On Asymptotic Global Error Estimation and Control of Finite Difference Solutions for Semilinear Parabolic Equations

The aim of this paper is to extend the global error estimation and control addressed in Lang and Verwer [SIAM J. Sci. Comput. 29, 2007] for initial value problems to finite difference solutions of semilinear parabolic partial differential equations. The approach presented there is combined with an estimation of the PDE spatial truncation error by Richardson extrapolation to estimate the overall error in the computed solution. Approximations of the error transport equations for spatial and temporal global errors are derived by using asymptotic estimates that neglect higher order error terms for sufficiently small step sizes in space and time. Asymptotic control in a discrete $L_2$-norm is achieved through tolerance proportionality and uniform or adaptive mesh refinement. Numerical examples are used to illustrate the reliability of the estimation and control strategies

Blended numerical schemes for the advection equation and conservation laws

In this paper we propose a method to couple two or more explicit numerical schemes approximating the same time-dependent PDE, aiming at creating new schemes which inherit advantages of the original ones. We consider both advection equations and nonlinear conservation laws. By coupling a macroscopic (Eulerian) scheme with a microscopic (Lagrangian) scheme, we get a new kind of multiscale numerical method

The use of chronosequences in studies of ecological succession and soil development

1. Chronosequences and associated space-for-time substitutions are an important and often necessary tool for studying temporal dynamics of plant communities and soil development across multiple time-scales. However, they are often used inappropriately, leading to false conclusions about ecological patterns and processes, which has prompted recent strong criticism of the approach. Here, we evaluate when chronosequences may or may not be appropriate for studying community and ecosystem development. 2. Chronosequences are appropriate to study plant succession at decadal to millennial time-scales when there is evidence that sites of different ages are following the same trajectory. They can also be reliably used to study aspects of soil development that occur between temporally linked sites over time-scales of centuries to millennia, sometimes independently of their application to shorter-term plant and soil biological communities. 3. Some characteristics of changing plant and soil biological communities (e.g. species richness, plant cover, vegetation structure, soil organic matter accumulation) are more likely to be related in a predictable and temporally linear manner than are other characteristics (e.g. species composition and abundance) and are therefore more reliably studied using a chronosequence approach. 4. Chronosequences are most appropriate for studying communities that are following convergent successional trajectories and have low biodiversity, rapid species turnover and low frequency and severity of disturbance. Chronosequences are least suitable for studying successional trajectories that are divergent, species-rich, highly disturbed or arrested in time because then there are often major difficulties in determining temporal linkages between stages. 5. Synthesis. We conclude that, when successional trajectories exceed the life span of investigators and the experimental and observational studies that they perform, temporal change can be successfully explored through the judicious use of chronosequences

Learning in a changing environment

Multiple cue probability learning studies have typically focused on stationary environments. We present three experiments investigating learning in changing environments. A fine-grained analysis of the learning dynamics shows that participants were responsive to both abrupt and gradual changes in cue-outcome relations. We found no evidence that participants adapted to these types of change in qualitatively different ways. Also, in contrast to earlier claims that these tasks are learned implicitly, participants showed good insight into what they learned. By fitting formal learning models, we investigated whether participants learned global functional relationships or made localized predictions from similar experienced exemplars. Both a local (the Associative Learning Model) and a global learning model (the novel Bayesian Linear Filter) fitted the data of the first two experiments. However, the results of Experiment 3, which was specifically designed to discriminate between local and global learning models, provided more support for global learning models. Finally, we present a novel model to account for the cue competition effects found in previous research and displayed by some of our participants