, T=0.Example Eikonal with an obstacle. We consider an Eikonal equation with an obstacle term, also taken from[3]:min(v t + |v x |, v − g(x)) = 0, t > 0, x ∈ [−1, 1],(3.20) v 0 (x) = 0.5 + sin(πx) x ∈ [−1, 1], (3.21)…”

mentioning

confidence: 99%

An Efficient Filtered Scheme for Some First Order Time-Dependent Hamilton--Jacobi Equations

Bokanowski¹,

Falcone²,

Sahu³

2016

SIAM J. Sci. Comput.

Self Cite

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“…Some of them deal with finite difference methods as the first-order monotone schemes proposed in the work of Crandall and Lions 28 or higher ones (Essentially Non Oscillatory (ENO) scheme in the work of Osher and Shu 29 and Weighted Essentially Non Oscillatory (WENO) scheme in the work of Zhang et al 30 ). A second class of methods concerns discontinuous Galerkin method, a direct method was proposed in the work of Cheng and Shu 31 and a scheme for front propagation with obstacles in the work of Bokanowski et al 32 Then, we can also consider semi-Lagrangian schemes, which are based on the discretization of the dynamic programming principle as developed in the works of Falcone and Ferretti 33,34 and Carlini et al, 35 for instance. A brief review of different efficient techniques that have been proposed can also be found in the work of Cacace et al 36 In this paper, we use the Ultra-Bee scheme (a finite difference type scheme) to solve the HJB equations (12)- (13).…”

Section: Ultra-bee Scheme For Hjb Resolutionmentioning

confidence: 99%

On a Hamilton‐Jacobi‐Bellman approach for coordinated optimal aircraft trajectories planning

Parzani

Puechmorel

2017

Optim Control Appl Methods

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Summary In the context of future air traffic management, an increasing importance is given to environmental considerations and especially fuel consumption. It is thus advisable to make an optimal use of external conditions knowledge, like wind or temperature, to reduce the total fuel needed to complete a flight. On the other hand, safety must be guaranteed all over the trajectory: encounters below the regulatory separation minima, termed as conflicts, must never occur. In this paper, we consider the problem of optimally planning conflict free aircraft trajectories, based on a minimal time criterion and taking into account an ambient wind field. Aircraft motions are restricted to the horizontal plane only to reduce the dimension of the problem and to avoid costly changes of flight level. Since global optimality for the whole set of aircraft is sought after, the admissible space is modeled after a Cartesian product of the individual, 2‐dimensional state spaces with forbidden configurations removed. A Hamilton‐Jacobi‐Bellman approach in the so‐called configuration space obtained that way is then applied to get a coordinated, conflict‐free optimal planning. One of the main advantages of this method is the insurance of getting a global optimum without having to rely on a complex initialization procedure. In this work, the Ultra‐Bee scheme is adapted to configuration spaces and solved in a 4‐dimensional plus time setting.

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“…Another recent reference [15] gives an overview on semi-Lagrangian schemes and the features of this class of methods. For a finite element approach we refer to [28], and for discontinuous Galerkin methods to [7]. Another class of antidiffusive methods have also been studied in [8,12].…”

Section: Discretization In Spacementioning

confidence: 99%

“…In the last decades, several theoretical and numerical developments in HJB theory led to powerful and efficient numerical approaches that can be used for control problems up to 6-dimensional problems [2,7,10,8,15,37,40]. For higher dimensional problems, various approaches have been studied in the literature, including model reduction or advanced numerical schemes as, e.g., sparse grids, see [11].…”

mentioning

confidence: 99%

Optimal feedback control for undamped wave equations by solving a HJB equation

2015

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Abstract. An optimal finite-time horizon feedback control problem for (semi-linear) wave equations is presented. The feedback law can be derived from the dynamic programming principle and requires to solve the evolutionary Hamilton-Jacobi-Bellman (HJB) equation. Classical discretization methods based on finite elements lead to approximated problems governed by ODEs in high dimensional spaces which makes the numerical resolution by the HJB approach infeasible. In the present paper, an approximation based on spectral elements is used to discretize the wave equation. The effect of noise is considered and numerical simulations are presented to show the relevance of the approach.

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A discontinuous Galerkin scheme for front propagation with obstacles

Cited by 12 publications

References 23 publications

An Efficient Filtered Scheme for Some First Order Time-Dependent Hamilton--Jacobi Equations

An Efficient Filtered Scheme for Some First Order Time-Dependent Hamilton--Jacobi Equations

On a Hamilton‐Jacobi‐Bellman approach for coordinated optimal aircraft trajectories planning

Optimal feedback control for undamped wave equations by solving a HJB equation

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