295 research outputs found
Centered-potential regularization for the advection upstream splitting method
International audienceThis paper is devoted to a centered IMEX scheme in a multidimensional framework for a wide class of multicomponent and isentropic flows. The proposed strategy is based on a regularized model where the advection velocity is modified by the gradient of the potential of the conservative forces in both mass and momentum equations. The stability of the scheme is ensured by the dissipation of mechanic energy, which stands for a mathematical entropy, under an advective CFL condition. The main physical properties, such as positivity, conservation of the total momentum, and conservation of the steady state at rest, are satisfied. In addition, asymptotic preserving properties in the regimes (“incompressible” and “acoustic”) are analyzed. Finally, several simulations are presented to illustrate our results in a simplified context of oceanic flows in one dimension
A hierarchy of Eulerian models for trajectory crossing in particle-laden turbulent flows over a wide range of Stokes numbers
With the large increase in available computational resources, large-eddy simulation (LES) of industrial configurations has become an efficient and tractable alternative to traditional multiphase turbulence models. Many applications involve a liquid or solid disperse phase carried by a gas phase (eg, fuel injection in automotive or aeronautical engines, fluidized beds, and alumina particles in rocket boosters)
A multi-fidelity machine learning based semi-Lagrangian finite volume scheme for linear transport equations and the nonlinear Vlasov-Poisson system
Machine-learning (ML) based discretization has been developed to simulate
complex partial differential equations (PDEs) with tremendous success across
various fields. These learned PDE solvers can effectively resolve the
underlying solution structures of interest and achieve a level of accuracy
which often requires an order-of-magnitude finer grid for a conventional
numerical method using polynomial-based approximations. In a previous work in
[13], we introduced a learned finite volume discretization that further
incorporates the semi-Lagrangian (SL) mechanism, enabling larger CFL numbers
for stability. However, the efficiency and effectiveness of such methodology
heavily rely on the availability of abundant high-resolution training data,
which can be prohibitively expensive to obtain. To address this challenge, in
this paper, we propose a novel multi-fidelity ML-based SL method for transport
equations. This method leverages a combination of a small amount of
high-fidelity data and sufficient but cheaper low-fidelity data. The approach
is designed based on a composite convolutional neural network architecture that
explore the inherent correlation between high-fidelity and low-fidelity data.
The proposed method demonstrates the capability to achieve a reasonable level
of accuracy, particularly in scenarios where a single-fidelity model fails to
generalize effectively. We further extend the method to the nonlinear
Vlasov-Poisson system by employing high order Runge-Kutta exponential
integrators. A collection of numerical tests are provided to validate the
efficiency and accuracy of the proposed method
Numerical scheme for multilayer shallow-water model in the low-Froude number regime
International audienceThe aim of this note is to present a multi-dimensional numerical scheme approximating the solutions of the multilayer shallow water model in the low Froude number regime. The proposed strategy is based on a regularized model where the advection velocity is modified with a pressure gradient in both mass and momentum equations. The numerical solution satisfy the dissipation of energy, which act for mathematical entropy, and the main physical properties required for simulations within oceanic flows. Résumé Schéma numérique pour lesmo eles de Saint-Venant multi-couchè a faible nombre de Froude. Le but de cette note est de présenter un schéma numérique multi-dimensionnel rapprochant les solutions dumo ele de Saint-Venant multi-couche en régime de faible nombre de Froude. La stratégie proposée est basée sur unmo ele régulariséò u la vitesse de transport est modifié par un gradient de pression dans le equations de la masse et de la quantité de mouvement. La solution numérique satisfait la dissipation denergie,jouantlerôledel'entropiedupointdevuemathématique,etlesprincipalespropriétésphysiquesnécessairesauxsimulationsdanslecadredeecoulementsocéanique
Asymptotics, structure, and integration of sound-proof atmospheric flow equations
Relative to the full compressible flow equations, sound-proof models filter acoustic waves while maintaining advection and internal waves. Two well-known sound-proof models, an anelastic model by Bannon and Durran's pseudo-incompressible model, are shown here to be structurally very
close to the full compressible flow equations. Essentially, the anelastic model is obtained by suppressing @t in the mass continuity equation and slightly modifying the gravity term, whereas the pseudoincompressible model results from dropping @tp from the pressure equation.
For length scales small compared to the density and pressure scale heights, the anelastic model reduces to the Boussinesq approximation, while the pseudo-incompressible model approaches the zero Mach number, variable density
flow equations. Thus, for small scales, both models are asymptotically consistent with the full compressible
flow equations, yet the pseudo-incompressible model is more general in that it remains valid in the presence of large density variations. For the relatively small density variations found in typical atmosphere-ocean flows, both models are found to yield very similar results, with deviations between models much smaller than deviations obtained when using different numerical schemes for the same model. This in agreement with Smolarkiewicz and Dörnbrack (2007).
Despite these useful properties, neither model can be derived by a low-Mach number asymptotic expansion for length scales comparable to the pressure scale height, i.e., for the regime they were originally designed for. Derivations of these models via scale analysis ignore an asymptotic time scale separation between advection and internal waves. In fact, only the classical Ogura & Phillips model, which assumes weak stratication of the order of the Mach number squared, can be obtained as a
leading-order model from systematic low Mach number asymptotic analysis.
Issues of formal asymptotics notwithstanding, the close structural similarity of the anelastic and pseudo-incompressible models to the full compressible flow equations makes them useful limit systems in building computational models for atmospheric flows. In the second part of the paper we propose a second-order finite-volume projection method for the anelastic and pseudo-incompressible models that observes these structural similarities. The method is applied to test problems involving free convection in a neutral atmosphere, the breaking of orographic waves at high altitudes, and the
descent of a cold air bubble in the small-scale limit. The scheme is meant to serve as a starting point for the development of a robust compressible atmospheric flow solver in future work
Numerical modelling in a multiscale ocean
Systematic improvement in ocean modelling and prediction systems over the past several decades has resulted from several concurrent factors. The first of these has been a sustained increase in computational power, as summarized in Moore\u27s Law, without which much of this recent progress would not have been possible. Despite the limits imposed by existing computer hardware, however, significant accruals in system performance over the years have been achieved through novel innovations in system software, specifically the equations used to represent the temporal evolution of the oceanic state as well as the numerical solution procedures employed to solve them. Here, we review several recent approaches to system design that extend our capability to deal accurately with the multiple time and space scales characteristic of oceanic motion. The first two are methods designed to allow flexible and affordable enhancement in spatial resolution within targeted regions, relying on either a set of nested structured grids or, alternatively, a single unstructured grid. Finally, spatial discretization of the continuous equations necessarily omits finer, subgrid-scale processes whose effects on the resolved scales of motion cannot be neglected. We conclude with a discussion of the possibility of introducing subgrid-scale parameterizations to reflect the influences of unresolved processes
Reactive Flow and Transport Through Complex Systems
The meeting focused on mathematical aspects of reactive flow, diffusion and transport through complex systems. The research interest of the participants varied from physical modeling using PDEs, mathematical modeling using upscaling and homogenization, numerical analysis of PDEs describing reactive transport, PDEs from fluid mechanics, computational methods for random media and computational multiscale methods
A linear stability analysis of compressible hybrid lattice Boltzmann methods
An original spectral study of the compressible hybrid lattice Boltzmann
method (HLBM) on standard lattice is proposed. In this framework, the mass and
momentum equations are addressed using the lattice Boltzmann method (LBM),
while finite difference (FD) schemes solve an energy equation. Both systems are
coupled with each other thanks to an ideal gas equation of state. This work
aims at answering some questions regarding the numerical stability of such
models, which strongly depends on the choice of numerical parameters. To this
extent, several one- and two-dimensional HLBM classes based on different energy
variables, formulation (primitive or conservative), collision terms and
numerical schemes are scrutinized. Once appropriate corrective terms
introduced, it is shown that all continuous HLBM classes recover the
Navier-Stokes Fourier behavior in the linear approximation. However, striking
differences arise between HLBM classes when their discrete counterparts are
analysed. Multiple instability mechanisms arising at relatively high Mach
number are pointed out and two exhaustive stabilization strategies are
introduced: (1) decreasing the time step by changing the reference temperature
and (2) introducing a controllable numerical dissipation via
the collision operator. A complete parametric study reveals that only HLBM
classes based on the primitive and conservative entropy equations are found
usable for compressible applications. Finally, an innovative study of the
macroscopic modal composition of the entropy classes is conducted. Through this
study, two original phenomena, referred to as shear-to-entropy and
entropy-to-shear transfers, are highlighted and confirmed on standard
two-dimensional test cases.Comment: 49 pages, 23 figure
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