Semi-discrete Model Research Articles

Nonlinear damped partial differential equations and their uniform discretizations

We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time discretization parameters, by adding appropriate numerical viscosity terms. Our main arguments use the optimal-weight convexity method and uniform observability inequalities with respect to the discretization parameters. We establish our results, first in the continuous setting, then for space semi-discrete models, and then for time semi-discrete models. The full discretization is then inferred from the previous results, by adapting the ideas to deal with linear systems.Our results cover, for instance, the Schrödinger equation with nonlinear damping, the nonlinear wave equation, the nonlinear plate equation, the nonlinear transport equation, as well as certain classes of equations with nonlocal terms.

Stochastic dynamics for inextensible fibers in a spatially semi-discrete setting

We investigate a spatially discrete surrogate model for the dynamics of a slender, elastic, inextensible fiber in turbulent flows. Deduced from a continuous space-time beam model for which no solution theory is available, it consists of a high-dimensional second order stochastic differential equation in time with a nonlinear algebraic constraint and an associated Lagrange multiplier term. We establish a suitable framework for the rigorous formulation and analysis of the semi-discrete model and prove existence and uniqueness of a global strong solution. The proof is based on an explicit representation of the Lagrange multiplier and on the observation that the obtained explicit drift term in the equation satisfies a one-sided linear growth condition on the constraint manifold. The theoretical analysis is complemented by numerical studies concerning the time discretization of our model. The performance of implicit Euler-type methods can be improved when using the explicit representation of the Lagrange multiplier to compute refined initial estimates for the Newton method applied in each time step.

Atomically informed nonlocal semi-discrete variational Peierls-Nabarro model for planar core dislocations

Prediction of Peierls stress associated with dislocation glide is of fundamental concern in understanding and designing the plasticity and mechanical properties of crystalline materials. Here, we develop a nonlocal semi-discrete variational Peierls-Nabarro (SVPN) model by incorporating the nonlocal atomic interactions into the semi-discrete variational Peierls framework. The nonlocal kernel is simplified by limiting the nonlocal atomic interaction in the nearest neighbor region, and the nonlocal coefficient is directly computed from the dislocation core structure. Our model is capable of accurately predicting the displacement profile, and the Peierls stress, of planar-extended core dislocations in face-centered cubic structures. Our model could be extended to study more complicated planar-extended core dislocations, such as <110> {111} dislocations in Al-based and Ti-based intermetallic compounds.

Discrete and Semidiscrete Models for AKNS Equation

Abstract We establish a discrete model for the Ablowitz–Kaup–Newell–Segur (AKNS) equation via generalised Cauchy matrix approach. Two semidiscrete AKNS equations are obtained by, respectively, introducing straight continuum limit and skew continuum limit. Some reductions are also discussed.

Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models

The existence of weak solutions and upper bounds for the blow-up time for time-discrete parabolic-elliptic Keller-Segel models for chemotaxis in the two-dimensional whole space are proved. For various time discretizations, including the implicit Euler, BDF, and Runge-Kutta methods, the same bounds for the blow-up time as in the continuous case are derived by discrete versions of the virial argument. The theoretical results are illustrated by numerical simulations using an upwind finite-element method combined with second-order time discretizations.

Augmentative biocontrol when natural enemies are subject to Allee effects.

Intraspecific interactions such as Allee effects are key properties that can guide population management. This contribution considers component Allee effects that are elementary mechanisms leading to declines of fitness at the population scale, i.e. demographic Allee effects. It especially focuses on the consequences of such properties in predator populations, and investigates their repercussions in a biological control context. A modelling framework able to account for reproductive and/or foraging component Allee effects is proposed. From this, four models of augmentative biological control, corresponding to the periodic introduction of natural enemies, have been investigated. This is done using semi-discrete models: ordinary differential equations are used to depict predator-prey dynamics and a discrete equation describes the abrupt augmentation of predators at periodic intervals. In that context, stability of a prey-free solution corresponding to pest eradication has been analyzed. It has been found that rare but large introductions should be preferred over frequent and small ones, when Allee effects influence predator populations. In particular, the occurrence of foraging, rather than reproducing, Allee effects significantly hinders pest eradication. Cases where the pest-free solution is locally, but not globally, stable were also observed and were shown to be favored by the occurrence of reproductive Allee effects among predators.

A Semi-Discrete Landweber–Kaczmarz Method for Cone Beam Tomography and Laminography Exploiting Geometric Prior Information

We present a semi-discrete Landweber–Kaczmarz method for solving linear ill-posed problems and its application to Cone Beam tomography and laminography. Using a basis function-type discretization in the image domain, we derive a semi-discrete model of the underlying scanning system. Based on this model, the proposed method provides an approximate solution of the reconstruction problem, i.e. reconstructing the density function of a given object from its projections, in suitable subspaces equipped with basis function-dependent weights. This approach intuitively allows the incorporation of additional information about the inspected object leading to a more accurate model of the X-rays through the object. Also, physical conditions of the scanning geometry, like flat detectors in computerized tomography as used in non-destructive testing applications as well as non-regular scanning curves e.g. appearing in computed laminography (CL) applications, are directly taken into account during the modeling process. Finally, numerical experiments of a typical CL application in three dimensions are provided to verify the proposed method. The introduction of geometric prior information leads to a significantly increased image quality and superior reconstructions compared to standard iterative methods.

Peierls stress in face-centered-cubic metals predicted from an improved semi-discrete variation Peierls-Nabarro model

In order to quantitatively predict Peierls stress, a semi-discrete variational Peierls-Nabarro model is improved by incorporating an additional gradient energy term into the energy functional. This gradient energy term is designed to effectively represent the influence of both the discreteness of atoms and the quick variations of the displacement profile in the dislocation core. Using face-centered-cubic metals as a model system for validation, we obtain a more accurate prediction of the displacement profile across the slip plane, and consequent precise Peierls stress, within a few times the prediction from molecular dynamics calculations.

Approximation of the controls for the linear beam equation

This article deals with the approximation of the boundary controls of a 1-D linear equation modeling the transversal vibrations of a hinged beam using a finite-difference space semi-discrete scheme. Due to the high frequency numerical spurious oscillations, the semi-discrete model is not uniformly controllable with respect to the mesh size and the convergence of the approximate controls corresponding to initial data in the finite energy space cannot be guaranteed. In this paper we analyze how do the initial data to be controlled and their discretization affect the result of the approximation process. We prove that the convergence of the scheme is ensured if the continuous initial data are sufficiently regular or if the highest frequencies of their discretization have been filtered out. In both cases, the minimal weighted \(L^2\)-norm discrete controls are shown to be convergent to the corresponding continuous one when the mesh size tends to zero.

Mathematical aspects of the meteorological interpretation of satellite hyperspectral infrared measurements part I: statement of the inverse problem for estimation of the cloud absorption vertical profile

ABSTRACTSatellite high spectral resolution infrared measurements provide information for cloud vertical characterization when optically semi-transparent clouds at high altitude cover cloud layers at lower altitudes. It is an important issue because such atmospheric conditions are common and clouds are characterized by large-scale vertical development. An approximation radiative transfer model for a cloudy atmosphere is introduced. Cloud particle absorption of infrared (IR) radiation depends on the spectral frequency. So, the effective cloud parameters such as amount (absorption) and height, derived from IR spectral measurements, will be spectral functions as well. The degree of uncertainty in the determination of effective cloud parameters cannot be eliminated by increasing the number of spectral measurements. A cloud model should have extra degrees of freedom to address the spectral and spatial (vertical) variability of cloud absorption. A semi-discrete multilevel cloud model is used to describe the perturbation of the outgoing IR thermal radiation caused by cloudiness in the field of view of a satellite instrument. The model delineates cloudiness in a number of layers at fixed heights. Each layer (level) is characterized by the effective cloud absorption. An inverse problem of cloud absorption vertical profile (CAVP) estimation is described. The estimate of an effective cloud absorption profile is considered as predictor for identification of cloud presence at specific atmospheric layers. The problem is numerically examined for real satellite IR spectral measurements and solution estimates are compared with lidar measurements. Results show that the resulting estimate of CAVPs provides a realistic characterization of cloud top and cloud vertical scale.

The evolution of plant virus transmission pathways

The evolution of plant virus transmission pathways is studied through transmission via seed, pollen, or a vector. We address the questions: under what circumstances does vector transmission make pollen transmission redundant? Can evolution lead to the coexistence of multiple virus transmission pathways? We restrict the analysis to an annual plant population in which reproduction through seed is obligatory. A semi-discrete model with pollen, seed, and vector transmission is formulated to investigate these questions. We assume vector and pollen transmission rates are frequency-dependent and density-dependent, respectively. An ecological stability analysis is performed for the semi-discrete model and used to inform an evolutionary study of trade-offs between pollen and seed versus vector transmission. Evolutionary dynamics critically depend on the shape of the trade-off functions. Assuming a trade-off between pollen and vector transmission, evolution either leads to an evolutionarily stable mix of pollen and vector transmission (concave trade-off) or there is evolutionary bi-stability (convex trade-off); the presence of pollen transmission may prevent evolution of vector transmission. Considering a trade-off between seed and vector transmission, evolutionary branching and the subsequent coexistence of pollen-borne and vector-borne strains is possible. This study contributes to the theory behind the diversity of plant–virus transmission patterns observed in nature.

Approximation of the controls for the beam equation with vanishing viscosity

We consider a finite difference semi-discrete scheme for the approximation of the boundary controls of a 1-D equation modeling the transversal vibrations of a hinged beam. It is known that, due to the high frequency numerical spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural setting. Consequently, the convergence of the approximate controls corresponding to initial data in the finite energy space cannot be guaranteed. We prove that, by adding a vanishing numerical viscosity, the uniform controllability property and the convergence of the scheme is ensured.

Alternative stable states in a stage-structured consumer–resource biomass model with niche shift and seasonal reproduction

We formulate and analyze a stage-structured consumer–resource biomass model, in which consumers reproduce in a pulsed event at the beginning of a growing season and furthermore go through a niche shift during their life history. We show that the resulting semi-discrete model can exhibit two stable states that can be characterized as a development-controlled state and a reproduction-controlled state. Varying resource availabilities and varying the extent of the niche shift determines whether juveniles or adults are more limited by their resource(s) and can lead to switches between the alternative stable states. Furthermore, we quantify the persistence of the consumer population and the occurrence of the two alternative stable states as a function of resource availabilities and extent of the niche shift. All the results show that irrespective of the type of reproduction of the consumers (continuous or seasonal), the stage-structured model will exhibit alternative stable states as long as development of the juvenile stage and reproduction of the adult stage are both resource-dependent.

Class-Based Storage Assignment in a Unit-Load Warehouse Employing AS/RS with Inventory Space Allocation Considering Product Specific Setup to Holding Cost Ratio

Many authors have used continuous optimization models for class-based storage allocation for unit load warehouses employing AS/RS system assuming the same setup to holding cost ratio for all products in their EOQ inventory policy. Accordingly, an ABC curve based on demand ordering of products is used to determine pallet turnovers in their travel distance models which is not applicable in many practical situations and can be inaccurate. This paper develops a new model to determine two/three class optimal partitions while relaxing assumption of constant setup to holding cost ratio in the EOQ formula and as a consequence uses an ABC curve based on decreasing ranking of product turnovers rather than that based on demand. Computational experience shows that this new continuous model approximates pick travel distances which are very close to as compared to that of a semi-discrete model (SDM) proposed in this paper. Further computational experiments are also provided comparing traditional models with our formulations.

Optimization of non-hydrostatic Euler model for water waves

The distribution of variables in the water column controls the dispersion properties of non-hydrostatic models. Because solving the Poisson equation is the most time consuming part of a non-hydrostatic model, it is highly desirable to reduce the number of unknowns in the water column by placing them at optimal locations. The paper presents the analytical dispersion relationship of a non-hydrostatic Euler model for water waves. The phase speed of linear waves simulated by the semi-discretized Euler model can be expressed as a rational polynomial function of the dimensionless water depth, kh, and the thicknesses of layers encompassing the water column become optimizable parameters in this function. The dispersion error is obtained by comparing this phase speed against the exact solution based on the linear wave theory. It is shown that for a given dispersion error (e.g. 1%), the range of kh can be extended if the layer thicknesses are optimally selected. The optimal two- and three-layer distributions for the aforementioned Euler model are provided. The Euler model with the optimized layer distribution shows good linear dispersion properties up to kh≈9 with two layers, and kh≈49.5 with three layers. The derived phase speed was tested against both numerical and exact solutions of standing waves for various cases. Excellent agreement was achieved. The model was also tested using the fifth-order Stokes theory for nonlinear standing waves. The phase speed of nonlinear waves follows a similar trend of the derived phase speed although it deviates proportionally with the increase of wave steepness ka0. Thus, the optimal layer distribution can also be applied to nonlinear waves within a range of ka0 and kh. The optimization method is applicable to other non-hydrostatic Euler models for water waves.

Steady-state traffic flow on a ring road with up- and down-slopes

Steady-state traffic flow on a ring road with up- and down-slopes is investigated using a semi-discrete model. By exploiting the relations between the semi-discrete and continuum models, a steady-state solution is uniquely determined for a given total number of vehicles on a ring road. The solution is exact and always stable with respect to the first-order continuum model, whereas it is a good approximation with respect to the semi-discrete model provided that the involved equilibrium constant states are linearly stable. In other cases, the instability of one or more equilibria could trigger stop-and-go waves propagating in certain sections of road or throughout the ring road. The indicated results are reasonable and thus physically significant for better understanding of real-world traffic flow on inhomogeneous roads, such as those with junctions or bottlenecks.

An Entropic Scheme for an Angular Moment Model for the Classical Fokker-Planck-Landau Equation of Electrons

AbstractIn plasma physics domain, the electron transport is described with the Fokker-Planck-Landau equation. The direct numerical solution of the kinetic equation is usually intractable due to the large number of independent variables. That is why we propose in this paper a new model whose derivation is based on an angular closure in the phase space and retains only the energy of particles as kinetic dimension. To find a solution compatible with physics conditions, the closure of the moment system is obtained under a minimum entropy principle. This model is proved to satisfy the fundamental properties like a H theorem. Moreover an entropic discretization in the velocity variable is proposed on the semi-discrete model. Finally, we validate on numerical test cases the fundamental properties of the full discrete model.

STABILITY AND NEIMARK-SACKER BIFURCATION OF A SEMI-DISCRETE POPULATION MODEL

In this paper, a semi-discrete model is derived for a nonlinear simple population model, and its stability and bifurcation are investigated by invoking a key lemma we present. Our results display that a Neimark-Sacker bifurcation occurs in the positive fixed point of this system under certain parametric conditions. By using the Center Manifold Theorem and bifurcation theory, the stability of invariant closed orbits bifurcated is also obtained. The numerical simulation results not only show the correctness of our theoretical analysis, but also exhibit new and interesting dynamics of this system, which do not exist in its corresponding continuous version.

Discrete Models of Fluids: Spatial Averaging, Closure, and Model Reduction

We consider semidiscrete ODE models of single-phase fluids and two-fluid mixtures. In the presence of multiple fine-scale heterogeneities, the size of these ODE systems can be very large. Spatial averaging is then a useful tool for reducing computational complexity of the problem. The averages satisfy exact balance equations of mass, momentum, and energy. These equations do not form a satisfactory continuum model because evaluation of stress and heat flux requires solving the underlying ODEs. To produce continuum equations that can be simulated without resolving microscale dynamics, we recently proposed a closure method based on the use of regularized deconvolution. Here we continue the investigation of deconvolution closure with the long term objective of developing consistent computational upscaling for multiphase particle methods. The structure of the fine-scale particle solvers is reminiscent of molecular dynamics. For this reason we use nonlinear averaging introduced for atomistic systems by Noll, Hardy, and Murdoch-Bedeaux. We also consider a simpler linear averaging originally developed in large eddy simulation of turbulence. We present several simple but representative examples of spatially averaged ODEs, where the closure error can be analyzed. Based on this analysis we suggest a general strategy for reducing the relative error of approximate closure. For problems with periodic highly oscillatory material parameters we propose a spectral boosting technique that augments the standard deconvolution and helps to correctly account for dispersion effects. We also conduct several numerical experiments, one of which is a complete mesoscale simulation of a stratified two-fluid flow in a channel. In this simulation, the operation count per coarse time step scales sublinearly with the number of particles.

Controllability of the space semi-discrete approximation for the beam equation

The aim of this work is to study of the numerical approximation of the controls for the hinged beam equation. A consequence of the numerical spurious high frequencies is the lack of the uniform controllability property of the semi-discrete model for the beam equation, in the classical setting. We solve this deficiency by adding a vanishing numerical viscosity term, which will damp out these high frequencies. An approximation algorithm based on the conjugate gradient method and some numerical experiments are presented.