1,720,993 research outputs found
Linearly Implicit Approximations of Diffusive Relaxation Systems
No full textDiffusive relaxation systems provide a general framework to approximate nonlinear diffusion problems, also in the degenerate case (Aregba-Driollet et al. in Math. Comput. 73(245):63-94, 2004; Boscarino et al. in Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, 2011; Cavalli et al. in SIAM J. Sci. Comput. 34:A137-A160, 2012; SIAM J. Numer. Anal. 45(5):2098-2119, 2007; Naldi and Pareschi in SIAM J. Numer. Anal. 37:1246-1270, 2000; Naldi et al. in Surveys Math. Indust. 10(4):315-343, 2002). Their discretization is usually obtained by explicit schemes in time coupled with a suitable method in space, which inherits the standard stability parabolic constraint. In this paper we combine the effectiveness of the relaxation systems with the computational efficiency and robustness of the implicit approximations, avoiding the need to resolve nonlinear problems and avoiding stability constraints on time step. In particular we consider an implicit scheme for the whole relaxation system except for the nonlinear source term, which is treated though a suitable linearization technique. We give some theoretical stability results in a particular case of linearization and we provide insight on the general case. Several numerical simulations confirm the theoretical results and give evidence of the stability and convergence also in the case of nonlinear degenerate diffusion
Linearly implicit schemes for convection-diffusion equations
No full textWe present a family of schemes for the approximation of one dimensional convection-diffusion equations. It is based on a linearization technique that allows to treat explicitly the hyperbolic term and linearly implicitly the parabolic one. This avoids the parabolic stability constraint of explicit schemes, and does not require any non-linear solver for the implicit problem. We present several numerical simulations to show the effectiveness
of the proposed schemes and to investigate their stability, convergence and accuracy. In particular, since the proposed schemes provide to be accurate for both smooth and non-smooth solutions, they turn out to be attractive for adaptivit
Expectation formation in an overlapping generation model with production
No full textIn this paper we investigate the dynamic properties of an overlapping generations' model with capital accumulation, in which agents work in both periods of life. We compare three different expectation mechanisms: Perfect foresight, myopic foresight, and adaptive expectations, focusing, in particular, on this last one. We show that the steady state is the same under each mechanism, and we prove its global stability for perfectly foresighted agents. After investigating local stability conditions under myopic expectations, we study in detail the case of adaptive expectations. We show that, under both reduced rationality mechanisms, if the share of time devoted to labor in the second period of life is large enough, periodic and complex dynamics can occur. Moreover, deepening the investigation through numerical simulations, we study the global stability behavior under adaptive expectations. Such complex scenarios also include the coexistence between the stable steady state and a periodic or chaotic attractor, giving rise to multistability, which does not arise under myopic expectations. Finally, we provide some considerations about the possibility for the agents to improve their forecasts by observing the forecasting error time series
Monopoly models with time-varying demand function
No full textWe study a family of monopoly models for markets characterized by time-varying demand functions, in which a boundedly rational agent chooses output levels on the basis of a gradient adjustment mechanism. After presenting the model for a generic framework, we analytically study the case of cyclically alternating demand functions. We show that both the perturbation size and the agent's reactivity to profitability variation signals can have counterintuitive roles on the resulting period-2 cycles and on their stability. In particular, increasing the perturbation size can have both a destabilizing and a stabilizing effect on the resulting dynamics. Moreover, in contrast with the case of time-constant demand functions, the agent's reactivity is not just destabilizing, but can improve stability, too. This means that a less cautious behavior can provide better performance, both with respect to stability and to achieved profits. We show that, even if the decision mechanism is very simple and is not able to always provide the optimal production decisions, achieved profits are very close to those optimal. Finally, we show that in agreement with the existing empirical literature, the price series obtained simulating the proposed model exhibit a significant deviation from normality and large volatility, in particular when underlying deterministic dynamics become unstable and complex
Effect of price elasticity of demand in monopolies with gradient adjustment
No full textWe study a monopolistic market characterized by a constant elasticity demand function, in which the firm technology is described by a linear total cost function. The firm is assumed to be boundedly rational and to follow a gradient rule to adjust the production level in order to optimize its profit. We focus on what happens on varying the price elasticity of demand, studying the effect on the equilibrium stability. We prove that, depending on the relation between the market size and the marginal cost, two different scenarios are possible, in which elasticity has either a stabilizing or a mixed stabilizing/destabilizing effect
A multiscale time model with piecewise constant argument for a boundedly rational monopolist
No full textWe 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
High Order Relaxed Schemes for Nonlinear Reaction Diffusion Problems
No full textDifferent relaxation approximations to partial differential equations, including conservation
laws, Hamilton-Jacobi equations, convection-diffusion problems, gas dynamics problems, have
been recently proposed. The present paper focuses onto diffusive relaxed schemes for the numer-
ical approximation of nonlinear reaction diffusion equations. High order methods are obtained
by coupling ENO and WENO schemes for space discretization with IMEX schemes for time
integration, where the implicit part can be explicitly solved at a linear cost. To illustrate the
high accuracy and good properties of the proposed numericalschemes, also in the degenerate
case, we consider various examples in one and two dimensions: the Fisher-Kolmogoroff equation,
the porous-Fisher equation and the porous medium equation with strong absorption
Complex dynamics and multistability with increasing rationality in market games
No full textIn this work we study oligopoly models in which firms adopt decision mechanisms based on best response techniques with different rationality degrees. Firms are also assumed to face resource or financial constraints in adjusting their production levels, so that, from time to time, they can only increase or decrease their strategy by a bounded quantity. We consider different families of oligopolies of generic sizes, characterized by heterogeneous compositions with respect to the rationality degrees of firms. We analytically study the local stability of the equilibrium depending on the oligopoly size and composition and through numerical simulations we investigate the possible dynamics arising when trajectories do not converge toward the equilibrium. We show that in this case complex dynamics can arise, and this is due to both the loss of stability of the equilibrium and to the emergence of multiple attractors, with the stable steady state coexisting with a different, periodic or chaotic, attractor. In particular, we show that multistability phenomena occur when the overall degree of rationality of the oligopoly is increased. Finally, we investigate the effect of non-convergent dynamics on the realized profits
Nonlinear dynamics and convergence speed of heterogeneous Cournot duopolies involving best response mechanisms with different degrees of rationality
No full textIn this paper, we propose and compare three heterogeneous Cournotian duopolies, in which players adopt best response mechanisms based on different degrees of rationality. The economic setting we assume is described by an isoelastic demand function with constant marginal costs. In particular, we study the effect of the rationality degree on stability and convergence speed to the equilibrium output. We study conditions required to converge to the Nash equilibrium and the possible route to destabilization when such conditions are violated, showing that a more elevated degree of rationality of a single player does not always guarantee an improved stability. We show that the considered duopolies exhibit either a flip or a Neimark–Sacker bifurcation. In particular, in heterogeneous oligopolies models, the Neimark–Sacker bifurcation usually arises in the presence of a player adopting gradient-like decisional mechanisms, and not best response heuristic, as shown in the present case. Moreover, we show that the cost ratio crucially influences not only the size of the stability region, but also the speed of convergence toward the equilibrium
A Cournot duopoly game with heterogeneous players: Nonlinear dynamics of the gradient rule versus local monopolistic approach
No full textWe 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
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