Singularity embedding method in potential flow calculations

The so-called H-type mesh is used in a finite-element (or finite-volume) calculation of the potential flow past an airfoil. Due to coordinate singularity at the leading edge, a special singular trial function is used for the elements neighboring the leading edge. The results using the special singular elements are compared to those using the regular elements. It is found that the unreasonable pressure distribution obtained by the latter is removed by the embedding of the singular element. Suggestions to extend the present method to transonic cases are given

A Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin for Diffusion

We introduce a new approach to high-order accuracy for the numerical solution of diffusion problems by solving the equations in differential form using a reconstruction technique. The approach has the advantages of simplicity and economy. It results in several new high-order methods including a simplified version of discontinuous Galerkin (DG). It also leads to new definitions of common value and common gradient quantities at each interface shared by the two adjacent cells. In addition, the new approach clarifies the relations among the various choices of new and existing common quantities. Fourier stability and accuracy analyses are carried out for the resulting schemes. Extensions to the case of quadrilateral meshes are obtained via tensor products. For the two-point boundary value problem (steady state), it is shown that these schemes, which include most popular DG methods, yield exact common interface quantities as well as exact cell average solutions for nearly all cases

Partner orbits and action differences on compact factors of the hyperbolic plane. Part I: Sieber-Richter pairs

Physicists have argued that periodic orbit bunching leads to universal spectral fluctuations for chaotic quantum systems. To establish a more detailed mathematical understanding of this fact, it is first necessary to look more closely at the classical side of the problem and determine orbit pairs consisting of orbits which have similar actions. In this paper we specialize to the geodesic flow on compact factors of the hyperbolic plane as a classical chaotic system. We prove the existence of a periodic partner orbit for a given periodic orbit which has a small-angle self-crossing in configuration space which is a `2-encounter'; such configurations are called `Sieber-Richter pairs' in the physics literature. Furthermore, we derive an estimate for the action difference of the partners. In the second part of this paper [13], an inductive argument is provided to deal with higher-order encounters.Comment: to appear on Nonlinearit

The Time-Honored Friendship: A History of Vietnamese-Algerian Relations (1946-2015)

In 1958, the newly established Democratic Republic of Vietnam initiated a top secret program to ship a “large quantity” of submachine guns disguised as commercial goods to Algeria to assist the Front de libération nationale in its struggle for independence from French colonial rule. In 1973, Algeria leveraged its position as the host of the fourth Summit of the Non-Aligned Movement to issue a draft resolution requesting that all member nations pledge diplomatic support to the Việt Cộng, contribute to Vietnam’s post-war reconstruction, and demand the wholescale withdrawal of foreign troops from the Southeast Asian nation. At the close of 2015, Vietnam and Algeria celebrated the first commercial oil flow from the joint Vietnamese-Algerian Bir Seba oil project, located in the Algerian Sahara. Despite such events indicating that there exists a long and rich history of Vietnamese-Algerian relations, there has been no scholarship documenting it. Responding to this gap in scholarship, this project, “The Time-Honored Friendship,” pieces together the history of Vietnamese-Algerian relations from the beginning of the Indochina War in 1946 to the present day. In doing so, it proposes that the relationship can be divided into three distinct eras: anti-colonial solidarity (1946-1962); socialist, anti-imperial brotherhood (1962-1986); and joint ventures in economic liberalization (1986-2015). Corresponding with these three proposed eras of Vietnamese-Algerian engagement, this project is divided into three main sections. The first section, “The Era of Anti-Colonial Solidarity (1946-1962),” argues that the Vietnamese and Algerian people understood each others’ struggles against French colonial rule as extensions of their own and supported each other accordingly. Acts of solidarity were not merely initiated at the state level by political elite, but were also overwhelmingly driven from the grassroots during both the Indochina and Algerian wars of independence. The second section, “The Era of Socialist, Anti-Colonial Solidarity (1962-1986),” asserts that having both secured their formal independence from France, Vietnam and Algeria were eager to engage with one another through official bilateral relations. They premised their official relationship on their common adherence to the socialist creed and on supporting each other in securing economic sovereignty from the neo-colonial West. The third section, “The Era of Joint Ventures in Economic Liberalization (1986-2015),” details the drastic turn in the Vietnamese-Algerian relationship from being premised on revolutionary struggle against colonialism in all its forms to being premised on mutual economic growth through foreign investment, increased bilateral trade, and technical cooperation for the sake of reaching parity with the developed world. Rather than collaborating to fend off the corrupting influences of the West, the two nations came to embrace liberalization, and worked together to navigate a post-Cold War capitalist order. The conclusion entreats scholars of all disciplines to both build on this project’s findings and to explore how other formerly colonized peoples around the world, united in common oppression under distant European powers, engaged with each other in the quest for a more nuanced trans-regional scholarship

New results from H.E.S.S. observations of galaxy clusters

Clusters of galaxies are believed to contain a significant population of cosmic rays. From the radio and probably hard X-ray bands it is known that clusters are the spatially most extended emitters of non-thermal radiation in the Universe. Due to their content of cosmic rays, galaxy clusters are also potential sources of VHE (>100 GeV) gamma rays. Recently, the massive, nearby cluster Abell 85 has been observed with the H.E.S.S. experiment in VHE gamma rays with a very deep exposure as part of an ongoing campaign. No significant gamma-ray signal has been found at the position of the cluster. The non-detection of this object with H.E.S.S. constrains the total energy of cosmic rays in this system. For a hard spectral index of the cosmic rays of -2.1 and if the cosmic-ray energy density follows the large scale gas density profile, the limit on the fraction of energy in these non-thermal particles with respect to the total thermal energy of the intra-cluster medium is 8% for this particular cluster. This value is at the lower bounds of model predictions.Comment: 4 pages, one figure, invited talk at the 2nd Heidelberg workshop: "High-Energy Gamma-rays and Neutrinos from Extra-Galactic Sources", January 13 - 16, 2009, to be published in Int. J. Mod. Phys.

On High-Order Upwind Methods for Advection

In the fourth installment of the celebrated series of five papers entitled "Towards the ultimate conservative difference scheme", Van Leer (1977) introduced five schemes for advection, the first three are piecewise linear, and the last two, piecewise parabolic. Among the five, scheme I, which is the least accurate, extends with relative ease to systems of equations in multiple dimensions. As a result, it became the most popular and is widely known as the MUSCL scheme (monotone upstream-centered schemes for conservation laws). Schemes III and V have the same accuracy, are the most accurate, and are closely related to current high-order methods. Scheme III uses a piecewise linear approximation that is discontinuous across cells, and can be considered as a precursor of the discontinuous Galerkin methods. Scheme V employs a piecewise quadratic approximation that is, as opposed to the case of scheme III, continuous across cells. This method is the basis for the on-going "active flux scheme" developed by Roe and collaborators. Here, schemes III and V are shown to be equivalent in the sense that they yield identical (reconstructed) solutions, provided the initial condition for scheme III is defined from that of scheme V in a manner dependent on the CFL number. This equivalence is counter intuitive since it is generally believed that piecewise linear and piecewise parabolic methods cannot produce the same solutions due to their different degrees of approximation. The finding also shows a key connection between the approaches of discontinuous and continuous polynomial approximations. In addition to the discussed equivalence, a framework using both projection and interpolation that extends schemes III and V into a single family of high-order schemes is introduced. For these high-order extensions, it is demonstrated via Fourier analysis that schemes with the same number of degrees of freedom per cell, in spite of the different piecewise polynomial degrees, share the same sets of eigenvalues and thus, have the same stability and accuracy. Moreover, these schemes are accurate to order 21, which is higher than the expected order of

Collocation and Galerkin Time-Stepping Methods

We study the numerical solutions of ordinary differential equations by one-step methods where the solution at tn is known and that at t(sub n+1) is to be calculated. The approaches employed are collocation, continuous Galerkin (CG) and discontinuous Galerkin (DG). Relations among these three approaches are established. A quadrature formula using s evaluation points is employed for the Galerkin formulations. We show that with such a quadrature, the CG method is identical to the collocation method using quadrature points as collocation points. Furthermore, if the quadrature formula is the right Radau one (including t(sub n+1)), then the DG and CG methods also become identical, and they reduce to the Radau IIA collocation method. In addition, we present a generalization of DG that yields a method identical to CG and collocation with arbitrary collocation points. Thus, the collocation, CG, and generalized DG methods are equivalent, and the latter two methods can be formulated using the differential instead of integral equation. Finally, all schemes discussed can be cast as s-stage implicit Runge-Kutta methods