In this paper we investigate the variable coefficient two-sided fractional diffusion, advection, reaction equations on a bounded interval. It is known that the fractional diffusion operator may lose coercivity due to the variable coefficient, which makes both the mathematical and numerical analysis challenging. To resolve this issue, we design appropriate test and trial functions to prove the inf-sup condition of the variable coefficient fractional diffusion, advection, reaction operators in suitable function spaces. Based on this property, we prove the well-posedness and regularity of the solutions, as well as analyze the Petrov–Galerkin approximation scheme for the proposed model. Numerical experiments are presented to substantiate the theoretical findings and to compare the behaviors of different models.

The codes needed to reproduce the results of the paper. For every figure there is a folder with the corresponding code.

A new high spectral efficient SSB-OOFDM DWDM transmission system has been experimentally demonstrated. The proposed transmitter employs paired optical channels consisting of two SSB modulated OFDM signals using opposite sidebands in order to allow an efficient use of the spectrum with optical carriers separation under 10 GHz. Moreover, different paired channels are multiplexed into the 25 GHz grid DWDM fiber transmission link. Optical carrier spacing of 8.75 GHz in paired channels has been demonstrated allowing 40.8 Gb/s signal transmission rate over a 25 GHz paired channel bandwidth.

A minimal requirement for simulating multi-scale systems is to reproduce the statistical behavior of the slow variables. In particular, a good numerical method should accurately aproximate the probability density function of the continuous-time slow variables. In this note we use results from homogenization and from backward error analysis to quantify how errors of time integrators affect the mean behavior of trajectories. We show that numerical simulations converge, not to the exact probability density function (pdf) of the homogenized multi-scale system, but rather to that of the homogenized modified equations following from backward error analysis. Using homogenization theory we find that the observed statistical bias is exacerbated for multi-scale systems driven by fast chaotic dynamics that decorrelate insufficiently rapidly. This suggests that to resolve the statistical behavior of trajectories in certain multi-scale systems solvers of sufficiently high order are necessary. Alternatively, backward error analysis suggests the form of an amended vector field that corrects the lowest order bias in Euler's method. The resulting scheme, a second order Taylor method, avoids any statistical drift bias. We corroborate our analysis with a numerical example.

The paper presents a new state estimation algorithm for a bilinear equation representing the Fourier-Galerkin (FG) approximation of the Navier-Stokes (NS) equations on a torus in ℝ 2 . This state equation is subject to uncertain but bounded noise in the input (Kolmogorov forcing) and initial conditions, and its output is incomplete and contains bounded noise. The algorithm designs a time-dependent gain such that the estimation error converges to zero exponentially. The sufficient condition for the existence of the gain are formulated in the form of algebraic Riccati equations. To demonstrate the results we apply the proposed algorithm to the reconstruction a chaotic fluid flow from incomplete and noisy data.

Models used in simulation may give accurate short-term trajectories but distort long-term (statistical) properties. In this work, we augment a given approximate model with a control law (a ‘thermostat’) that gently perturbs the dynamical system to target a thermodynamic state consistent with a set of prescribed (possibly evolving) observations. As proof of concept, we provide an example involving a point vortex fluid model on the sphere, for which we show convergence of equilibrium quantities (in the stationary case) and the ability of the thermostat to dynamically track a transient state.

Models used in simulation may give accurate short-term trajectories but distort long-term (statistical) properties. In this work, we augment a given approximate model with a control law (a ‘thermostat’) that gently perturbs the dynamical system to target a thermodynamic state consistent with a set of prescribed (possibly evolving) observations. As proof of concept, we provide an example involving a point vortex fluid model on the sphere, for which we show convergence of equilibrium quantities (in the stationary case) and the ability of the thermostat to dynamically track a transient state.

We explore the direct modification of the pseudo-spectral truncation of two-dimensional, incompressible fluid dynamics to maintain a prescribed kinetic energy spectrum. The method provides a means of simulating fluid states with defined spectral properties, for the purpose of matching simulation statistics to given information, arising from observations, theoretical prediction or high-fidelity simulation. In the scheme outlined here, Nosé–Hoover thermostats, commonly used in molecular dynamics, are introduced as feedback controls applied to energy shells of the Fourier-discretized Navier–Stokes equations. As we demonstrate in numerical experiments, the dynamical properties (quantified using autocorrelation functions) are only modestly perturbed by our device, while ensemble dispersion is significantly enhanced compared with simulations of a corresponding truncation incorporating hyperviscosity.

A key challenge for an efficient splitting technique is defining the importance function. If the rare event set consists of multiple separated subsets this challenge becomes bigger since the most likely path to the rare event set may be very different from the most likely path to an intermediate level. We propose to mitigate this problem of path deviation by estimating the subset probabilities separately using a modified splitting technique. We compare the proposed separated splitting technique with a standard splitting technique by estimating the probability of entering either of two separated intervals on the real line. The squared relative error of the estimator is shown to be significantly higher when using standard splitting than when using separated splitting. We show that this difference increases if the rare event probability becomes smaller, illustrating the advantage of the separated splitting technique.

We consider dynamical systems evolving near an equilibrium statistical state where the interest is in modelling long term behavior that is consistent with thermodynamic constraints. We adjust the distribution using an entropy-optimizing formulation that can be computed on-the- fly, making possible partial corrections using incomplete information, for example measured data or data computed from a different model (or the same model at a different scale). We employ a thermostatting technique to sample the target distribution with the aim of capturing relavant statistical features while introducing mild dynamical perturbation (thermostats). The method is tested for a point vortex fluid model on the sphere, and we demonstrate both convergence of equilibrium quantities and the ability of the formulation to balance stationary and transient- regime errors.

In this paper we propose a state estimation approach for linear parabolic Partial Differential Equations (PDE) with uncertain parameters. It is based on an extension of the Galerkin projection method. The extended method models projection coefficients, representing the state of the PDE in some basis, by means of a Differential-Algebraic Equation (DAE). The original estimation problem for the PDE is then recast as a state estimation problem for the constructed DAE using a linear continuous minimax filter. We develop a discretization method of order p to go from continuous to discrete time to realize a numerical simulation. The time discretisation method preserves quadratic invariants, thus ensuring that the state estimation error, which is proved to hold for the continuous filter, is preserved for the discrete filter during the numerical simulation. To conclude we demonstrate the efficacy of the proposed method by applying it to the tracking of a discharged pollutant slick in a 2D fluid.

As intermittent renewable energy penetrates electrical power grids more and more, assessing grid reliability is of increasing concern for grid operators. Monte Carlo simulation is a robust and popular technique to estimate indices for grid reliability, but the involved computational intensity may be too high for typical reliability analyses. We show that various reliability indices can be expressed as expectations depending on the rare event probability of a so-called power curtailment, and explain how to extend a Crude Monte Carlo grid reliability analysis with an existing rare event splitting technique. The squared relative error of index estimators can be controlled, whereas orders of magnitude less workload is required than when using an equivalent Crude Monte Carlo method. We show further that a bad choice for the time step size or for the importance function may endanger this squared relative error.

Thermal bath coupling mechanisms as utilized in molecular dynamics are applied to partial differential equation models. Working from a semi-discrete (Fourier mode) formulation for the Burgers–Hopf or Korteweg–de Vries equation, we introduce auxiliary variables and stochastic perturbations in order to drive the system to sample a target ensemble which may be a Gibbs state or, more generally, any smooth distribution defined on a constraint manifold. We examine the ergodicity of approaches based on coupling of the heat bath to the high wave numbers, with the goal of controlling the ensemble through the fast modes. We also examine different thermostat methods in the extent to which dynamical properties are corrupted in order to accurately compute the average of a desired observable with respect to the invariant distribution. The principal observation of this paper is that convergence to the invariant distribution can be achieved by thermostatting just the highest wave number, while the evolution of the slowest modes is little affected by such a thermostat.

Climate change is expected to cause higher discharge levels in the river Rhine at the Dutch-German border. In this study group project that was commissioned by Rijkswaterstaat, we investigate the possibility of flooding the Rijnstrangen area as a protective measure. We identify three subproblems. We first analyze the data recorded by Rijkswaterstaat and estimate the likelihood and the duration of extremely large discharges at the German border into the river. Next, we investigate how a change in discharge levels affects the water height in the first 35 kilometer section in the Netherlands. Finally we study the design of weirs and floodgates to allow diverting a sufficiently large amount of water flow from the river into the retention area. Our statistical analysis shows that an extreme discharge level is expected to occur once every 1250 years and to last for about three and a half days. Our numerical flow model shows the water height reaches equilibrium on a time scale that is much smaller than the one on which flooding occurs. The flow can thus be considered quasi-stationary. Passive weirs finally are shown to be too long to be feasible. Actively controlled floodgates are therefore recommended.

We study a Hamiltonian toy model for a Lagrangian fluid parcel in the semi-geostrophic limit which exhibits slow and fast dynamics. We first reinject unresolved fast dynamics into the deterministic equation through a stochastic parametrization that respects the conservation of the energy of the deterministic system. In a second step we use stochastic singular perturbation theory to derive an effective reduced stochastic differential equation for the slow dynamics. We verify the results in numerical simulations.

Physics-compatible numerical methods are methods that aim to preserve key mathematical and physical properties of continuum physics models in their finite-dimensional algebraic representations. They include methods which preserve properties such as energy, monotonicity, maximum principles, symmetries, and involutions of the continuum models. Examples are mimetic methods for spatial discretizations, variational and geometric integrators, conservative finite-volume and finite-element methods, etc. Research on physics-compatible numerical methods is rapidly becoming a major research thrust across multiple disciplines within the broader area of computational science and engineering. Our principal goal in arranging this issue was to provide readers with a representative sample rather than a comprehensive survey of this flourishing field. As a result, we welcomed papers with a more pronounced review flavor as well as papers with a more substantial formal mathematical content than what is common for the Journal of Computational Physics. We hope that the resulting special issue will prove to be informative and useful for all researchers interested in the current state-of-the-art in physics-compatible numerical methods. We thank all people who have helped us in preparing this special issue: the reviewers, the technical editors of the Journal of Computational Physics, and most of all the authors.

We develop a hydrostatic Hamiltonian particle-mesh (HPM) method for efficient long-term numerical integration of the atmosphere. In the HPM method, the hydrostatic approximation is interpreted as a holonomic constraint for the vertical position of particles. This can be viewed as defining a set of vertically buoyant horizontal meshes, with the altitude of each mesh point determined so as to satisfy the hydrostatic balance condition and with particles modelling horizontal advection between the moving meshes. We implement the method in a vertical-slice model and evaluate its performance for the simulation of idealized linear and nonlinear orographic flow in both dry and moist environments. The HPM method is able to capture the basic features of the gravity wave to a degree of accuracy comparable with that reported in the literature. The numerical solution in the moist experiment indicates that the influence of moisture on wave characteristics is represented reasonably well and the reduction of momentum flux is in good agreement with theoretical analysis. Copyright (c) 2011 Royal Meteorological Society

In this note we study the asymptotic limit of large variance in a stochastically perturbed thermostat model, the Nosé-Hoover-Langevin device. We show that in this limit, the model reduces to a Langevin equation with one-dimensional Wiener process, and that the perturbation is in the direction of the conjugate momentum vector. Numerical experiments with a double well potential corroborate the asymptotic analysis.