Cost-effective Multiresolution schemes for Shock
 Computations

Chiavassa, Guillaume; Donat, Rosa; Martı́nez-Gavara, Anna

doi:10.1051/proc/2009052

Cited by 5 publications

(7 citation statements)

References 29 publications

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“…In order to achieve this goal within realistic delay the HPC aspect of the method must be enriched using adaptive mesh refinement. The multiresolution method developed in [5] and further tested and extended in [6] or [11] is currently implemented in this new configuration, where the presence of source terms and integral terms will require special considerations as exemplified for instance in [2].…”

Section: Resultsmentioning

confidence: 99%

“…In practice we choose ∆a = ∆γ. The discretization step ∆γ and the cellular maturity threshold γ s must be chosen so that the interfaces where the speed coefficients (2,3) are discontinuous fall on grid points covering the computational domain of size [N c , 1]. Denoting by N m the number of grid cells by half granulosa cell cycle, we set N γ = 2N m and ∆a = ∆γ = 1/N γ and we introduce the dedicated 2 notations for grid points (a k , γ l ) and mesh centers (a k+1/2 , γ l+1/2 ) a k = k∆a, a k+1/2 = (k + 1/2)∆a, for…”

Section: Discretizationmentioning

confidence: 99%

“…It appears in the explicit in time approximations of the speeds (2), and (3) at the center of the mesh g n k,l = g(a k+ 1 2 , γ l+ 1 2 , u n f ), h n k,l = h(a k+ 1 2 , γ l+ 1 2 , u n f ). Note that the transmission conditions (10) and (11) on the interfaces where these speed coefficients (2) and 3are discontinuous are exactly treated with the method described in the article of Godlewski and Raviart ( [10]). The numerical fluxes (25) are designed using a limiter strategy.…”

Section: Convective Partmentioning

confidence: 99%

“…( Denoting by E the evolution operator, that consists in steps (2) and 3, the second order in time is achieved by a second order Runge-Kutta method (Heun)…”

Section: Parallel Algorithm and Improvement To Order 2 In Timementioning

confidence: 99%

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Numerical simulation of the selection process of the ovarian follicles

et al. 2012

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This paper presents the design and implementation of a numerical method to simulate a multiscale model describing the selection process in ovarian follicles. The PDE model consists in a quasi-linear hyperbolic system of large size, namely N f × N f , ruling the time evolution of the cell density functions of N f follicles (in practice N f is of the order of a few to twenty). These equations are weakly coupled through the sum of the first order moments of the density functions. The timedependent equations make use of two structuring variables, age and maturity, which play the roles of space variables. The problem is naturally set over a compact domain of R 2. The formulation of the time-dependent controlled transport coefficients accounts for available biological knowledge on follicular cell kinetics. We introduce a dedicated numerical scheme that is amenable to parallelization, by taking advantage of the weak coupling. Numerical illustrations assess th e relevance of the proposed method both in term of accuracy and HPC achievements. Résumé. Ce document présente la conception et l'implémentation d'une méthode numérique servant a simuler un modèle multiéchelle décrivant le processus de sélection des follicules ovariens. Le modèle EDP consiste en un système hyperbolique quasi linéaire de grande taille, typiquement N f × N f , gouvernant l'évolution des fonctions de densité cellulaire pour N f follicules (en pratique N f est de l'ordre de quelques-unsà une vingtaine). Ceséquations d'évolution utilisent deux variables structurantes, l'âge et la maturité, qui jouent le rôle de variables d'espace. Le problème est naturellement posé sur un domaine compact de R 2. La formulation du transportà coefficients variables au cours du temps en fonction du contrôle est issue des connaissances disponibles sur la cinétique cellulaire au sein des follicules ovariens. Nous présentons un schéma numérique dédié au problème parallélisable en tirant parti du faible couplage. Des simulations numériques permettent d'évaluer la pertinence de la méthode proposée tant en terme de précision que de calcul haute performance.

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Section: Resultsmentioning

confidence: 99%

Section: Discretizationmentioning

confidence: 99%

Section: Convective Partmentioning

confidence: 99%

“…( Denoting by E the evolution operator, that consists in steps (2) and 3, the second order in time is achieved by a second order Runge-Kutta method (Heun)…”

Section: Parallel Algorithm and Improvement To Order 2 In Timementioning

confidence: 99%

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Numerical simulation of the selection process of the ovarian follicles

et al. 2012

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“…The TE algorithm in [1,11] is based on Harten's Multiresolution Framework (MRF), which provides a set of tools to manage data in a multi-scale setting. In the last two decades, the MRF has been successfully applied in various contexts, and in particular in the design of adaptive schemes for the numerical solution of conservation laws and systems [6][7][8]13]. The TE algorithm follows a strategy similar to that used in the cost-effective schemes described in [7] and used in [6,8,13].…”

Section: Introductionmentioning

confidence: 99%

High-accuracy approximation of piecewise smooth functions using the Truncation and Encode approach

Donat

López-Ureña

2017

Applied Mathematics and Nonlinear Sciences

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In the present work, we analyze a technique designed by Geraci et al. in [1,11] named the Truncate and Encode (TE) strategy. It was presented as a non-intrusive method for steady and non-steady Partial Differential Equations (PDEs) in Uncertainty Quantification (UQ), and as a weakly intrusive method in the unsteady case.We analyze the TE algorithm applied to the approximation of functions, and in particular its performance for piecewise smooth functions. We carry out some numerical experiments, comparing the performance of the algorithm when using different linear and non-linear interpolation techniques and provide some recommendations that we find useful in order to achieve a high performance of the algorithm.

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On the Use of Monotonicity-Preserving Interpolatory Techniques in Multilevel Schemes for Balance Laws

Baeza,

Donat,

Martínez-Gavara

2024

Commun. Appl. Math. Comput.

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Cost-effective multilevel techniques for homogeneous hyperbolic conservation laws are very successful in reducing the computational cost associated to high resolution shock capturing numerical schemes. Because they do not involve any special data structure, and do not induce savings in memory requirements, they are easily implemented on existing codes and are recommended for 1D and 2D simulations when intensive testing is required. The multilevel technique can also be applied to balance laws, but in this case, numerical errors may be induced by the technique. We present a series of numerical tests that point out that the use of monotonicity-preserving interpolatory techniques eliminates the numerical errors observed when using the usual 4-point centered Lagrange interpolation, and leads to a more robust multilevel code for balance laws, while maintaining the efficiency rates observed for hyperbolic conservation laws.

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Cost-effective Multiresolution schemes for Shock Computations

Cited by 5 publications

References 29 publications

Numerical simulation of the selection process of the ovarian follicles

Numerical simulation of the selection process of the ovarian follicles

High-accuracy approximation of piecewise smooth functions using the Truncation and Encode approach

On the Use of Monotonicity-Preserving Interpolatory Techniques in Multilevel Schemes for Balance Laws

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