An admissibility and asymptotic preserving scheme for systems of conservation laws with source term on 2D unstructured meshes with high-order MOOD reconstruction

Blachère, Florian; Turpault, Rodolphe

doi:10.1016/j.cma.2017.01.012

Cited by 13 publications

(15 citation statements)

References 59 publications

(116 reference statements)

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“…Here we use the 1st order accurate FV scheme. While the MOOD paradigm is in use in several codes see for instance [16,5,2,9,11,3,33,6,17,41,4,43], some weaknesses can be pointed. First the a posteriori MOOD loop breaks the parallel efficiency because some cells demand more attention than others: they are possibly recomputed several times while (most) others are accepted at the end of the very first MOOD iteration.…”

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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“…On the other hand, the θ 1 and θ 2 coefficients are still defined by (39). To conclude, the rigorous analysis (admissibility, asymptotic-preserving property...) of the proposed second-order type extension is not easy, see for instance [7], and it is postponed to a forthcoming study.…”

Section: Accuracy Enhancementmentioning

confidence: 99%

“…In practice, the admissibility is checked at each time step. In case the numerical solution is not admissible, it is recomputed using the classical scheme with no reconstruction, in the spirit of the MOOD approach (see for instance [7] and the references therein).…”

Section: Accuracy Enhancementmentioning

confidence: 99%

An Antidiffusive HLL Scheme for the Electronic $M_1$ Model in the Diffusion Limit

Chalons¹,

Guisset²

2018

Multiscale Model. Simul.

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In this work, an asymptotic-preserving scheme is proposed for the electronic M 1 model in the diffusion limit. A very simple modification of the HLL numerical viscosity is considered in order to capture the correct asymptotic limit in the diffusion limit. This alteration also ensures the admissibility of the numerical solution under a suitable CFL condition. Interestingly, it is proved that the new scheme can also be understood as a Godunov-type scheme based on a suitable approximate Riemann solver. Various numerical test cases are performed and the results are compared with a standard HLL scheme and an explicit discretisation of the limit diffusion equation.

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“…Different from classical a priori limiting schemes, the Multi-dimensional Optimal Order Detection (MOOD), also known as a posteriori limiting scheme, was proposed in [10,19,21], and has been employed in various numerical formulations [7,20,6] and applications [11,22,8,36,4,24,61,56] and more recently [23,57,27]. In MOOD algorithm, some criterions or detectors were designed based on the computational stability and physical properties.…”

Section: Introductionmentioning

confidence: 99%

Solution property preserving reconstruction BVD+MOOD scheme for compressible euler equations with source terms and detonations

et al. 2020

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In "Solution Property Reconstruction for Finite Volume scheme: a BVD+MOOD framework", Int. J. Numer. Methods Fluids, 2020, we have designed a novel solution property preserving reconstruction, so-called multi-stage BVD-MOOD scheme. The scheme is able to maintain a high accuracy in smooth profile, eliminate the oscillations in the vicinity of discontinuity, capture sharply discontinuity and preserve some physical properties like the positivity of density and pressure for the Euler equations of compressible gas dynamics. In this paper, we present an extension of this approach for the compressible Euler equations supplemented with source terms (e.g., gravity, chemical reaction). One of the main challenges when simulating these models is the occurrence of negative density or pressure during the time evolution, which leads to a blow-up of the computation. General compressible Euler equations with different type of source terms are considered as models for physical situations such as detonation waves. Then, we illustrate the performance of the proposed scheme via a numerical test suite including genuinely demanding numerical tests. We observe that the present scheme is able to preserve the physical properties of the numerical solution still ensuring robustness and accuracy when and where appropriate.

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An admissibility and asymptotic preserving scheme for systems of conservation laws with source term on 2D unstructured meshes with high-order MOOD reconstruction

Cited by 13 publications

References 59 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

An Antidiffusive HLL Scheme for the Electronic $M_1$ Model in the Diffusion Limit

Solution property preserving reconstruction BVD+MOOD scheme for compressible euler equations with source terms and detonations

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