Piecewise Parabolic Method on a Local Stencil for Magnetized Supersonic Turbulence Simulation

Stable, accurate, divergence-free simulation of magnetized supersonic turbulence is a severe test of numerical MHD schemes and has been surprisingly difficult to achieve due to the range of flow conditions present. Here we present a new, higher order-accurate, low dissipation numerical method which requires no additional dissipation or local "fixes" for stable execution. We describe PPML, a local stencil variant of the popular PPM algorithm for solving the equations of compressible ideal magnetohydrodynamics. The principal difference between PPML and PPM is that cell interface states are evolved rather that reconstructed at every timestep, resulting in a compact stencil. Interface states are evolved using Riemann invariants containing all transverse derivative information. The conservation laws are updated in an unsplit fashion, making the scheme fully multidimensional. Divergence-free evolution of the magnetic field is maintained using the higher order-accurate constrained transport technique of Gardiner and Stone. The accuracy and stability of the scheme is documented against a bank of standard test problems drawn from the literature. The method is applied to numerical simulation of supersonic MHD turbulence, which is important for many problems in astrophysics, including star formation in dark molecular clouds. PPML accurately reproduces in three-dimensions a transition to turbulence in highly compressible isothermal gas in a molecular cloud model. The low dissipation and wide spectral bandwidth of this method make it an ideal candidate for direct turbulence simulations.Comment: 28 pages, 18 figure

Viscous profiles in models of collective movement with negative diffusivity

In this paper, we consider an advection\u2013diffusion equation, in one space dimension, whose diffusivity can be negative. Such equations arise in particular in the modeling of vehicular traffic flows or crowds dynamics, where a negative diffusivity simulates aggregation phenomena. We focus on traveling-wave solutions that connect two states whose diffusivity has different signs; under some geometric conditions, we prove the existence, uniqueness (in a suitable class of solutions avoiding plateaus) and sharpness of the corresponding profiles. Such results are then extended to the case of end states where the diffusivity is positive, but it becomes negative in some interval between them. Also the vanishing viscosity limit is considered. At last, we provide and discuss several examples of diffusivities that change sign and show that our conditions are satisfied for a large class of them in correspondence of real data

Large Scale Stochastic Dynamics

Equilibrium statistical mechanics studies random fields distributed according to a Gibbs probability measure. Such random fields can be equipped with a stochastic dynamics given by a Markov process with the correspondingly high-dimensional state space. One particular case are stochastic partial differential equations suitably regularized. Another common version is to consider the evolution of random fields taking only values 0 or 1. The workshop was concerned with an understanding of qualitative properties of such high-dimensional Markov processes. Of particular interest are nonreversible dynamics for which the stationary measures are determined only through the dynamics and not given a priori (as would be the case for reversible dynamics). As a general observation, properties on a large scale do not depend on the precise details of the local updating rules. Such kind of universality was a guiding theme of our workshop

[Book of abstracts]

USPFAPESPCAPESICMC Summer Meeting on Differential Equations (2015 São Carlos

Singular Pseudodifferential Operators, Symmetrizers, and Oscillatory Multidimensional Shocks

We introduce a calculus of singular pseudodifferential operators (SPOs) depending on wavelength ε and use them to solve three different types of singular quasilinear hyperbolic systems. Such systems arise in nonlinear geometric optics and also, for example, in the study of incompressible limits and of nonlinear wave equations with small nonlinear terms or small data. The SPOs act in both slow and fast variables and are singular not only because their symbols have finite regularity and depend on , but also because their derivatives fail to decay in the usual way in the dual variables. There is a necessarily crude calculus with large parameter (e.g., residual operators are just bounded on L2), but the calculus admits the proof of Garding inequalities and enables us to symmetrize and sometimes even diagonalize the singular systems being considered by microlocalizing simultaneously in both slow and fast variables. The paper culminates in a proof of the existence of oscillatory multidimensional shocks on a fixed time interval independent of the wavelength ε as ε→0. The use of SPOs allows us to eliminate the small divisor assumptions made in earlier work and also to construct more general oscillatory solutions in which elliptic boundary layers are present on one or both sides of the shock

Singular Pseudodifferential Operators, Symmetrizers, and Oscillatory Multidimensional Shocks

We introduce a calculus of singular pseudodifferential operators (SPOs) depending on wavelength ε and use them to solve three different types of singular quasilinear hyperbolic systems. Such systems arise in nonlinear geometric optics and also, for example, in the study of incompressible limits and of nonlinear wave equations with small nonlinear terms or small data. The SPOs act in both slow and fast variables and are singular not only because their symbols have finite regularity and depend on , but also because their derivatives fail to decay in the usual way in the dual variables. There is a necessarily crude calculus with large parameter (e.g., residual operators are just bounded on L2), but the calculus admits the proof of Garding inequalities and enables us to symmetrize and sometimes even diagonalize the singular systems being considered by microlocalizing simultaneously in both slow and fast variables. The paper culminates in a proof of the existence of oscillatory multidimensional shocks on a fixed time interval independent of the wavelength ε as ε→0. The use of SPOs allows us to eliminate the small divisor assumptions made in earlier work and also to construct more general oscillatory solutions in which elliptic boundary layers are present on one or both sides of the shock