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We revisit the postprocessing algorithm and give a justification from a classical truncation analysis point of view. We assume a perturbation expansion for the high frequency mode component of solutions to the underlying equation. Keeping terms to certain orders, we then generate approximate systems which correspond to numerical schemes. We show that the first two leading order methods are in fact the postprocessed Galerkin and postprocessed nonlinear Galerkin methods, respectively. Hence postprocessed Galerkin is a natural leading order method, more natural than the standard Galerkin method, for approximating solutions of parabolic dissipative PDEs. The analysis is presented in the framework of the two-dimensional Navier--Stokes equation (NSE); however, similar analysis may be done for any parabolic, dissipative nonlinear PDE.The truncation analysis is based on asymptotic estimates (in time) for the low and high mode components. We also introduce and investigate an alternative postprocessing scheme, whic...

A conservative forward-in-time numerical technique to improve the efficiency of a fully compressible wildfire model is presented. The technique is based on a method of averaging (MOA), which allows the costly advective terms based on a synchronous advection algorithm (e.g., the monotonicity of scalar fields is preserved) to be computed on a time step several times larger than would be dictated by the speed of the fastest waves. The MOA technique is explicit and does not require the use of either direct or iterative solvers to invert a matrix; instead the governing equations are solved to first order within an inner loop in which time-averaged quantities are obtained for use in a more costly outer loop. A linearized stability analysis of the entire scheme, including the interaction of gravity wave propagation and material motion, is presented and numerical stability for a wide range of physical and numerical parameters is demonstrated. Convergence studies are used to verify that the overall method maintains second-order accuracy. A model to simulate the propagation of the firefront in wildfires is described, and several calculations are provided to illustrate the application and advantages of the MOA in a problem that includes many complex physical processes.

In this paper we will survey our results on the Camassa–Holm equations and their relation to turbulence as discussed in S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi, S. Wynne, The Camassa–Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett 81 (1998) 5338. S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi, S. Wynne, A connection between the Camassa–Holm equations and turbulent flows in channels and pipes, Phys. Fluids, in press. In particular we will provide a more detailed mathematical treatment of those equations for pipe flows which yield accurate predictions of turbulent flow profiles for very large Reynolds numbers. There are many facts connecting the Camassa–Holm equations to turbulent fluid flows. The dimension of the attractor agrees with the heuristic argument based on the Kolmogorov statistical theory of turbulence. The statistical properties of the energy spectrum agree in numerical simulation with the Kolmogorov power law. Furthermore, comparison of mean flow profiles for turbulent flow in channels and pipes given by experimental and numerical data show acceptable agreement with the profile of the corresponding solution of the Camassa–Holm equations.

We propose the viscous Camassa-Holm equations as a closure approximation for the Reynolds-averaged equations of the incompressible Navier-Stokes fluid. This approximation is tested on turbulent channel and pipe flows with steady mean. Analytical solutions for the mean velocity and the Reynolds shear stress are consistent with experiments in most of the flow region.

Natural resource managers are concerned that domestic livestock grazing is negatively impacting subtropical Yungas forests, and subsistence-scale livestock owners report annual cattle deaths due to inadequate native forage in the Yungas of Jujuy province, Argentina. This study was carried out to assist livestock owners and landowners in the Jujuy Model Forest to move towards more sustainable livestock grazing. Annual herbaceous forage production was measured within grazing exclosures, and annual forage from woody plants was estimated at six sites within each of the deciduous forest, anthropogenic pasture, and highland pasture ecological zones. The anthropogenic pasture had significantly greater annual forage production and recommended stocking rates than the deciduous forest. A high proportion of unpalatable plants within the anthropogenic pasture limited potential forage production. Annual forage production in the highland pasture was not directly measured due to uncertainty regarding what portion of tussock grasses is annual growth. Forest structure, as measured by tree basal area, density, frequency, and importance value, suggests that heavy livestock grazing in the anthropogenic pasture, along with firewood and timber harvesting, has led to changes in arboreal composition and decreased sapling density in relation to the deciduous forest. Indicators of soil quality measured along four transects at each study site, including soil cover, organic C, total N , and penetration resistance, suggested that land use in the anthropogenic pasture is leading to declines in soil quality in relation to the deciduous forest. Potential solutions to current challenges related to livestock grazing include reducing livestock numbers to recommended stocking rates, and implementation of deferred, rotational grazing. There is interest among livestock owners and landowners in broadening their understanding of range management principles, and learning how to improve animal health and productivity. Agroforestry systems could provide benefits such as rehabilitation of degraded anthropogenic pastures, income diversification, increased food security, and improved demarcation of pastures. Open, unregulated access to public land, and precarious land tenancy for livestock owners, are the two most significant challenges to sustainable livestock grazing. The success of proposed solutions depends upon further community-based learning, and increased cooperation among land owners and livestock owners.

Summary. We extend the idea of the post-processing Galerkin method, in the context of dissipative evolution equations, to the nonlinear Galerkin, the filtered Galerkin, and the filtered nonlinear Galerkin methods. In general, the post-processing algorithm takes advantage of the fact that the error committed in the lower modes of the nonlinear Galerkin method (and Galerkin method), for approximating smooth, bounded solutions, is much smaller than the total error of the method. In each case, an improvement in accuracy is obtained by post-processing these more accurate lower modes with an appropriately chosen, highly accurate, approximate inertial manifold (AIM). We present numerical experiments that support the theoretical improvements in accuracy. Both the theory and computations are presented in the framework of a two dimensional reaction-diffusion system with polynomial nonlinearity. However, the algorithm is very general and can be implemented for other dissipative evolution systems. The computations clearly show the post-processed filtered Galerkin method to be the most efficient method.