Removing the stability limit of the time–space domain explicit finite difference schemes for acoustic modeling with stability condition-based spatial operators

The time step and grid spacing in explicit finite-difference (FD) modeling are constrained by the Courant-Friedrichs-Lewy (CFL) condition. Recently, it has been found that spatial FD coefficients may be designed through simultaneously minimizing the spatial dispersion error and maximizing the CFL number. This allows one to stably use a larger time step or a smaller grid spacing than usually possible. However, when using such a method, only second-order temporal accuracy is achieved. To address this issue, I propose a method to determine the spatial FD coefficients, which simultaneously satisfy the stability condition of the whole wavenumber range and the time–space domain dispersion relation of a given wavenumber range. Therefore, stable modeling can be performed with high-order spatial and temporal accuracy. The coefficients can adapt to the variation of velocity in heterogeneous models. Additionally, based on the hybrid absorbing boundary condition, I develop a strategy to stably and effectively suppress artificial reflections from the model boundaries for large CFL numbers. Stability analysis, accuracy analysis and numerical modeling demonstrate the accuracy and effectiveness of the proposed method.

On the courant-friedrichs-lewy condition for numerical solvers of the coagulation equation

Evolving the size distribution of solid aggregates challenges simulations of young stellar objects. Among other difficulties, generic formulae for stability conditions of explicit solvers provide severe constrains when integrating the coagulation equation for astrophysical objects. Recent numerical experiments have recently reported that these generic conditions may be much too stringent. By analysing the coagulation equation in the Laplace space, we explain why this is indeed the case and provide a novel stability condition which avoids time over-sampling.

Metapopulations with habitat modification

Across the tree of life, organisms modify their local environment, rendering it more or less hospitable for other species. Despite the ubiquity of these processes, simple models that can be used to develop intuitions about the consequences of widespread habitat modification are lacking. Here, we extend the classic Levins metapopulation model to a setting where each of n species can colonize patches connected by dispersal, and when patches are vacated via local extinction, they retain a “memory” of the previous occupant—modeling habitat modification. While this model can exhibit a wide range of dynamics, we draw several overarching conclusions about the effects of modification and memory. In particular, we find that any number of species may potentially coexist, provided that each is at a disadvantage when colonizing patches vacated by a conspecific. This notion is made precise through a quantitative stability condition, which provides a way to unify and formalize existing conceptual models. We also show that when patch memory facilitates coexistence, it generically induces a positive relationship between diversity and robustness (tolerance of disturbance). Our simple model provides a portable, tractable framework for studying systems where species modify and react to a shared landscape.

Characterizing the universal rigidity of generic tensegrities

A tensegrity is a structure made from cables, struts, and stiff bars. A d-dimensional tensegrity is universally rigid if it is rigid in any dimension $$d'$$ d ′ with $$d'\ge d$$ d ′ ≥ d . The celebrated super stability condition due to Connelly gives a sufficient condition for a tensegrity to be universally rigid. Gortler and Thurston showed that super stability characterizes universal rigidity when the point configuration is generic and every member is a stiff bar. We extend this result in two directions. We first show that a generic universally rigid tensegrity is super stable. We then extend it to tensegrities with point group symmetry, and show that this characterization still holds as long as a tensegrity is generic modulo symmetry. Our strategy is based on the block-diagonalization technique for symmetric semidefinite programming problems, and our proof relies on the theory of real irreducible representations of finite groups.

Stability Test of Networked Systems with Multiple Delays

In this paper, we propose a sufficient stability condition for networked systems with multiple delays based on the 2-D polynomials and 2-D Hurwitz-Schur stability. The main advantage of the new stability condition is that it is applicable to the general case of networked systems with multiple, incommensurate delays yet numerically tractable. The characteristic polynomials of networked systems are mapping into 2-D hybrid polynomials, then to test the Hurwitz-Schur stability can. determine the networked systems, examples including system simulations verify the validity of the proposed test algorithms.

Seismic Analysis of a Limestone Rock Slope Through Numerical Modelling: Pseudo-Static vs. Non-Linear Dynamic Approach

In the present work, a seismic analysis was performed in advance on a limestone rock slope (height = 150 m) outcropping along the Tagliamento River valley, in the Friuli Venezia Giulia Region, north-eastern Italy. The analysed slope is characterised by strong rock mass damage, thus resulting in a critical stability condition (unstable volume = 110,000–200,000 m3). The seismic analysis was performed adopting the 2D finite difference method (FDM) and employing both a pseudo-static approach and a non-linear dynamic approach. Model outcomes demonstrate that the seismic motion induces internal, localised ruptures within the rock mass. Some important differences in the mechanical behaviour of the rock slope were highlighted, depending on the specific modelling approach assumed. When adopting a pseudo-static approach, the slope failure occurs for PGA values ranging between 0.056 g and 0.124 g, depending on the different initial static stability condition assumed for the slope (Strength Reduction Factor SRF = 1.00–1.15). According to the non-linear dynamic approach, the slope failure is achieved for PGA values varying between 0.056 g and 0.213 g. Pre-collapse slope displacements calculated with the pseudo-static approach (12–15 cm) are much more greater than those obtained through the non-linear dynamic approach (0.5–3 mm). The modelling results obtained through the non-linear dynamic analysis also testify that the seismic topographic amplification is 1.5 times the target acceleration at the slope face and 2.5 times the target acceleration at the slope toe.

Fear effect in discrete prey-predator model incorporating square root functional response

In this work, an interaction between prey and its predator involving the effect of fear in presence of the predator and the square root functional response is investigated. Fixed points and their stability condition are calculated. The conditions for the occurrence of some phenomena namely Neimark-Sacker, Flip, and Fold bifurcations are given. Base on some hypothetical data, the numerical simulations consist of phase portraits and bifurcation diagrams are demonstrated to picturise the dynamical behavior. It is also shown numerically that rich dynamics are obtained by the discrete model as the effect of fear.

The “Backward Looking” Effect in the Continuum Model Considering a New Backward Equilibrium Velocity Function.

In this paper, a new continuum traffic model is developed considering the backward-looking effect through a new positive backward equilibrium speed function. As compared with the conventional full velocity difference model, the backward equilibrium velocity function, which is largely acceptably grounded from mathematical and physical perspectives, plays an important role in significantly enhancing the stability of the traffic flow field. A linear stability condition is derived to demonstrate the flow neutralization capacity of the model, whereas the Korteweg–de Vries–Burgers equation and the attendant analytical solution are deduced using nonlinear analysis to observe the traffic flow behavior near the neutral stability condition. A numerical simulation, used to determine the flow stability enhancement efficiency of the model, is also conducted to verify the theoretical results.

Recent Advances on Boundary Conditions for Equations in Nonequilibrium Thermodynamics

This paper is concerned with modeling nonequilibrium phenomena in spatial domains with boundaries. The resultant models consist of hyperbolic systems of first-order partial differential equations with boundary conditions (BCs). Taking a linearized moment closure system as an example, we show that the structural stability condition and the uniform Kreiss condition do not automatically guarantee the compatibility of the models with the corresponding classical models. This motivated the generalized Kreiss condition (GKC)—a strengthened version of the uniform Kreiss condition. Under the GKC and the structural stability condition, we show how to derive the reduced BCs for the equilibrium systems as the classical models. For linearized problems, the validity of the reduced BCs can be rigorously verified. Furthermore, we use a simple example to show how thus far developed theory can be used to construct proper BCs for equations modeling nonequilibrium phenomena in spatial domains with boundaries.