Spectral Analysis of High Order Continuous FEM for Hyperbolic PDEs on Triangular Meshes: Influence of Approximation, Stabilization, and Time-Stepping

In this work we study various continuous finite element discretization for two dimensional hyperbolic partial differential equations, varying the polynomial space (Lagrangian on equispaced, Lagrangian on quadrature points (Cubature) and Bernstein), the stabilization techniques (streamline-upwind Petrov–Galerkin, continuous interior penalty, orthogonal subscale stabilization) and the time discretization (Runge–Kutta (RK), strong stability preserving RK and deferred correction). This is an extension of the one dimensional study by Michel et al. (J Sci Comput 89(2):31, 2021. https://doi.org/10.1007/s10915-021-01632-7), whose results do not hold in multi-dimensional frameworks. The study ranks these schemes based on efficiency (most of them are mass-matrix free), stability and dispersion error, providing the best CFL and stabilization coefficients. The challenges in two-dimensions are related to the Fourier analysis. Here, we perform it on two types of periodic triangular meshes varying the angle of the advection, and we combine all the results for a general stability analysis. Furthermore, we introduce additional high order viscosity to stabilize the discontinuities, in order to show how to use these methods for tests of practical interest. All the theoretical results are thoroughly validated numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that Cubature elements combined with SSPRK and OSS stabilization is the most promising combination

Spectral Analysis of Continuous FEM for Hyperbolic PDEs: Influence of Approximation, Stabilization, and Time-Stepping

We study continuous finite element dicretizations for one dimensional hyperbolic partial differential equations. The main contribution of the paper is to provide a fully discrete spectral analysis, which is used to suggest optimal values of the CFL number and of the stabilization parameters involved in different types of stabilization operators. In particular, we analyze the streamline-upwind Petrov–Galerkin stabilization technique, the continuous interior penalty (CIP) stabilization method and the orthogonal subscale stabilization (OSS). Three different choices for the continuous finite element space are compared: Bernstein polynomials, Lagrangian polynomials on equispaced nodes, and Lagrangian polynomials on Gauss-Lobatto cubature nodes. For the last choice, we only consider inexact quadrature based on the formulas corresponding to the degrees of freedom of the element, which allows to obtain a fully diagonal mass matrix. We also compare different time stepping strategies, namely Runge–Kutta (RK), strong stability preserving RK (SSPRK) and deferred correction time integration methods. The latter allows to alleviate the computational cost as the mass matrix inversion is replaced by the high order correction iterations. To understand the effects of these choices, both time-continuous and fully discrete Fourier analysis are performed. These allow to compare all the different combinations in terms of accuracy and stability, as well as to provide suggestions for optimal values discretization parameters involved. The results are thoroughly verified numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that cubature elements combined with SSPRK and CIP or OSS stabilization are the most promising combinations

Spectral Analysis of Continuous FEM for Hyperbolic PDEs: Influence of Approximation, Stabilization, and Time-Stepping

International audienceWe study continuous finite element dicretizations for one dimensional hyperbolic partial differential equations. The main contribution of the paper is to provide a fully discrete spectral analysis, which is used to suggest optimal values of the CFL number and of the stabilization parameters involved in different types of stabilization operators. In particular, we analyze the streamline-upwind Petrov-Galerkin (SUPG) stabilization technique, the continuous interior penalty (CIP) stabilization method and the local projection stabilization (LPS). Three different choices for the continuous finite element space are compared: Bernstein polynomials, Lagrangian polynomials on equispaced nodes, and Lagrangian polynomials on Gauss-Lobatto cubature nodes. For the last choice, we only consider inexact quadrature based on the formulas corresponding to the degrees of freedom of the element, which allows to obtain a fully diagonal mass matrix. We also compare different time stepping strategies, namely Runge-Kutta (RK), strong stability preserving RK (SSPRK) and deferred correction time integration methods. The latter allows to alleviate the computational cost as the mass matrix inversion is replaced by the high order correction iterations. To understand the effects of these choices, both time-continuous and fully discrete Fourier analysis are performed. These allow to compare all the different combinations in terms of accuracy and stability, as well as to provide suggestions for optimal values discretization parameters involved. The results are thoroughly verified numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that cubature elements combined with SSPRK and CIP or LPS stabilization are the most promising combinations

Spectral analysis of high order continuous FEM for hyperbolic PDEs on triangular meshes: influence of approximation, stabilization, and time-stepping

arXiv admin note: text overlap with arXiv:2103.16158International audienceIn this work we study various continuous finite element discretization for two dimensional hyperbolic partial differential equations, varying the polynomial space (Lagrangian on equispaced, Lagrangian on quadrature points (Cubature) and Bernstein), the stabilization techniques (streamline-upwind Petrov-Galerkin, continuous interior penalty, orthogonal subscale stabilization) and the time discretization (Runge-Kutta (RK), strong stability preserving RK and deferred correction). This is an extension of the one dimensional study by Michel S. et al J. Sci. Comput. (2021), whose results do not hold in multi-dimensional frameworks. The study ranks these schemes based on efficiency (most of them are mass-matrix free), stability and dispersion error, providing the best CFL and stabilization coefficients. The challenges in two-dimensions are related to the Fourier analysis. Here, we perform it on two types of periodic triangular meshes varying the angle of the advection, and we combine all the results for a general stability analysis. Furthermore, we introduce additional high order viscosity to stabilize the discontinuities, in order to show how to use these methods for tests of practical interest. All the theoretical results are thoroughly validated numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that Cubature elements combined with SSPRK and OSS stabilization is the most promising combination

Hypersonic flows for reentry problems

INRIA and GAMNI/SMAI have organized a Workshop open to the international scientific community on hypersonic flows for reentry problems which was held in Palais des Congres d'Antibes in January 1990 (part I) and April 1991 (part II). This workshop was focussed on the issues of validation of numerical methodolies for the computation of high-Mach number flows, and gathered experts in Scientifical computing, fluid mechanics and experimentalists associated with the R and D Hermes program. Several volumes are devoted to the results and the detailed syntheses. This report presents a general synthesis of the motivations for this initiative, the development of the meetings, and the main conclusions drawn. A shorter version of this document has served as notes corresponding to a lecture given at an AGARD meeting

Near-to mid-IR spectral purity transfer with a tunable frequency comb: methanol frequency metrology over a record frequency span

We report the development and operation of a frequency-comb-assisted high-resolution mid-infrared molecular spectrometer combining high spectral purity, SI-traceability, wide tunability and high sensitivity. An optical frequency comb is used to transfer the spectral purity of a SI-traceable 1.54 $\mu$m metrology-grade frequency reference to a 10.3 $\mu$m quantum cascade laser (QCL). The near-infrared reference is operated at the French time/frequency metrology institute, calibrated there to primary frequency standards, and transferred to Laboratoire de Physique des Lasers via the REFIMEVE fiber network. The QCL exhibits a sub-10 --15 frequency stability from 0.1 to 10 s and its frequency is traceable to the SI with a total uncertainty better than 4 x 10 --14 after 1-s averaging time. We have developed the instrumentation allowing comb modes to be continuously tuned over 9 GHz resulting in a QCL of record spectral purity uninterruptedly tunable at the precision of the reference over an unprecedented span of 1.4 GHz. We have used our apparatus to conduct sub-Doppler spectroscopy of methanol in a multi-pass cell, demonstrating state-of-art frequency uncertainties down to the few kilohertz level. We have observed weak intensity resonances unreported so far, resolved subtle doublets never seen before and brought to light discrepancies with the HITRAN database. This demonstrates the potential of our apparatus for probing subtle internal molecular processes, building accurate spectroscopic models of polyatomic molecules of atmospheric or astrophysical interest, and carrying out precise spectroscopic tests of fundamental physics

Quantum cascade laser frequency stabilisation at the sub-Hz level

Quantum Cascade Lasers (QCL) are increasingly being used to probe the mid-infrared "molecular fingerprint" region. This prompted efforts towards improving their spectral performance, in order to reach ever-higher resolution and precision. Here, we report the stabilisation of a QCL onto an optical frequency comb. We demonstrate a relative stability and accuracy of 2x10-15 and 10-14, respectively. The comb is stabilised to a remote near-infrared ultra-stable laser referenced to frequency primary standards, whose signal is transferred via an optical fibre link. The stability and frequency traceability of our QCL exceed those demonstrated so far by two orders of magnitude. As a demonstration of its capability, we then use it to perform high-resolution molecular spectroscopy. We measure absorption frequencies with an 8x10-13 relative uncertainty. This confirms the potential of this setup for ultra-high precision measurements with molecules, such as our ongoing effort towards testing the parity symmetry by probing chiral species

The LNE-SYRTE clock ensemble and applications

International audienc

The LNE-SYRTE clock ensemble and applications

International audienc

Evaluation des performances de la fontaine atomique PHARAO, Participation à l'étude de l'horloge spatiale PHARAO

The performances of an atomic frequency standard depend drastically on the observation time of the atoms. The interrogation of laser cooled atoms allows to obtain about half a second interaction time in a fountain geometry. This duration could be much more varied in absence of gravity, and would allow a better trade-off between stability and accuracy. The application of this principle is the aim of the PHARAO project, that should attend to the ACES mission planned in 2006 onboard the International Space Station. The first part of this thesis deals with the cold Cs133 PHARAO fountain. This clock stems from the transformation of a space clock prototype previewsly tested in microgravity. A detailed evaluation of the whole frequency shifts has been carried out, reaching a 7,7 10^(-16) accuracy and a 1,7 10^(-13)tau^(-1/2) short term stability. These values are obtained for 4 10^(5) detected atoms, that provides a good stability-accuracy trade-off. This transportable fountain, built at BNM-SYRTE, has been operating at MPQ in Munich (Germany). The collaboration between the 2 laboratories gave a ~10 improvement factor on the measurement accuracy (1,8 10^(-14)) for the 1S-2S two photons hydrogen transition. In a second part of this thesis, we present the characterisation of 2 elements of the PHARAO space clock: the construction of an etalon extended cavity laser and the test of the phase symmetry between the two interrogating areas of the space cavity.Les performances d'un étalon de fréquence atomique dépendent étroitement du temps d'observation des atomes. L'interrogation d'atomes refroidis par laser dans une géométrie de fontaine permet d'obtenir des temps d'interaction d'une demie seconde. Cette durée peut être variée sur une plus large gamme en l'absence de gravité terrestre, dans un environnement spatial et doit fournir un meilleur compromis stabilité-exactitude. L'application de ce principe constitue une des motivations principales du projet PHARAO, qui participera à la mission spatiale ACES, prévue en 2006 à bord de la station spatiale internationale. La première partie de ce travail de thèse porte sur l'étude de la fontaine PHARAO, fonctionnant avec des atomes de Cs133. Cette horloge est issue de la transformation d'un prototype d'horloge spatiale testé en micro-gravité. Une évaluation détaillée de l'ensemble des déplacements de fréquence mène à une exactitude relative de 7,7 10^(-16), essentiellement limitée par les collisions entre atomes froids. Sa stabilité de fréquence est de 1,7 10^(-13)tau^(-1/2). Ces valeurs sont obtenues pour un fonctionnement avec 4 10^(5) atomes détectés fournissant un bon compromis stabilité-exactitude. Cette horloge transportable construite au BNM-SYRTE, a également fonctionné au MPQ à Munich (Allemagne). La collaboration entre les deux laboratoires a permis une amélioration d'un facteur ~10 sur l'exactitude de la mesure (1,8 10^(-14)) de la transition à deux photons 1S-2S de l'atome d'hydrogène. Ce mémoire présente dans une seconde partie, la caractérisation de sous ensembles de l'horloge spatiale PHARAO, en particulier la construction d'un prototype de diode laser à cavité étendue et le test de la symétrie de phase entre les deux zones d'interrogation de la cavité spatiale