The convergence rate of approximate solutions for nonlinear scalar conservation laws

The convergence rate is discussed of approximate solutions for the nonlinear scalar conservation law. The linear convergence theory is extended into a weak regime. The extension is based on the usual two ingredients of stability and consistency. On the one hand, the counterexamples show that one must strengthen the linearized L(sup 2)-stability requirement. It is assumed that the approximate solutions are Lip(sup +)-stable in the sense that they satisfy a one-sided Lipschitz condition, in agreement with Oleinik's E-condition for the entropy solution. On the other hand, the lack of smoothness requires to weaken the consistency requirement, which is measured in the Lip'-(semi)norm. It is proved for Lip(sup +)-stable approximate solutions, that their Lip'convergence rate to the entropy solution is of the same order as their Lip'-consistency. The Lip'-convergence rate is then converted into stronger L(sup p) convergence rate estimates

Diffusion Paths: Fixed Points, Periodicity and Chaos

It is common to recognize that ideas, technology and information disseminate across the economy following some kind of diffusion pattern. Typically, the process of adopting a new piece of knowledge will be translated into an s-shaped trajectory for the adoption rate. This type of process of diffusion tends to be stable in the sense that convergence from any initial state towards the long-term scenario in which all the potential adopters enter in contact with the innovation is commonly guaranteed. Here, we introduce a mechanism under which stability of the diffusion process does not necessarily hold. When the perceived law of motion concerning the evolution of the number of potential adopters differs from the actual law of motion, and agents try to learn this law resorting to an adaptive learning rule, nonlinear long-term outcomes might emerge: the percentage of individuals accepting the innovation in the long-run may be a varying value that evolves according to some cyclical (periodic or a-periodic) pattern. The concept of nonlinear diffusion that is addressed is applied to a problem of information and monetary policy.Diffusion, Nonlinearities, Chaos, Stability, Adaptive learning, Monetary policy.

A robust and efficient hybrid solver for crystal plasticity

Conventional crystal plasticity (CP) solvers are based on a Newton-Raphson (NR) approach which use an initial guess for the free variables (often stress) to be solved. These solvers are limited by a finite interval of convergence and often fail when the free variable falls outside this interval. Solution failure results in the reduction of the time increment to be solved, thus convergence of the CP solver is a bottleneck which determines the computational cost of the simulation. The numerical stability of the slip law in its inverted form offers a solver that isn't vulnerable to poor pre-conditioning (initial guess) and can be used to progress to a solution from a stable starting point (i.e., from zero slip rate γ˙pk=0 s−1). In this paper, a novel formulation that enables the application of the slip law in its inverted form is introduced; this treats all slip systems as independent by approximating the Jacobian as a diagonal matrix, thus overcomes ill-defined and singular Jacobians associated with previous approaches. This scheme was demonstrated to offer superior robustness and convergence rate for a case with a single slip system, however the convergence rate for extreme cases with several active slip systems was relatively poor. Here, we introduce a novel ‘hybrid scheme’ that first uses the reverse scheme for the first stage of the solution, and then transitions to the forward scheme to complete the solution at a higher convergence rate. Several examples are given for pointwise calculations, followed by CPFEM simulations for FCC copper and HCP Zircaloy-4, which demonstrated solver performance in practise. The performance of simulations using the hybrid scheme was shown to require six to nine times fewer increments compared to the conventional forward scheme solver based on a free variable of stress and initial guess based on a fully elastic increment

Diffusion paths: fixed points, periodicity and chaos

It is common to recognize that ideas, technology and information disseminate across the economy following some kind of diffusion pattern. Typically, the process of adopting a new piece of knowledge will be translated into an s-shaped trajectory for the adoption rate. This type of process of diffusion tends to be stable in the sense that convergence from any initial state towards the long-term scenario in which all the potential adopters enter in contact with the innovation is commonly guaranteed. Here, we introduce a mechanism under which stability of the diffusion process does not necessarily hold. When the perceived law of motion concerning the evolution of the number of potential adopters differs from the actual law of motion, and agents try to learn this law resorting to an adaptive learning rule, nonlinear long-term outcomes might emerge: the percentage of individuals accepting the innovation in the long-run may be a varying value that evolves according to some cyclical (periodic or a-periodic) pattern. The concept of nonlinear diffusion that is addressed is applied to a problem of information and monetary policy

Buoyancy instability of homologous implosions

I consider the hydrodynamic stability of imploding gases as a model for inertial confinement fusion capsules, sonoluminescent bubbles and the gravitational collapse of astrophysical gases. For oblate modes under a homologous flow, a monatomic gas is governed by the Schwarzschild criterion for buoyant stability. Under buoyantly unstable conditions, fluctuations experience power-law growth in time, with a growth rate that depends upon mean flow gradients and is independent of mode number. If the flow accelerates throughout the implosion, oblate modes amplify by a factor (2C)^(|N0| ti)$, where C is the convergence ratio of the implosion, N0 is the initial buoyancy frequency and ti is the implosion time scale. If, instead, the implosion consists of a coasting phase followed by stagnation, oblate modes amplify by a factor exp(pi |N0| ts), where N0 is the buoyancy frequency at stagnation and ts is the stagnation time scale. Even under stable conditions, vorticity fluctuations grow due to the conservation of angular momentum as the gas is compressed. For non-monatomic gases, this results in weak oscillatory growth under conditions that would otherwise be buoyantly stable; this over-stability is consistent with the conservation of wave action in the fluid frame. By evolving the complete set of linear equations, it is demonstrated that oblate modes are the fastest-growing modes and that high mode numbers are required to reach this limit (Legendre mode l > 100 for spherical flows). Finally, comparisons are made with a Lagrangian hydrodynamics code, and it is found that a numerical resolution of ~30 zones per wavelength is required to capture these solutions accurately. This translates to an angular resolution of ~(12/l) degrees, or < 0.1 degree to resolve the fastest-growing modes.Comment: 10 pages, 3 figures, accepted for publication in the Journal of Fluid Mechanics Rapid