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

Oberman, Adam M.; Salvador, Tiago

doi:10.1016/j.jcp.2014.12.039

Cited by 22 publications

(18 citation statements)

References 28 publications

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“…Using analogous reasoning it is straightforward to show that for the other two cases of the square stencil we can obtain (25), (22), respectively. In fact, it is enough to develop the points on Γ along ±η∆ using ∇ ± f , which are always well defined.…”

Section: Proof Of Point (Ii)mentioning

confidence: 90%

Multidimensional smoothness indicators for first-order Hamilton-Jacobi equations

Falcone

Paolucci

Tozza

2020

Journal of Computational Physics

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The lack of smoothness is a common feature of weak solutions of nonlinear hyperbolic equations and is a crucial issue in their approximation. This has motivated several efforts to define appropriate indicators, based on the values of the approximate solutions, in order to detect the most troublesome regions of the domain. This information helps to adapt the approximation scheme in order to avoid spurious oscillations when using high-order schemes. In this paper we propose a genuinely multidimensional extension of the WENO procedure in order to overcome the limitations of indicators based on dimensional splitting. Our aim is to obtain new regularity indicators for problems in 2D and apply them to a class of "adaptive filtered" schemes for first order evolutive Hamilton-Jacobi equations. According to the usual procedure, filtered schemes are obtained by a simple coupling of a high-order scheme and a monotone scheme. The mixture is governed by a filter function F and by a switching parameter ε n = ε n (∆t, ∆x) > 0 which goes to 0 as (∆t, ∆x) is going to 0. The adaptivity is related to the smoothness indicators and allows to tune automatically the switching parameter ε n j in time and space. Several numerical tests on critical situations in 1D and 2D are presented and confirm the effectiveness of the proposed indicators and the efficiency of our scheme.

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Section: Proof Of Point (Ii)mentioning

confidence: 90%

Multidimensional smoothness indicators for first-order Hamilton-Jacobi equations

Falcone

Paolucci

Tozza

2020

Journal of Computational Physics

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“…where we chose the value b = 1.05 in order to make the function approach better the value 1 for x = −1, 1. Finally, we recall also the filter defined in [18] as After extensive computations, we noticed that the results obtained with our adaptive filtered (AF) scheme are not sensitive with respect to changes in regularity of the filter function, even with very large transition phases. That is probably because, as we will see in the next section, the parameter ε n is designed to obtain the property (F1) whenever possible, then in regions of regularity of the solution the argument of F lies most probably in [−1, 1], where all the filter functions are practically the same.…”

Section: Filter Functionmentioning

confidence: 99%

“…In recent years a general approach to the construction of high-order methods using filters has been proposed by Lions and Souganidis in [17] and further developed by Oberman and Salvador [18]. Let us remind that a typical feature of a filtered scheme S F is that at the node x j the scheme is combination of a high-order scheme S A and a monotone scheme S M according to a filter function F .…”

Section: Introductionmentioning

confidence: 99%

Adaptive Filtered Schemes for First Order Hamilton-Jacobi Equations

Falcone

Paolucci

Tozza

2019

Lecture Notes in Computational Science and Engineering

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We consider a class of "filtered" schemes for first order time dependent Hamilton-Jacobi equations and prove a general convergence result for this class of schemes. A typical filtered scheme is obtained mixing a high-order scheme and a monotone scheme according to a filter function F which decides where the scheme has to switch from one scheme to the other. A crucial role for this switch is played by a parameter ε = ε(∆t, ∆x) > 0 which goes to 0 as (∆t, ∆x) is going to 0 and does not depend on the time t n . The tuning of this parameter in the code is rather delicate and has an influence on the global accuracy of the filtered scheme. Here we introduce an adaptive and automatic choice of ε = ε n (∆t, ∆x) at every iteration modifying the classical set up. The adaptivity is controlled by a smoothness indicator which selects the regions where we modify the regularity threshold ε n . A convergence result and some error estimates for the new adaptive filtered scheme are proved, this analysis relies on the properties of the scheme and of the smoothness indicators. Finally, we present some numerical tests to compare the adaptive filtered scheme with other methods.

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“…The crucial point is the coupling between the two schemes which is obtained via a filter function which selects which scheme has to be applied at a node of the grid in order to guarantee (under appropriate assumptions) a global convergence. The construction of these schemes is rather simple as explained by Oberman and Salvador [19] because one can couple various numerical methods and leave the filter function decide the switch between the two schemes. A general convergence result has been proved by Bokanowski, Falcone and Sahu in [5] and recently improved by Falcone, Paolucci and Tozza [11] with an adaptive and automatic choice of the parameter governing the switch.…”

Section: Introductionmentioning

confidence: 99%

A High-Order Scheme for Image Segmentation via a Modified Level-Set Method

Falcone¹,

Paolucci²,

Tozza³

2020

SIAM J. Imaging Sci.

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In this paper we propose an accurate method for image segmentation following the level-set approach. The method is based on an adaptive "filtered" scheme recently introduced by the authors. The main feature of the scheme is the possibility to stabilize an a priori unstable high-order scheme via a filter function which allows to combine a high-order scheme in the regularity regions and a monotone scheme elsewhere, in presence of singularities. The filtered scheme considered in this paper uses the local Lax-Friedrichs scheme as monotone scheme and the Lax-Wendroff scheme as high-order scheme but other couplings are possible. Moreover, we introduce also a modified velocity function for the level-set model used in segmentation, this velocity allows to obtain more accurate results with respect to other velocities proposed in the literature. Some numerical tests on synthetic and real images confirm the accuracy of the proposed method and the advantages given by the new velocity.

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

Cited by 22 publications

References 28 publications

Multidimensional smoothness indicators for first-order Hamilton-Jacobi equations

Multidimensional smoothness indicators for first-order Hamilton-Jacobi equations

Adaptive Filtered Schemes for First Order Hamilton-Jacobi Equations

A High-Order Scheme for Image Segmentation via a Modified Level-Set Method

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