Fausto Cavalli

High-Order Relaxation Schemes for Nonlinear Degenerate Diffusion Problems

Several relaxation approximations to partial differential equations have been recently proposed. Examples include conservation laws, Hamilton-Jacobi equations, convection-diffusion problems, and gas dynamics problems. The present paper focuses on diffusive relaxation schemes for the numerical approximation of nonlinear parabolic equations. These schemes are based on a suitable semilinear hyperbolic system with relaxation terms. High-order methods are obtained by coupling ENO and weighted essentially nonoscillatory (WENO) schemes for space discretization with implicit-explicit (IMEX) schemes for time integration. Error estimates and a convergence analysis are developed for semidiscrete schemes with a numerical analysis for fully discrete relaxed schemes. Various numerical results in one and two dimensions illustrate the high accuracy and good properties of the proposed numerical schemes, also in the degenerate case. These schemes can be easily implemented on parallel computers and applied to more general systems of nonlinear parabolic equations in two- and three-dimensional cases.

A Cournot duopoly game with heterogeneous players

We analyze a duopolistic Cournotian game with firms producing a homogeneous good, isoelastic demand function and linear total cost functions. In this economic setting, the traditional dynamic adjustment based on the classical best reply mechanism is very demanding in terms of rationality and information set. Therefore, in the competition we study, both the players adopt decisional mechanisms which are based on a reduced degree of rationality, being the agents supposed to have only limited informational and computational capabilities. We assume that the first player adopts a gradient rule mechanism, while the second one adjusts his output level according to a Local Monopolistic Approximation. We provide local stability conditions in terms of marginal costs ratio and complex dynamics are investigated. In particular, we show that two different routes to complicated dynamics are possible: a cascade of flip bifurcations leading to periodic cycles (and chaos) and the Neimark-Sacker bifurcation, which results in an attractive invariant closed curve.

3D simulations of early blood vessel formation

Blood vessel networks form by spontaneous aggregation of individual cells migrating toward vascularization sites (vasculogenesis). A successful theoretical model of two-dimensional experimental vasculogenesis has been recently proposed, showing the relevance of percolation concepts and of cell cross-talk (chemotactic autocrine loop) to the understanding of this self-aggregation process. Here we study the natural 3D extension of the computational model proposed earlier, which is relevant for the investigation of the genuinely three-dimensional process of vasculogenesis in vertebrate embryos. The computational model is based on a multidimensional Burgers equation coupled with a reaction diffusion equation for a chemotactic factor and a mass conservation law. The numerical approximation of the computational model is obtained by high order relaxed schemes. Space and time discretization are performed by using TVD schemes and, respectively, IMEX schemes. Due to the computational costs of realistic simulations, we have implemented the numerical algorithm on a cluster for parallel computation. Starting from initial conditions mimicking the experimentally observed ones, numerical simulations produce network-like structures qualitatively similar to those observed in the early stages of in vivo vasculogenesis. We develop the computation of critical percolative indices as a robust measure of the network geometry as a first step towards the comparison of computational and experimental data.

A multiscale time model with piecewise constant argument for a boundedly rational monopolist

We present a dynamic model for a boundedly rational monopolist who, in a partially known environment, follows a rule-of-thumb learning process. We assume that the production activity is continuously carried out and that the costly learning activity only occurs periodically at discrete time periods, so that the resulting dynamical model consists of a piecewise constant argument differential equation. Considering general demand, cost and agent’s reactivity functions, we show that the behavior of the differential model is governed by a nonlinear discrete difference equation. Differently from the classical model with smooth argument, unstable, complex dynamics can arise. The main novelty consists in showing that the occurrence of such dynamics is caused by the presence of multiple (discrete and continuous) time scales and depends on size of the time interval between two consecutive learning processes, in addition to the agent’s reactivity and the sensitivity of the marginal profit.

A simulation environment for directional sensing as a phase separation process

The ability of eukaryotic cells to navigate along spatial gradients of extracellular guidance cues is crucial for embryonic development, tissue regeneration, and cancer progression. One proposed model for chemotaxis is a phosphoinositide-based phase separation process, which takes place at the plasma membrane upon chemoattractant stimulation and triggers directional motility of eukaryotic cells. Here, we make available virtual-cell software that allows the execution and spatiotemporal analysis of in silico chemotaxis experiments, in which the user can control physical and chemical parameters as well as the number and position of chemoattractant sources.

Effects of Size, Composition, and Evolutionary Pressure in Heterogeneous Cournot Oligopolies with Best Response Decisional Mechanisms

We study heterogeneous Cournot oligopolies of variable sizes and compositions, in which the firms have different degrees of rationality, being either rational firms with perfect foresight or naive best response firms with static expectations. Each oligopoly can be described using its size and composition, that is, the fraction of firms that are rational. We take into account two frameworks, one in which the decisional rules are exogenously assigned and the other in which the firms may change their heuristics. We consider a switching mechanism based on a logit rule, where the switching propensity is regulated by a parameter which represents the evolutionary pressure. In the fixed fractions setting, we prove that, in general, the composition has a stabilizing effect, while increasing the oligopoly size leads to instability. However, we show that, for particular parameters settings, stability is not affected by the composition or the firms number. Similarly, in the evolutionary fractions setting, we analytically prove that when marginal costs are identical, increasing the evolutionary pressure has a destabilizing effect. Nevertheless, focusing on particular examples with different marginal costs we are able to show that evolutionary pressure may also have a stabilizing or a neutral role.

A cobweb model for electricity markets

In this paper we study some dynamical features of electricity markets modelling demand and supply by means of a nonlinear cobweb model. We consider a demand function periodically perturbed to take into account the real world seasonalities (daily, weekly and yearly) while supply function can include a stochastic term that is able to encompass possible shocks like outages and plants unavailability. Using adaptive expectations we investigate the effects on equilibrium prices in a perturbed and in an unperturbed model considering peak and off-peak market configurations. The model we propose is based on a well known partial equilibrium model which is used to describe price dynamics of non-storable goods. The novelty we introduce is a sigmoid supply function and a periodically perturbed demand function. Our simulative investigations confirm that the model is able to reproduce several effects observed in real price dynamics. We also focus on how the dynamics can be influenced by altering some market rules set by regulators, like price caps and floors. In particular, we show that taking into account periodical perturbations in the demand function can lead to the anticipation, with respect to the classical model, of chaotic dynamics. Moreover, we study how the dynamic is influenced if negative prices are allowed.

Real and financial market interactions in a multiplier-accelerator model: Nonlinear dynamics, multistability and stylized facts

In the present paper, we investigate the dynamics of a model in which the real part of the economy, described within a multiplier-accelerator framework, interacts with a financial market with heterogeneous speculators, in order to study the channels through which the two sectors influence each other. Employing analytical and numerical tools, we investigate stability conditions as well as bifurcations and possible periodic, quasi-periodic, and chaotic dynamics, enlightening how the degree of market interaction, together with the accelerator parameter and the intervention of the fiscal authority, may affect the business cycle and the course of the financial market. In particular, we show that even if the steady state is locally stable, multistability phenomena can occur, with several and complex dynamic structures coexisting with the steady state. Finally, simulations reveal that the proposed model is able to explain several statistical properties and stylized facts observed in real financial markets, including persistent high volatility, fat-tailed return distributions, volatility clustering, and positive autocorrelation of absolute returns.

Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion Equations

In this work we present finite element approximations of relaxed systems for nonlinear diffusion problems, which can also tackle the cases of degenerate and strongly degenerate diffusion equations. Relaxation schemes take advantage of the replacement of the original partial differential equation (PDE) with a semilinear hyperbolic system of equations, with a stiff source term, tuned by a relaxation parameter

A Model of Monopoly with Lags in the Planning and Production Activity

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