A Self-organizing Adaptive-resolution Particle Method with Anisotropic Kernels

AbstractAdaptive-resolution particle methods reduce the computational cost for problems that develop a wide spectrum of length scales in their solution. Concepts from self-organization can be used to determine suitable particle distributions, sizes, and numbers at runtime. If the spatial derivatives of the function strongly depend on the direction, the computational cost and the required number of particles can be further reduced by using anisotropic particles. Anisotropic particles have ellipsoidal influence regions (shapes) that are locally aligned with the direction of smallest variation of the function. We present a framework that allows consistent evaluation of linear differential operators on arbitrary distributions of anisotropic particles. We further extend the concept of particle self-organization to anisotropic particles, where also the directions and magnitudes of anisotropy are self-adapted. We benchmark the accuracy and efficiency of the method in a number of 2D and 3D test cases

Simulations of Kinetic Electrostatic Electron Nonlinear (KEEN) Waves with Variable Velocity Resolution Grids and High-Order Time-Splitting

KEEN waves are nonlinear, non-stationary, self-organized asymptotic states in Vlasov plasmas outside the scope or purview of linear theory constructs such as electron plasma waves or ion acoustic waves. Nonlinear stationary mode theories such as those leading to BGK modes also do not apply. The range in velocity that is strongly perturbed by KEEN waves depends on the amplitude and duration of the ponderomotive force used to drive them. Smaller amplitude drives create highly localized structures attempting to coalesce into KEEN waves. These cases have much more chaotic and intricate time histories than strongly driven ones. The narrow range in which one must maintain adequate velocity resolution in the weakly driven cases challenges xed grid numerical schemes. What is missing there is the capability of resolving locally in velocity while maintaining a coarse grid outside the highly perturbed region of phase space. We here report on a new Semi-Lagrangian Vlasov-Poisson solver based on conservative non-uniform cubic splines in velocity that tackles this problem head on. An additional feature of our approach is the use of a new high-order time-splitting scheme which allows much longer simulations per computational e ort. This is needed for low amplitude runs which take a long time to set up KEEN waves, if they are able to do so at all. The new code's performance is compared to uniform grid simulations and the advantages quanti ed. The birth pains associated with KEEN waves which are weakly driven is captured in these simulations. These techniques allow the e cient simulation of KEEN waves in multiple dimensions which will be tackled next as well as generalizations to Vlasov-Maxwell codes which are essential to understanding the impact of KEEN waves in practice

Semi-Lagrangian particle methods for hyperbolic problems

International audienceParticle methods are often associated with frequent remeshing to maintain the regularity of the particle distribution and the accuracy of the method. In that case particle methods can be seen as forward, conservative, semi-lagrangian methods.In the case of systems, pressure terms are separated from advection terms and semi-lagrangian particle methods are a particular case of flux splitting methods. For CFL numbers smaller than one, they can also be interpreted as finite-volumemethods.In the linear case, the related finite-volume methods are rather classical (e.g. Lax-Wendroff or Beam-Warming schemes) but in the non-linear case one obtains less conventional methods, in particular with good entropy properties. On the other hand, the analogy with finite-volume methods allows to borrow from the finite-volume world several technics that can be translatedinto particle remeshing schemes. Among these technics, limiters allow to deriveTVD particle methods for which convergence can be proved, at least in the scalar case, whereas convergence for totally grid-free particles (e.g. the so-called SPH methods) remains an essentially open question. However, the main practical interest of particle methods lies in the possibility to use non-CFL constrained time-steps, a possibility which still remains largely unexplored for non-linear systems.In this talk I will address these questions. I will also discuss the derivation of high order methods for linear hyperbolic equations and their application for the Direct Numerical Simulation of turbulent transport

Development and validation of hybrid grid-based and grid-free computational VπLES method

A novel hybrid grid-based and grid-free computational method which is called VπLES is proposed and validated over several benchmark cases. VπLES splits the flow structures into large scale ones, resolved on the grid (Eulerian approach), and small scale ones, represented by vortex particles (Lagrangian approach). Two transport equations for grid and particle solution are derived which are dynamically coupled through the existence of coupling terms in each of them. The method resembles LES with an effort to directly reproduce the subgrid motion at least in the statistical sense

High-Resolution Mathematical and Numerical Analysis of Involution-Constrained PDEs

Partial differential equations constrained by involutions provide the highest fidelity mathematical models for a large number of complex physical systems of fundamental interest in critical scientific and technological disciplines. The applications described by these models include electromagnetics, continuum dynamics of solid media, and general relativity. This workshop brought together pure and applied mathematicians to discuss current research that cuts across these various disciplines’ boundaries. The presented material illuminated fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions. The scope of the material covered was broad, and the discussions conducted during the workshop were lively and far-reaching

Towards a solution of the closure problem for convective atmospheric boundary-layer turbulence

We consider the closure problem for turbulence in the dry convective atmospheric boundary layer (CBL). Transport in the CBL is carried by small scale eddies near the surface and large plumes in the well mixed middle part up to the inversion that separates the CBL from the stably stratified air above. An analytically tractable model based on a multivariate Delta-PDF approach is developed. It is an extension of the model of Gryanik and Hartmann [1] (GH02) that additionally includes a term for background turbulence. Thus an exact solution is derived and all higher order moments (HOMs) are explained by second order moments, correlation coefficients and the skewness. The solution provides a proof of the extended universality hypothesis of GH02 which is the refinement of the Millionshchikov hypothesis (quasi- normality of FOM). This refined hypothesis states that CBL turbulence can be considered as result of a linear interpolation between the Gaussian and the very skewed turbulence regimes. Although the extended universality hypothesis was confirmed by results of field measurements, LES and DNS simulations (see e.g. [2-4]), several questions remained unexplained. These are now answered by the new model including the reasons of the universality of the functional form of the HOMs, the significant scatter of the values of the coefficients and the source of the magic of the linear interpolation. Finally, the closures 61 predicted by the model are tested against measurements and LES data. Some of the other issues of CBL turbulence, e.g. familiar kurtosis-skewness relationships and relation of area coverage parameters of plumes (so called filling factors) with HOM will be discussed also

Wind-Driven Gas Networks and Star Formation in Galaxies: Reaction-Advection Hydrodynamic Simulations

The effects of wind-driven star formation feedback on the spatio-temporal organization of stars and gas in galaxies is studied using two-dimensional intermediate-representational quasi-hydrodynamical simulations. The model retains only a reduced subset of the physics, including mass and momentum conservation, fully nonlinear fluid advection, inelastic macroscopic interactions, threshold star formation, and momentum forcing by winds from young star clusters on the surrounding gas. Expanding shells of swept-up gas evolve through the action of fluid advection to form a ``turbulent'' network of interacting shell fragments whose overall appearance is a web of filaments (in two dimensions). A new star cluster is formed whenever the column density through a filament exceeds a critical threshold based on the gravitational instability criterion for an expanding shell, which then generates a new expanding shell after some time delay. A filament- finding algorithm is developed to locate the potential sites of new star formation. The major result is the dominance of multiple interactions between advectively-distorted shells in controlling the gas and star morphology, gas velocity distribution and mass spectrum of high mass density peaks, and the global star formation history. The gas morphology observations of gas in the LMC and in local molecular clouds. The frequency distribution of present-to-past average global star formation rate, the distribution of gas velocities in filaments (found to be exponential), and the cloud mass spectra (estimated using a structure tree method), are discussed in detail.Comment: 40 pp, 15 eps figs, mnras style, accepted for publication in MNRAS, abstract abridged, revisions in response to referee's comment

OIL SPILL MODELING FOR IMPROVED RESPONSE TO ARCTIC MARITIME SPILLS: THE PATH FORWARD

Maritime shipping and natural resource development in the Arctic are projected to increase as sea ice coverage decreases, resulting in a greater probability of more and larger oil spills. The increasing risk of Arctic spills emphasizes the need to identify the state-of-the-art oil trajectory and sea ice models and the potential for their integration. The Oil Spill Modeling for Improved Response to Arctic Maritime Spills: The Path Forward (AMSM) project, funded by the Arctic Domain Awareness Center (ADAC), provides a structured approach to gather expert advice to address U.S. Coast Guard (USCG) Federal On-Scene Coordinator (FOSC) core needs for decision-making. The National Oceanic & Atmospheric Administration (NOAA) Office of Response & Restoration (OR&R) provides scientific support to the USCG FOSC during oil spill response. As part of this scientific support, NOAA OR&R supplies decision support models that predict the fate (including chemical and physical weathering) and transport of spilled oil. Oil spill modeling in the Arctic faces many unique challenges including limited availability of environmental data (e.g., currents, wind, ice characteristics) at fine spatial and temporal resolution to feed models. Despite these challenges, OR&R’s modeling products must provide adequate spill trajectory predictions, so that response efforts minimize economic, cultural and environmental impacts, including those to species, habitats and food supplies. The AMSM project addressed the unique needs and challenges associated with Arctic spill response by: (1) identifying state-of-the-art oil spill and sea ice models, (2) recommending new components and algorithms for oil and ice interactions, (3) proposing methods for improving communication of model output uncertainty, and (4) developing methods for coordinating oil and ice modeling efforts

Cosmological Simulations with Dark Matter from beyond the Standard Model

We study the non-linear structure formation in cosmologies where the collision-less cold dark matter (CDM) is either replaced by interacting dark matter or (partly) replaced by a free-streaming non-cold dark matter component. We focus in the first case on models with a non-vanishing interaction cross-section between dark matter and radiation in the early Universe, i.e. photons (γCDM) and neutrinos (νCDM). We study the properties of the dark matter structures that form in the presence of the collisional damping using N-Body simulations. For their halo shapes, we find similar effects as for standard thermalized fermionic Warm Dark Matter (WDM). However, for the abundance of these structures, the interacting DM models are clearly distinguishable from WDM below the characteristic damping scale. We also have a closer look at dark matter halos that resemble those hosting the two main galaxies in our Local Group, the Milky Way (MW) and Andromeda (M31). By using a high-resolution zoom-simulation of Local Group-like environments, we reveal how the DM-radiation interactions help to ease certain CDM ”small scale problems”. Furthermore, the combination of these Local Group simulations with our previous cosmological simulations allows us to constrain the cross-section in our model by comparing the abundance of satellite galaxies in our Milky Way with the predictions for subhaloes. Thanks to the sensitivity of the subhalo abundance to the suppression of the primordial perturbations, even our most conservative constraints are orders of magnitude tighter than those previously obtained from CMB data. In the case of neutrinos or other non-cold dark matter, we study ways to predict numerically the evolution of this free-streaming component correctly. We identify shortcomings in all the previously proposed techniques we encountered in our studies of various models with massive neutrinos and come up with a new, adaptive Eulerian technique to treat the neutrino fluid accurately. In particular, we introduce our implementation called SEPARA. First test results for the code are presented while full cosmological simulations will be performed in the near future