Numerical integration of ordinary differential equations of various orders

Report describes techniques for the numerical integration of differential equations of various orders. Modified multistep predictor-corrector methods for general initial-value problems are discussed and new methods are introduced

The Gap-Tooth Method in Particle Simulations

We explore the gap-tooth method for multiscale modeling of systems represented by microscopic physics-based simulators, when coarse-grained evolution equations are not available in closed form. A biased random walk particle simulation, motivated by the viscous Burgers equation, serves as an example. We construct macro-to-micro (lifting) and micro-to-macro (restriction) operators, and drive the coarse time-evolution by particle simulations in appropriately coupled microdomains (teeth) separated by large spatial gaps. A macroscopically interpolative mechanism for communication between the teeth at the particle level is introduced. The results demonstrate the feasibility of a closure-on-demand approach to solving hydrodynamics problems

Multiscale analysis of re-entrant production lines: An equation-free approach

The computer-assisted modeling of re-entrant production lines, and, in particular, simulation scalability, is attracting a lot of attention due to the importance of such lines in semiconductor manufacturing. Re-entrant flows lead to competition for processing capacity among the items produced, which significantly impacts their throughput time (TPT). Such production models naturally exhibit two time scales: a short one, characteristic of single items processed through individual machines, and a longer one, characteristic of the response time of the entire factory. Coarse-grained partial differential equations for the spatio-temporal evolution of a "phase density" were obtained through a kinetic theory approach in Armbruster et al. [2]. We take advantage of the time scale separation to directly solve such coarse-grained equations, even when we cannot derive them explicitly, through an equation-free computational approach. Short bursts of appropriately initialized stochastic fine-scale simulation are used to perform coarse projective integration on the phase density. The key step in this process is lifting: the construction of fine-scale, discrete realizations consistent with a given coarse-grained phase density field. We achieve this through computational evaluation of conditional distributions of a "phase velocity" at the limit of large item influxes.Comment: 14 pages, 17 figure

Coarse Projective kMC Integration: Forward/Reverse Initial and Boundary Value Problems

In "equation-free" multiscale computation a dynamic model is given at a fine, microscopic level; yet we believe that its coarse-grained, macroscopic dynamics can be described by closed equations involving only coarse variables. These variables are typically various low-order moments of the distributions evolved through the microscopic model. We consider the problem of integrating these unavailable equations by acting directly on kinetic Monte Carlo microscopic simulators, thus circumventing their derivation in closed form. In particular, we use projective multi-step integration to solve the coarse initial value problem forward in time as well as backward in time (under certain conditions). Macroscopic trajectories are thus traced back to unstable, source-type, and even sometimes saddle-like stationary points, even though the microscopic simulator only evolves forward in time. We also demonstrate the use of such projective integrators in a shooting boundary value problem formulation for the computation of "coarse limit cycles" of the macroscopic behavior, and the approximation of their stability through estimates of the leading "coarse Floquet multipliers".Comment: Submitted to Journal of Computational Physic

ザンビア農村社会の脆弱性とレジリアンス – 土地所有制度と食料安全保障の観点から

The paper shows that pre-colonial ecologies of agricultural systems in some parts of rural Zambia were sustainable and resilient to prevailing environmental conditions, and were therefore able to ensure relative food security, under communal land tenure.However, colonial policies of land alienation and labour migration impacted negatively on food production systems of some ethnic groups like the citemene system of the Bemba and the flood plain cultivation system of the Lozi, making them extremely vulnerable due to the absence of large numbers of males. Paradoxically, the Tonga people in Southern Zambia responded positively to the introduction of modern methods of cultivation, exhibiting resilience by adapting and adopting the cultivation of hybrid maize and the ox-drawn plough. They also began to transform their land tenure system from being communal to become increasingly individualised.At independence in 1964, the UNIP government intervened strongly in promoting rural development (1964-1990), by subsidising maize production and by implementing protectionist policies to maintain communal tenure. However, food security could not be guaranteed, and the policies led to over dependence of small-scale farmers on government and on maize at the expense of other food crops.The introduction of neo-liberal policies (from 1991 to 2001) by the MMD government coupled with adverse weather conditions, made food production systems rather vulnerable to both policy and environmental shocks. However, efforts are being made (from 2001- to date) with the assistance of cooperating partners or the international community, the United Nations System and Non Governmental Organisations (NGOs), to continue with land tenure empowerment policies to ensure secure land tenure for both men and women, and make targeted interventions with partial subsidies to rebuild the resilience of rural society, so as to promote national and household food security

Parity-time symmetry from stacking purely dielectric and magnetic slabs

We show that Parity-time symmetry in matching electric permittivity to magnetic permeability can be established by considering an effective Parity operator involving both mirror symmetry and coupling between electric and magnetic fields. This approach extends the discussion of Parity-time symmetry to the situation with more than one material potential. We show that the band structure of a one-dimensional photonic crystal with alternating purely dielectric and purely magnetic slabs can undergo a phase transition between propagation modes and evanescent modes when the balanced gain/loss parameter is varied. The cross-matching between different material potentials also allows exceptional points of the constitutive matrix to appear in the long wavelength limit where they can be used to construct ultrathin metamaterials with unidirectional reflection