Primal dual mixed finite element methods for indefinite advection--diffusion equations

We consider primal-dual mixed finite element methods for the advection--diffusion equation. For the primal variable we use standard continuous finite element space and for the flux we use the Raviart-Thomas space. We prove optimal a priori error estimates in the energy- and the $L^2$-norms for the primal variable in the low Peclet regime. In the high Peclet regime we also prove optimal error estimates for the primal variable in the $H(div)$ norm for smooth solutions. Numerically we observe that the method eliminates the spurious oscillations close to interior layers that pollute the solution of the standard Galerkin method when the local Peclet number is high. This method, however, does produce spurious solutions when outflow boundary layer presents. In the last section we propose two simple strategies to remove such numerical artefacts caused by the outflow boundary layer and validate them numerically.Comment: 25 pages, 6 figures, 5 table

Finite element methods for deterministic simulation of polymeric fluids

In this work we consider a finite element method for solving the coupled Navier-Stokes (NS) and Fokker-Planck (FP) multiscale model that describes the dynamics of dilute polymeric fluids. Deterministic approaches such as ours have not received much attention in the literature because they present a formidable computational challenge, due to the fact that the analytical solution to the Fokker-Planck equation may be a function of a large number of independent variables. For instance, to simulate a non-homogeneous flow one must solve the coupled NS-FP system in which (for a 3-dimensional flow, using the dumbbell model for polymers) the Fokker-Planck equation is posed in a 6-dimensional domain. In this work we seek to demonstrate the feasibility of our deterministic approach. We begin by discussing the physical and mathematical foundations of the NS-FP model. We then present a literature review of relevant developments in computational rheology and develop our deterministic finite element based method in detail. Numerical results demonstrating the efficiency of our approach are then given, including some novel results for the simulation of a fully 3-dimensional flow. We utilise parallel computation to perform the large-scale numerical simulations

Computing transition rates for the 1-D stochastic Ginzburg--Landau--Allen--Cahn equation for finite-amplitude noise with a rare event algorithm

In this paper we compute and analyse the transition rates and duration of reactive trajectories of the stochastic 1-D Allen-Cahn equations for both the Freidlin-Wentzell regime (weak noise or temperature limit) and finite-amplitude white noise, as well as for small and large domain. We demonstrate that extremely rare reactive trajectories corresponding to direct transitions between two metastable states are efficiently computed using an algorithm called adaptive multilevel splitting. This algorithm is dedicated to the computation of rare events and is able to provide ensembles of reactive trajectories in a very efficient way. In the small noise limit, our numerical results are in agreement with large-deviation predictions such as instanton-like solutions, mean first passages and escape probabilities. We show that the duration of reactive trajectories follows a Gumbel distribution like for one degree of freedom systems. Moreover, the mean duration growths logarithmically with the inverse temperature. The prefactor given by the potential curvature grows exponentially with size. The main novelty of our work is that we also perform an analysis of reactive trajectories for large noises and large domains. In this case, we show that the position of the reactive front is essentially a random walk. This time, the mean duration grows linearly with the inverse temperature and quadratically with the size. Using a phenomenological description of the system, we are able to calculate the transition rate, although the dynamics is described by neither Freidlin--Wentzell or Eyring--Kramers type of results. Numerical results confirm our analysis

Coupling parameter and particle dynamics for adaptive sampling in Neural Galerkin schemes

Training nonlinear parametrizations such as deep neural networks to numerically approximate solutions of partial differential equations is often based on minimizing a loss that includes the residual, which is analytically available in limited settings only. At the same time, empirically estimating the training loss is challenging because residuals and related quantities can have high variance, especially for transport-dominated and high-dimensional problems that exhibit local features such as waves and coherent structures. Thus, estimators based on data samples from un-informed, uniform distributions are inefficient. This work introduces Neural Galerkin schemes that estimate the training loss with data from adaptive distributions, which are empirically represented via ensembles of particles. The ensembles are actively adapted by evolving the particles with dynamics coupled to the nonlinear parametrizations of the solution fields so that the ensembles remain informative for estimating the training loss. Numerical experiments indicate that few dynamic particles are sufficient for obtaining accurate empirical estimates of the training loss, even for problems with local features and with high-dimensional spatial domains

A model of diffusion in a potential well for the dynamics of the large-scale circulation in turbulent Rayleigh-Benard convection

Experimental measurements of properties of the large-scale circulation (LSC) in turbulent convection of a fluid heated from below in a cylindrical container of aspect ratio one are presented and used to test a model of diffusion in a potential well for the LSC. The model consists of a pair of stochastic ordinary differential equations motivated by the Navier-Stokes equations. The two coupled equations are for the azimuthal orientation theta_0, and for the azimuthal temperature amplitude delta at the horizontal midplane. The dynamics is due to the driving by Gaussian distributed white noise that is introduced to represent the action of the small-scale turbulent fluctuations on the large-scale flow. Measurements of the diffusivities that determine the noise intensities are reported. Two time scales predicted by the model are found to be within a factor of two or so of corresponding experimental measurements. A scaling relationship predicted by the model between delta and the Reynolds number is confirmed by measurements over a large experimental parameter range. The Gaussian peaks of probability distributions p(delta) and p(\dot\theta_0) are accurately described by the model; however the non-Gaussian tails of p(delta) are not. The frequency, angular change, and amplitude bahavior during cessations are accurately described by the model when the tails of the probability distribution of $\delta$ are used as experimental input.Comment: 17 pages, 17 figure

The Inertial Range of Turbulence in the Inner Heliosheath and in the Local Interstellar Medium

The governing mechanisms of magnetic field annihilation in the outer heliosphere is an intriguing topic. It is currently believed that the turbulent fluctuations pervade the inner heliosheath (IHS) and the Local Interstellar Medium (LISM). Turbulence, magnetic reconnection, or their reciprocal link may be responsible for magnetic energy conversion in the IHS.   As 1-day averaged data are typically used, the present literature mainly concerns large-scale analysis and does not describe inertial-cascade dynamics of turbulence in the IHS. Moreover, lack of spectral analysis make IHS dynamics remain critically understudied. Our group showed that 48-s MAG data from the Voyager mission are appropriate for a power spectral analysis over a frequency range of five decades, from 5e-8 Hz to 1e-2 Hz [Gallana et al., JGR 121 (2016)]. Special spectral estimation techniques are used to deal with the large amount of missing data (70%). We provide the first clear evidence of an inertial-cascade range of turbulence (spectral index is between -2 and -1.5). A spectral break at about 1e-5 Hz is found to separate the inertial range from the enegy-injection range (1/f energy decay). Instrumental noise bounds our investigation to frequencies lower than 5e-4 Hz. By considering several consecutive periods after 2009 at both V1 and V2, we show that the extension and the spectral energy decay of these two regimes may be indicators of IHS regions governed by different physical processes. We describe fluctuations’ regimes in terms of spectral energy density, anisotropy, compressibility, and statistical analysis of intermittency.   In the LISM, it was theorized that pristine interstellar turbulence may coexist with waves from the IHS, however this is still a debated topic. We observe that the fluctuating magnetic energy cascades as a power law with spectral index in the range [-1.35, -1.65] in the whole range of frequencies unaffected by noise. No spectral break is observed, nor decaying turbulence

A posteriori error estimates for streamline-diffusion and discontinuous Galerkin methods for the Vlasov-Maxwell system

This paper concerns a posteriori error analysis for the streamline diffusion(SD) finite element method for the one and one-half dimensional relativistic Vlasov–Maxwell system. The SD scheme yields a weak formulation, that corresponds to anadd of extra diffusion to, e.g. the system of equations having hyperbolic nature, orconvection-dominated convection diffusion problems. The a posteriori error estimatesrely on dual formulations and yield error controls based on the computable residuals.The convergence estimates are derived in negative norms, where the error is split intoan iteration and an approximation error and the iteration procedure is assumed to\ua0 converge

Bayesian inference for ocean transport and diffusivity fields from Lagrangian trajectory data

Eddy diffusion is commonly used to characterise subgrid-scale mixing of tracer quantities in geophysical fluid simulations. Limited by data availability, estimating eddy diffusivity from observational data remains challenging. This thesis describes a novel Bayesian framework to infer ocean transports and diffusivity fields from Lagrangian trajectories data. Modelling the Lagrangian trajectories by a stochastic differential equation whose transition density is given by the advection--diffusion equation, this framework produces a posterior probability distribution for the parameters defining the transport and diffusivity fields, enabling uncertainty quantification. In this thesis, Lagrangian trajectories from a three-layer quasigeostrophic double-gyre configuration are used to test the inference schemes. The double-gyre is a classic idealisation of large-scale ocean circulation. Mesoscale eddies are continuously produced and dissipated, leading to a complicated time-dependent flow which can plausibly be coarse-grained as diffusion and suitably used to validate the new inference schemes. Different approaches are implemented based on the Bayesian framework. In a local approach, the ocean domain is divided into an array of cells. Cell-wise defined linear velocity and constant diffusivity fields are inferred using the displacement data of Lagrangian particles originating from the cell. This approach proves capable of estimating the diffusivity in areas with a slow flow such that particles remain in the neighbourhood of their originating cell, but it fails to account for the particle trajectories straddling multiple cells in the considered time interval. An approach to correct the inference for particles straddling two cells is devised using large deviation theory, assuming the dominance of advective transport over diffusive transport. This approach successfully infers piecewise constant velocity and diffusivity using synthesised trajectories, especially in the limit of increasing dominance of advection. A global approach is then developed to infer velocity and diffusivity fields defined on the entire domain. This resolves the locality restriction on the trajectory data in the local approach. A naïve implementation of the global inference, however, involves an exceedingly large number of solutions to the advection--diffusion equation. A data coarse-graining approach is applied to overcome the computational challenge. The impact of the data coarse-graining procedures is quantified in the limit of large data. The global approach is applied to the double-gyre simulation data, using a finite volume method for the solution of advection--diffusion equation. The results demonstrate that the global approach enables a robust inference of the mean flow and diffusivity fields at varying sampling interva