Finite Spatial Domain Research Articles

Time-dependent dynamical energy analysis via convolution quadrature

Dynamical Energy Analysis was introduced in 2009 as a novel method for predicting high-frequency acoustic and vibrational energy distributions in complex engineering structures. In this paper we introduce the first time-dependent Dynamical Energy Analysis method. Time-domain models are important in numerous applications including sound simulation in room acoustics, predicting shock-responses in structural mechanics and modelling electromagnetic scattering from conductors. The first step is to reformulate Dynamical Energy Analysis in the time-domain by means of a convolution integral operator. We are then able to employ the Convolution Quadrature method to provide a link between the previous frequency-domain implementations of Dynamical Energy Analysis and fully time-dependent solutions by means of the Z-transform. By combining a modified multistep Convolution Quadrature approach for the time discretisation, together with Galerkin and Petrov-Galerkin methods for the space and momentum discretisations, respectively, we are able to accurately track the propagation of high-frequency transient signals through phase-space. The implementation here is detailed for finite two-dimensional spatial domains and we demonstrate the versatility of our approach by performing a range of numerical experiments for regular, non-convex and irregular geometries as well as different types of wave source.

Travelling modulating pulse solutions with small tails for a nonlinear wave equation in periodic media

Travelling modulating pulse solutions consist of a small amplitude pulse-like envelope moving with a constant speed and modulating a harmonic carrier wave. Such solutions can be approximated by solitons of an effective nonlinear Schrödinger equation arising as the envelope equation. We are interested in a rigorous existence proof of such solutions for a nonlinear wave equation with spatially periodic coefficients. Such solutions are quasi-periodic in a reference frame co-moving with the envelope. We use spatial dynamics, invariant manifolds, and near-identity transformations to construct such solutions on large domains in time and space. Although the spectrum of the linearised equations in the spatial dynamics formulation contains infinitely many eigenvalues on the imaginary axis or in the worst case the complete imaginary axis, a small denominator problem is avoided when the solutions are localised on a finite spatial domain with small tails in far fields.

A quantum algorithm for computing dispersal of submarine volcanic tephra

In this paper, we develop a quantum computing algorithm for solving the partial differential equation (PDE) for tephra dispersal through advection in the semi-infinite horizontal buoyant region of a submarine volcanic eruption. The concentration of pyroclastic particles in the fluid domain of a hydrothermal megaplume provides important information about the rate of volcanic energy release, mechanism of formation of the megaplume, and submarine depositional patterns. This work leveraging on previous works [F. Gaitan, NPJ Quantum Inf. 6, 61 (2020); F. Gaitan, Adv. Quantum Tech. 4, 2100055 (2021)] further opens up opportunities to solve wider classes of PDEs with different applications of interest. Some additional specific contributions of this work are transforming the semi-infinite spatial domain problem into a problem on a finite spatial domain for applying the quantum algorithm, and the investigation into the effect of spatial and temporal resolution on the solution of PDEs for the quantum algorithm. Furthermore, possible modification of the algorithm with different spatial discretization schemes has been presented and their influence and implications on the solution of the PDE have been discussed. Also, studies are conducted to examine the effect of regularity conditions in time and the presence of statistical noise in the spatial domain, on the solutions obtained using quantum algorithms. The study in this paper paves an important pathway to venture into other types of advection-diffusion problems.

Fully discrete Schwarz waveform relaxation analysis for the heat equation on a finite spatial domain

Schwarz waveform relaxation methods provide space-time parallelism for the solution of time dependent partial differential equations. The algorithms are differentiated by the choice of the transmission conditions enforced at the introduced space-time boundaries. Early results considered the theoretical analysis of these algorithms in the continuous and semi-discrete (in space) settings for various families of linear partial differential equations. Later, fully discrete results were obtained under the simplifying assumption of an infinite spatial domain. In this paper, we provide a first analysis of a fully discrete classical Schwarz Waveform algorithm for the one-dimensional heat equation on an arbitrary but finite number of bounded subdomains. The θ-method is chosen as the time integrator. Convergence results are given in both the infinity norm and two norm, with an explicit contraction given in the case of a uniform partitioning. The results are compared to the numerics and to the earlier theoretical results.

Stability of exact solutions of the (2 + 1)-dimensional nonlinear Schrödinger equation with arbitrary nonlinearity parameter κ

In this work, we consider the nonlinear Schrödinger equation (NLSE) in 2+1 dimensions with arbitrary nonlinearity exponent κ in the presence of an external confining potential. Exact solutions to the system are constructed, and their stability as we increase the ‘mass’ (i.e., the L 2 norm) and the nonlinearity parameter κ is explored. We observe both theoretically and numerically that the presence of the confining potential leads to wider domains of stability over the parameter space compared to the unconfined case. Our analysis suggests the existence of a stable regime of solutions for all κ as long as their mass is less than a critical value M*(κ). Furthermore, we find that there are two different critical masses, one corresponding to width perturbations and the other one to translational perturbations. The results of Derrick’s theorem are also obtained by studying the small amplitude regime of a four-parameter collective coordinate (4CC) approximation. A numerical stability analysis of the NLSE shows that the instability curve M*(κ) versus κ lies below the two curves found by Derrick’s theorem and the 4CC approximation. In the absence of the external potential, κ = 1 demarcates the separation between the blowup regime and the stable regime. In this 4CC approximation, for κ < 1, when the mass is above the critical mass for the translational instability, quite complicated motions of the collective coordinates are possible. Energy conservation prevents the blowup of the solution as well as confines the center of the solution to a finite spatial domain. We call this regime the ‘frustrated’ blowup regime and give some illustrations. In an appendix, we show how to extend these results to arbitrary initial ground state solution data and arbitrary spatial dimension d.

Forward and inverse cascades by exact resonances in surface gravity waves.

We study the energy transfer by exact resonances for deep-water surface gravity waves in a finite periodic spatial domain. Based on a kinematic model simulating the generation of active wave modes in a finite discrete wave number space S_{R}, we examine the possibility of direct and inverse cascades. More specifically, we set an initially excited region which iteratively spreads energy to wave modes in S_{R} through exact resonances. At each iteration, we first activate new modes from scale resonances (which generate modes with new lengths), then consider two bounding situations for angle resonances (which transfer energy at the same length scale): the lower bound where no angle resonance is included and the upper bound where all modes with the same length as any active mode are excited. Such a strategy is essential to enable the computation for a large domain S_{R} with the maximum wave number R∼10^{3}. We show that for both direct and inverse cascades, the modal propagation to the boundaries of S_{R} can be established when the initially excited region is sufficiently large; otherwise a frozen turbulence state occurs, with a sharp transition between the two regimes especially for the direct cascade. Through a study on the structure of resonant quartets, the mechanism associated with the sharp transition and the role of angular energy transfer in the cascades are elucidated.

Modelling and numerical synchronization of chaotic system with fractional-order operator

Abstract Numerical solution of nonlinear chaotic fractional in space reaction–diffusion system is considered in this paper on a large but finite spatial domain size x ∈ [0, L] for L ≫ 0, x = x(x, y) and t ∈ [0, T]. The classical order chaotic ordinary differential equation is formulated by introducing the second-order spatial fractional derivative with order β ∈ (1, 2]. This second order spatial derivative is modelled by using the definition of the Riesz fractional derivative. The method of approximation combines the Fourier spectral method with the novel exponential time difference schemes. The proposed technique is known to have gained spectral accuracy over finite difference schemes. Applicability and suitability of the suggested methods are tested on Rössler chaotic system of recurring interests in one and two dimensions.

Realizing ultra-stable SnO2 anodes via in-situ formed confined space for volume expansion

Tin dioxide (SnO2) is a promising anode material for lithium-ion and sodium-ion batteries. However, the huge volume expansion during cycling makes it far away from practical application. Herein, we report a template-free “heterogeneous carbonization” strategy for fabricating SnO2/carbon/void/carbon nanofibers (SCVC) compound membrane with finite spatial domain in carbon matrix as buffer layer for volume changes of SnO2. Reduced graphene oxide (rGO) is also imported into the SCVC membrane, acting as conductive binders to construct a free-standing SCVC-rGO electrode with exceptionally conductive network. The resultant SCVC-rGO anodes in lithium/sodium-ion batteries could deliver excellent electrochemical performances of higher reversible capacity, longer lifespan and superior rate capability. As lithium-ion battery anode, the SCVC-rGO electrode shows specific capacities of 950 mA h g−1 after 400 cycles at a current of 0.5 A g−1, and 824 mA h g−1 after 650 cycles at 1 A g−1. In sodium-ion batteries, the SCVC-rGO electrode gives rise to the specific capacity of 407 mA h g−1 after 2000 cycles at the current of 1 A g−1 with exceptional capacity retention.

Generalized finite difference method with irregular mesh for a class of three-dimensional variable-order time-fractional advection-diffusion equations

Fractional advection-diffusion equation, as a generalization of classical advection-diffusion equation, has been always mentioned to simulate anomalous diffusion in porous media. This work introduces a meshless generalized finite difference method (GFDM) to solve a class of three-dimensional variable-order time fractional advection-diffusion equation (TFADE) in finite domains. Three examples with known analytic solutions in different domains are given to demonstrate that the method is accurate and stable. To reduce computational and storage cost, we discretize the time derivative terms of TFADE by a fast finite difference method (FFDM) based on sum-of-exponentials (SOE) approximation. Meanwhile, discretizing space derivative terms, GFDM generates a linear equation set including function values of neighboring nodes with various weight coefficients. Then the partial derivatives of TFADE are indicated as the linear system above. Also, this paper investigates the irregular mesh in the finite spatial domain, which is more closely meets the description of practice problems. Numerical results indicate that models with irregular mesh can also be simulated by GFDM which maintains high accuracy. Furthermore, the method is stable and accurate in solving three-dimensional irregular domain problems, where the relative errors can be less than 0.01%. This paper shows that FFDM based on SOE approximation can improve computational efficiency, and GFDM can flexibly and efficiently solve three-dimensional variable-order and variable-coefficient TFADE.

On small-time similarity-solution behaviour in the solidification shrinkage of binary alloys

In the one-dimensional solidification of a binary alloy undergoing shrinkage, there is a relative motion between solid and liquid phases in the mushy zone, leading to the possibility of macrosegregation; thus, the problem constitutes an invaluable benchmark for the testing of numerical codes that model these phenomena. Here, we revisit an earlier obtained solution for this problem, that was posed on a semi-infinite spatial domain and valid for the case of low superheat, with a view to extending it to the more general situation of a finite spatial domain, arbitrarily large superheat and both eutectic and non-eutectic solidification. We find that a similarity solution is available for short times which contains a boundary layer on the liquid side of the mush–liquid interface; this solution is believed to constitute the correct initial condition for the subsequent numerical solution of the full non-similar problem, which is deferred to future work.

A Stationary Spatio‐Temporal GARCH Model

We introduce a lagged nearest‐neighbour, stationary spatio‐temporal generalized autoregressive conditional heteroskedasticity (GARCH) model on an infinite spatial grid that opens for GARCH innovations in a space‐time ARMA model. This is illustrated by a real data application to a classical dataset of sea surface temperature anomalies in the Pacific Ocean. The model and its translation invariant neighbourhood system are wrapped around a torus forming a model with finite spatial domain, which we call circular spatio‐temporal GARCH. Such a model could be seen as an approximation of the infinite one and simulation experiments show that the circular estimator with a straightforward bias correction performs well on such non‐circular data. Since the spatial boundaries are tied together, the well‐known boundary issue in spatial statistical modelling is effectively avoided. We derive stationarity conditions for these circular processes and study the spatio‐temporal correlation structure through an ARMA representation. We also show that the matrices defined by a vectorized version of the model are block circulants. The maximum quasi‐likelihood estimator is presented and we prove its strong consistency and asymptotic normality by generalizing results from univariate GARCH theory.

Learning a Local Hamiltonian from Local Measurements.

Recovering an unknown Hamiltonian from measurements is an increasingly important task for certification of noisy quantum devices and simulators. Recent works have succeeded in recovering the Hamiltonian of an isolated quantum system with local interactions from long-ranged correlators of a single eigenstate. Here, we show that such Hamiltonians can be recovered from local observables alone, using computational and measurement resources scaling linearly with the system size. In fact, to recover the Hamiltonian acting on each finite spatial domain, only observables within that domain are required. The observables can be measured in a Gibbs state as well as a single eigenstate; furthermore, they can be measured in a state evolved by the Hamiltonian for a long time, allowing us to recover a large family of time-dependent Hamiltonians. We derive an estimate for the statistical recovery error due to approximation of expectation values using a finite number of samples, which agrees well with numerical simulations.

The diffusion equation with nonlocal data

We study the diffusion (or heat) equation on a finite 1-dimensional spatial domain, but we replace one of the boundary conditions with a “nonlocal condition”, through which we specify a weighted average of the solution over the spatial interval. We provide conditions on the regularity of both the data and weight for the problem to admit a unique solution, and also provide a solution representation in terms of contour integrals. The solution and well-posedness results rely upon an extension of the Fokas (or unified) transform method to initial–nonlocal value problems for linear equations; the necessary extensions are described in detail. Despite arising naturally from the Fokas transform method, the uniqueness argument appears to be novel even for initial–boundary value problems.

Synaptic efficacy shapes resource limitations in working memory

Working memory (WM) is limited in its temporal length and capacity. Classic conceptions of WM capacity assume the system possesses a finite number of slots, but recent evidence suggests WM may be a continuous resource. Resource models typically assume there is no hard upper bound on the number of items that can be stored, but WM fidelity decreases with the number of items. We analyze a neural field model of multi-item WM that associates each item with the location of a bump in a finite spatial domain, considering items that span a one-dimensional continuous feature space. Our analysis relates the neural architecture of the network to accumulated errors and capacity limitations arising during the delay period of a multi-item WM task. Networks with stronger synapses support wider bumps that interact more, whereas networks with weaker synapses support narrower bumps that are more susceptible to noise perturbations. There is an optimal synaptic strength that both limits bump interaction events and the effects of noise perturbations. This optimum shifts to weaker synapses as the number of items stored in the network is increased. Our model not only provides a circuit-based explanation for WM capacity, but also speaks to how capacity relates to the arrangement of stored items in a feature space.

Lagrangian and Eulerian Description of Bed Load Transport

AbstractSediment particles transported as bed load undergo alternating periods of motion and rest, particularly at weak flow intensity. Bed load transport can be investigated by either following the motion of individual particles (Lagrangian approach) or observing the phenomenon at prescribed locations (Eulerian approach). In this paper, the Lagrangian and Eulerian descriptions are merged into a unifying framework that includes definitions for quantities used to describe the kinematics of particle motion, as well as the relationships among these quantities. The alternation of motion and rest is represented by two complementary descriptions: (i) proportion of motion, indicating either the relative time spent in motion by an individual particle or the relative number of moving particles; (ii) persistence of motion, indicating the extent to which the process consists of relatively few long periods of motion or of many short ones. The framework only involves first moments of the key quantities. The conceptual developments are tested against results from an experiment with weak bed load transport, demonstrating the soundness of the approach. From an operational point of view, a Lagrangian observation is difficult to perform, since the particle motion is usually investigated for finite spatial domains (e.g., a measurement window within a laboratory or natural reach). Strategies to overcome such limitations are described, suggesting the possibility of obtaining unbiased mean values for Lagrangian descriptors. The proposed framework can be used in any study aimed at parameterizing the kinematic properties of bed load particles as functions of the hydrodynamic conditions.

Optimal Schwarz waveform relaxation for fractional diffusion-wave equations

We introduce domain decomposition methods of Schwarz waveform relaxation (WR) type for fractional diffusion-wave equations. We show that the Dirichlet transmission conditions among the subdomains lead to slow convergence. So, we construct optimal transmission conditions at the artificial interfaces and we prove that optimal Schwarz WR methods on N subdomains converge in N iterations both on infinite spatial domains and on finite spatial domains. We also propose optimal transmission conditions when the original problem is spatially discretized and we prove the same result found in the continuous case.

Viscoelastic subdiffusion in a random Gaussian environment.

Viscoelastic subdiffusion governed by a fractional Langevin equation is studied numerically in a random Gaussian environment modeled by stationary Gaussian potentials with decaying spatial correlations. This anomalous diffusion is archetypal for living cells, where cytoplasm is known to be viscoelastic and a spatial disorder also naturally emerges. We obtain some first important insights into it within a model one-dimensional study. Two basic types of potential correlations are studied: short-range exponentially decaying and algebraically slow decaying with an infinite correlation length, both for a moderate (several kBT, in the units of thermal energy), and strong (5-10kBT) disorder. For a moderate disorder, it is shown that on the ensemble level viscoelastic subdiffusion can easily overcome the medium's disorder. Asymptotically, it is not distinguishable from the disorder-free subdiffusion. However, a strong scatter in single-trajectory averages is nevertheless seen even for a moderate disorder. It features a weak ergodicity breaking, which occurs on a very long yet transient time scale. Furthermore, for a strong disorder, a very long transient regime of logarithmic, Sinai-type diffusion emerges. It can last longer and be faster in the absolute terms for weakly decaying correlations as compared with the short-range correlations. Residence time distributions in a finite spatial domain are of a generalized log-normal type and are reminiscent also of a stretched exponential distribution. They can be easily confused for power-law distributions in view of the observed weak ergodicity breaking. This suggests a revision of some experimental data and their interpretation.

A decomposition approach via Fourier sine transform for valuing American knock-out options with rebates

We present an innovative decomposition approach for computing the price and the hedging parameters of American knock-out options with a time-dependent rebate. Our approach is built upon: (i) the Fourier sine transform applied to the partial differential equation with a finite time-dependent spatial domain that governs the option price, and (ii) the decomposition technique that partitions the price of the option into that of the European counterpart and an early exercise premium. Our analytic representations can generalize a number of existing decomposition formulas for some European-style and American-style options. A complexity analysis of the method, together with numerical results, show that the proposed approach is significantly more efficient than the state-of-the-art adaptive finite difference methods, especially in dealing with spot prices near the barrier. Numerical results are also examined in order to provide new insight in the significant effects of the rebate on the option price, the hedging parameters, and the optimal exercise boundary.

Propagating wave correlations in complex systems

We describe a novel approach for computing wave correlation functions inside finite spatial domains driven by complex and statistical sources. By exploiting semiclassical approximations, we provide explicit algorithms to calculate the local mean of these correlation functions in terms of the underlying classical dynamics. By defining appropriate ensemble averages, we show that fluctuations about the mean can be characterised in terms of classical correlations. We give in particular an explicit expression relating fluctuations of diagonal contributions to those of the full wave correlation function. The methods have a wide range of applications both in quantum mechanics and for classical wave problems such as in vibro-acoustics and electromagnetism. We apply the methods here to simple quantum systems, so-called quantum maps, which model the behaviour of generic problems on Poincaré sections. Although low-dimensional, these models exhibit a chaotic classical limit and share common characteristics with wave propagation in complex structures.

Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order

In this paper, some nonlinear space-fractional order reaction-diffusion equations (SFORDE) on a finite but large spatial domain x ∈ [0, L], x=x(x,y,z) and t ∈ [0, T] are considered. Also in this work, the standard reaction-diffusion system with boundary conditions is generalized by replacing the second-order spatial derivatives with Riemann-Liouville space-fractional derivatives of order α, for 0 < α < 2. Fourier spectral method is introduced as a better alternative to existing low order schemes for the integration of fractional in space reaction-diffusion problems in conjunction with an adaptive exponential time differencing method, and solve a range of one-, two- and three-components SFORDE numerically to obtain patterns in one- and two-dimensions with a straight forward extension to three spatial dimensions in a sub-diffusive (0 < α < 1) and super-diffusive (1 < α < 2) scenarios. It is observed that computer simulations of SFORDE give enough evidence that pattern formation in fractional medium at certain parameter value is practically the same as in the standard reaction-diffusion case. With application to models in biology and physics, different spatiotemporal dynamics are observed and displayed.