Velocity-based moving mesh methods for nonlinear partial differential equations

This article describes a number of velocity-based moving mesh numerical methods formultidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.

Inverting recurrent neural networks for internal model control of nonlinear systems

In this paper, we show how a set of recently derived theoretical results for recurrent neural networks can be applied to the production of an internal model control system for a nonlinear plant. The results include determination of the relative order of a recurrent neural network and invertibility of such a network. A closed loop controller is produced without the need to retrain the neural network plant model. Stability of the closed-loop controller is also demonstrated.

Identification of nonlinear systems with a dynamic recurrent neural network

Two approaches are presented to calculate the weights for a Dynamic Recurrent Neural Network (DRNN) in order to identify the input-output dynamics of a class of nonlinear systems. The number of states of the identified network is constrained to be the same as the number of states of the plant.

State estimation using model order reduction for unstable systems

The problem of state estimation occurs in many applications of fluid flow. For example, to produce a reliable weather forecast it is essential to find the best possible estimate of the true state of the atmosphere. To find this best estimate a nonlinear least squares problem has to be solved subject to dynamical system constraints. Usually this is solved iteratively by an approximate Gauss–Newton method where the underlying discrete linear system is in general unstable. In this paper we propose a new method for deriving low order approximations to the problem based on a recently developed model reduction method for unstable systems. To illustrate the theoretical results, numerical experiments are performed using a two-dimensional Eady model – a simple model of baroclinic instability, which is the dominant mechanism for the growth of storms at mid-latitudes. It is a suitable test model to show the benefit that may be obtained by using model reduction techniques to approximate unstable systems within the state estimation problem.

Evidence of dispersion relations for the nonlinear response of the Lorenz 63 system

Along the lines of the nonlinear response theory developed by Ruelle, in a previous paper we have proved under rather general conditions that Kramers-Kronig dispersion relations and sum rules apply for a class of susceptibilities describing at any order of perturbation the response of Axiom A non equilibrium steady state systems to weak monochromatic forcings. We present here the first evidence of the validity of these integral relations for the linear and the second harmonic response for the perturbed Lorenz 63 system, by showing that numerical simulations agree up to high degree of accuracy with the theoretical predictions. Some new theoretical results, showing how to derive asymptotic behaviors and how to obtain recursively harmonic generation susceptibilities for general observables, are also presented. Our findings confirm the conceptual validity of the nonlinear response theory, suggest that the theory can be extended for more general non equilibrium steady state systems, and shed new light on the applicability of very general tools, based only upon the principle of causality, for diagnosing the behavior of perturbed chaotic systems and reconstructing their output signals, in situations where the fluctuation-dissipation relation is not of great help.

A low-order model investigation of the analysis of gravity waves in the ensemble Kalman filter.

The behavior of the ensemble Kalman filter (EnKF) is examined in the context of a model that exhibits a nonlinear chaotic (slow) vortical mode coupled to a linear (fast) gravity wave of a given amplitude and frequency. It is shown that accurate recovery of both modes is enhanced when covariances between fast and slow normal-mode variables (which reflect the slaving relations inherent in balanced dynamics) are modeled correctly. More ensemble members are needed to recover the fast, linear gravity wave than the slow, vortical motion. Although the EnKF tends to diverge in the analysis of the gravity wave, the filter divergence is stable and does not lead to a great loss of accuracy. Consequently, provided the ensemble is large enough and observations are made that reflect both time scales, the EnKF is able to recover both time scales more accurately than optimal interpolation (OI), which uses a static error covariance matrix. For OI it is also found to be problematic to observe the state at a frequency that is a subharmonic of the gravity wave frequency, a problem that is in part overcome by the EnKF.However, error in themodeled gravity wave parameters can be detrimental to the performance of the EnKF and remove its implied advantages, suggesting that a modified algorithm or a method for accounting for model error is needed.

Nonlinear stability and the saturation of instabilities to axisymmetric vortices

Nonlinear stability theorems are presented for axisymmetric vortices under the restriction that the disturbance is independent of either the azimuthal or the axial coordinate. These stability theorems are then used, in both cases, to derive rigorous upper bounds on the saturation amplitudes of instabilities. Explicit examples of such bounds are worked out for some canonical profiles. The results establish a minimum order for the dependence of saturation amplitude on supercriticality, and are thereby suggestive as to the nature of the bifurcation at the stability threshold.

A spectral view of nonlinear fluxes and stationary-transient interaction in the atmosphere

Nonlinear spectral transfers of kinetic energy and enstrophy, and stationary-transient interaction, are studied using global FGGE data for January 1979. It is found that the spectral transfers arise primarily from a combination, in roughly equal measure, of pure transient and mixed stationary-transient interactions. The pure transient interactions are associated with a transient eddy field which is approximately locally homogeneous and isotropic, and they appear to be consistently understood within the context of two-dimensional homogeneous turbulence. Theory based on spatial wale separation concepts suggests that the mixed interactions may be understood physically, to a first approximation, as a process of shear-induced spectral transfer of transient enstrophy along lines of constant zonal wavenumber. This essentially conservative enstrophy transfer generally involves highly nonlocal stationary-transient energy conversions. The observational analysis demonstrates that the shear-induced transient enstrophy transfer is mainly associated with intermediate-scale (zonal wavenumber m > 3) transients and is primarily to smaller (meridional) scales, so that the transient flow acts as a source of stationary energy. In quantitative terms, this transient-eddy rectification corresponds to a forcing timescale in the stationary energy budget which is of the same order of magnitude as most estimates of the damping timescale in simple stationary-wave models (5 to 15 days). Moreover, the nonlinear interactions involved are highly nonlocal and cover a wide range of transient scales of motion.

A finite element method for nonlinear elliptic problems

We present a Galerkin method with piecewise polynomial continuous elements for fully nonlinear elliptic equations. A key tool is the discretization proposed in Lakkis and Pryer, 2011, allowing us to work directly on the strong form of a linear PDE. An added benefit to making use of this discretization method is that a recovered (finite element) Hessian is a byproduct of the solution process. We build on the linear method and ultimately construct two different methodologies for the solution of second order fully nonlinear PDEs. Benchmark numerical results illustrate the convergence properties of the scheme for some test problems as well as the Monge–Amp`ere equation and the Pucci equation.

Efficient fully nonlinear data assimilation for geophysical fluid dynamics

A potential problem with Ensemble Kalman Filter is the implicit Gaussian assumption at analysis times. Here we explore the performance of a recently proposed fully nonlinear particle filter on a high-dimensional but simplified ocean model, in which the Gaussian assumption is not made. The model simulates the evolution of the vorticity field in time, described by the barotropic vorticity equation, in a highly nonlinear flow regime. While common knowledge is that particle filters are inefficient and need large numbers of model runs to avoid degeneracy, the newly developed particle filter needs only of the order of 10-100 particles on large scale problems. The crucial new ingredient is that the proposal density cannot only be used to ensure all particles end up in high-probability regions of state space as defined by the observations, but also to ensure that most of the particles have similar weights. Using identical twin experiments we found that the ensemble mean follows the truth reliably, and the difference from the truth is captured by the ensemble spread. A rank histogram is used to show that the truth run is indistinguishable from any of the particles, showing statistical consistency of the method.

A nonlinear oscillator describing storm track variability

We construct a two-variable model which describes the interaction between local baroclinicity and eddy heat flux in order to understand aspects of the variance in storm tracks. It is a heuristic model for diabatically forced baroclinic instability close to baroclinic neutrality. The two-variable model has the structure of a nonlinear oscillator. It exhibits some realistic properties of observed storm track variability, most notably the intermittent nature of eddy activity. This suggests that apparent threshold behaviour can be more accurately and succinctly described by a simple nonlinearity. An analogy is drawn with triggering of convective events.

Reduced-order dynamics of the Martian atmospheric dynamics

In this paper we explore the possibility of deriving low-dimensional models of the dynamics of the Martian atmosphere. The analysis consists of a Proper Orthogonal Decomposition (POD) of the atmospheric streamfunction after first decomposing the vertical structure with a set of eigenmodes. The vertical modes were obtained from the quasi-geostrophic vertical structure equation. The empirical orthogonal functions (EOFs) were optimized to represent the atmospheric total energy. The total energy was used as the criterion to retain those modes with large energy content and discard the rest. The principal components (PCs) were analysed by means of Fourier analysis, so that the dominant frequencies could be identified. It was possible to observe the strong influence of the diurnal cycle and to identify the motion and vacillation of baroclinic waves.

Fractional order and non-linear system identification algorithms for biomedical applications

We discuss the modelling of dielectric responses of amorphous biological samples. Such samples are commonly encountered in impedance spectroscopy studies as well as in UV, IR, optical and THz transient spectroscopy experiments and in pump-probe studies. In many occasions, the samples may display quenched absorption bands. A systems identification framework may be developed to provide parsimonious representations of such responses. To achieve this, it is appropriate to augment the standard models found in the identification literature to incorporate fractional order dynamics. Extensions of models using the forward shift operator, state space models as well as their non-linear Hammerstein-Wiener counterpart models are highlighted. We also discuss the need to extend the theory of electromagnetically excited networks which can account for fractional order behaviour in the non-linear regime by incorporating nonlinear elements to account for the observed non-linearities. The proposed approach leads to the development of a range of new chemometrics tools for biomedical data analysis and classification.

Coherent EISCAT Svalbard Radar spectra from the dayside cusp/cleft and their implications for transient field-aligned currents

Naturally enhanced incoherent scatter spectra from the vicinity of the dayside cusp/cleft, interpreted as being due to plasma turbulence driven by short bursts of intense field-aligned current, are compared with high-resolution narrow-angle auroral images and meridian scanning photometer data. Enhanced spectra have been observed on many occasions in association with nightside aurora, but there has been only one report of such spectra seen in the cusp/cleft region. Narrow-angle images show considerable change in the aurora on timescales shorter than the 10-s radar integration period, which could explain spectra observed with both ion lines simultaneously enhanced. Enhanced radar spectra are generally seen inside or beside regions of 630-nm auroral emission, indicative of sharp F region conductivity gradients, but there appears also to be a correlation with dynamic, small-scale auroral forms of order 100 m and less in width.

Advances in the study of boundary value problems for nonlinear integrable PDEs

In this review I summarise some of the most significant advances of the last decade in the analysis and solution of boundary value problems for integrable partial differential equations in two independent variables. These equations arise widely in mathematical physics, and in order to model realistic applications, it is essential to consider bounded domain and inhomogeneous boundary conditions. I focus specifically on a general and widely applicable approach, usually referred to as the Unified Transform or Fokas Transform, that provides a substantial generalisation of the classical Inverse Scattering Transform. This approach preserves the conceptual efficiency and aesthetic appeal of the more classical transform approaches, but presents a distinctive and important difference. While the Inverse Scattering Transform follows the "separation of variables" philosophy, albeit in a nonlinear setting, the Unified Transform is a based on the idea of synthesis, rather than separation, of variables. I will outline the main ideas in the case of linear evolution equations, and then illustrate their generalisation to certain nonlinear cases of particular significance.