a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

Clain, Stéphane; Loubère, Raphaël; Machado, Gaspar J.

doi:10.1007/s10444-017-9556-6

Cited by 9 publications

(23 citation statements)

References 55 publications

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“…Second, the MOOD paradigm has a fundamental explicit nature due to the a posteriori check and the local degree decrementing. Even if some recent works explore its extension to implicit schemes [10,27], a fully a priori version would be more convenient and presumably more efficient even in parallel. Third, the numerical admissible detection criteria based on a relaxed DMP in the detection is not based on a rigorous theoretical and mathematical base and demands one parameter to be fixed.…”

Section: Introductionmentioning

confidence: 99%

A Priori Neural Networks Versus A Posteriori MOOD Loop: A High Accurate 1D FV Scheme Testing Bed

2020

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In this work we present an attempt to replace an a posteriori MOOD loop used in a high accurate Finite Volume (FV) scheme by a trained artificial Neural Network (NN). The MOOD loop, by decrementing the reconstruction polynomial degrees, ensures accuracy, essentially non-oscillatory, robustness properties and preserves physical features. Indeed it replaces the classical a priori limiting strategy by an a posteriori troubled cell detection, supplemented with a local time-step re-computation using a lower order FV scheme (ie lower polynomial degree reconstructions). We have trained shallow NNs made of only two so-called hidden layers and few perceptrons which a priori produces an educated guess (classification) of the appropriate polynomial degree to be used in a given cell knowing the physical and numerical states in its vicinity. We present a proof of concept in 1D. The strategy to train and use such NNs is described on several 1D toy models: scalar advection and Burgers' equation, the isentropic Euler and radiative M1 systems. Each toy model brings new difficulties which are enlightened on the obtained numerical solutions. On these toy models, and for the proposed test cases, we observe that an artificial NN can be trained and substituted to the a posteriori MOOD loop in mimicking the numerical admissibility criteria and predicting the appropriate polynomial degree to be employed safely. The physical admissibility criteria is however still dealt with the a posteriori MOOD loop. Constructing a valid training data set is of paramount importance, but once available, the numerical scheme supplemented with NN produces promising results in this 1D setting.

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Section: Introductionmentioning

confidence: 99%

A Priori Neural Networks Versus A Posteriori MOOD Loop: A High Accurate 1D FV Scheme Testing Bed

2020

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“…In the present work, the a posteriori paradigm is used. In particular, we use the multidimensional Optimal Order Detection (MOOD) limiting procedure [24,32,33,[43][44][45][46]. In this a posteriori method, a candidate solution is obtained by solving the system of equations using the most accurate method at disposal without any stabilization mechanism.…”

Section: Mood Method: a Posteriori Stabilizationmentioning

confidence: 99%

A Well-Balanced SPH-ALE Scheme for Shallow Water Applications

et al. 2021

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In this work, a new discretization of the source term of the shallow water equations with non-flat bottom geometry is proposed to obtain a well-balanced scheme. A Smoothed Particle Hydrodynamics Arbitrary Lagrangian-Eulerian formulation based on Riemann solvers is presented to solve the SWE. Moving-Least Squares approximations are used to compute high-order reconstructions of the numerical fluxes and, stability is achieved using the a posteriori MOOD paradigm. Several benchmark 1D and 2D numerical problems are considered to test and validate the properties and behavior of the presented schemes.

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“…In [4] numerical experimentation has been carried on advection, Bürgers', and Euler equations, and we have demonstrated that the scheme manages to retrieve smooth solutions with optimal order of accuracy but also irregular solutions without spurious oscillations. The a posteriori MOOD loop allows to produce a valid solution by polynomial degree decrementing.…”

Section: Introductionmentioning

confidence: 97%

“…In a recent work [4] we have designed a new family of high order accurate Finite Volume (FV) schemes. High-accuracy is achieved thanks to polynomial reconstructions on centered stencils while stability and robustness are gained by an a posteriori Multidimensional Optimal Order Detection (MOOD) method which controls the cell polynomial degree, eliminating non-physical oscillations in the vicinity of discontinuities by a reduction up to degree zero when needed.…”

Section: Introductionmentioning

confidence: 99%

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a posteriori stabilized sixth-order finite volume scheme with adaptive stencil construction -- Basics for the 1D steady-state hyperbolic equations

Machado¹,

Clain²,

Loubère³

2021

Preprint

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We propose an adaptive stencil construction for high order accurate finite volume schemes a posteriori stabilized devoted to solve one-dimensional steady-state hyperbolic equations. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructions while stability is provided with an a posteriori MOOD method which controls the cell polynomial degree for eliminating non-physical oscillations in the vicinity of discontinuities. We supplemented this scheme with a stencil construction allowing to reduce even further the numerical dissipation. The stencil is shifted away from troubles (shocks, discontinuities, etc.) leading to less oscillating polynomial reconstructions. Experimented on linear, Bürgers', and Euler equations, we demonstrate that the adaptive stencil technique manages to retrieve smooth solutions with optimal order of accuracy but also irregular ones without spurious oscillations. Moreover we numerically show that the approach allows to reduce the dissipation still maintaining the essentially non-oscillatory behavior.

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a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

Cited by 9 publications

References 55 publications

A Priori Neural Networks Versus A Posteriori MOOD Loop: A High Accurate 1D FV Scheme Testing Bed

A Priori Neural Networks Versus A Posteriori MOOD Loop: A High Accurate 1D FV Scheme Testing Bed

A Well-Balanced SPH-ALE Scheme for Shallow Water Applications

a posteriori stabilized sixth-order finite volume scheme with adaptive stencil construction -- Basics for the 1D steady-state hyperbolic equations

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