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Arc length based WENO scheme for Hamilton-Jacobi Equations
In this article, novel smoothness indicators are presented for calculati...
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A High-Order Scheme for Image Segmentation via a modified Level-Set method
The method is based on an adaptive "filtered" scheme recently introduced...
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Simple smoothness indicator WENO-Z scheme for hyperbolic conservation laws
The advantage of WENO-JS5 scheme [ J. Comput. Phys. 1996] over the WENO-...
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An order-adaptive compact approximation Taylor method for systems of conservation laws
We present a new family of high-order shock-capturing finite difference ...
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A novel method for constructing high accurate and robust WENO-Z type scheme
A novel method for constructing robust and high-order accurate weighted ...
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Nonlinear Quality of Life Index
We present details of the analysis of the nonlinear quality of life inde...
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Improving accuracy of the fifth-order WENO scheme by using the exponential approximation space
The aim of this study is to develop a novel WENO scheme that improves th...
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Multidimensional smoothness indicators for first-order Hamilton-Jacobi equations
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(Δ t,Δ x)>0 which goes to 0 as (Δ t,Δ 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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