H. De Sterck

TRANSONIC HYDRODYNAMIC ESCAPE OF HYDROGEN FROM EXTRASOLAR PLANETARY ATMOSPHERES

Hydrodynamic escape is an important process in the formation and evolution of planetary atmospheres. Transonic steady state solutions of the time-independent hydrodynamic equations are difficult to find because of the existence of a singularity point. A numerical model is developed to study the hydrodynamic escape of neutral gas from planetary atmospheres by solving the time-dependent hydrodynamic equations. The model is validated against an analytical solution of the escape from an isothermal atmosphere. The model uses a two-dimensional energy deposition calculation instead of the single-layer heating assumption, which is not sufficiently accurate for hydrodynamic escapefrom ahydrogen-richplanetaryatmosphere.Whenapplied totheatmospheresofextrasolar planets, themodel results are in good agreement with observations of the transiting extrasolar planet HD 209458b. The model predicts that hydrogen is escaping from HD 209458b at a maximum rate of 6 ; 10 10 gs � 1 . The extrasolar planet is stable under the hydrodynamic escape of hydrogen. The rate of hydrogen hydrodynamic escape from other possible extrasolar planets is investigated using the model. The importance of hydrogen hydrodynamic escape for the long-term evolution of extrasolar planets is discussed. Simulation shows that through hydrodynamic escape of hydrogen, a planet at the orbit of Mercury (0.4 AU) and with 0.5 Uranus mass can lose about 10% of its mass within 850 million yr if the solar EUV radiation is 10 times the present level. This calculation provides an indication of how Mercury may have evolved during the early days of the solar system. Subject heading gs: planetary systems — planets and satellites: general

Smoothed Aggregation Multigrid for Markov Chains

A smoothed aggregation multigrid method is presented for the numerical calculation of the stationary probability vector of an irreducible sparse Markov chain. It is shown how smoothing the interpolation and restriction operators can dramatically increase the efficiency of aggregation multigrid methods for Markov chains that have been proposed in the literature. The proposed smoothing approach is inspired by smoothed aggregation multigrid for linear systems, supplemented with a new lumping technique that assures well-posedness of the coarse-level problems: the coarse-level operators are singular M-matrices on all levels, resulting in strictly positive coarse-level corrections on all levels. Numerical results show how these methods lead to nearly optimal multigrid efficiency for an extensive set of test problems, both when geometric and algebraic aggregation strategies are used.

Multilevel Adaptive Aggregation for Markov Chains, with Application to Web Ranking

A multilevel adaptive aggregation method for calculating the stationary probability vector of an irreducible stochastic matrix is described. The method is a special case of the adaptive smoothed aggregation and adaptive algebraic multigrid methods for sparse linear systems and is also closely related to certain extensively studied iterative aggregation/disaggregation methods for Markov chains. In contrast to most existing approaches, our aggregation process does not employ any explicit advance knowledge of the topology of the Markov chain. Instead, adaptive agglomeration is proposed that is based on the strength of connection in a scaled problem matrix, in which the columns of the original problem matrix at each recursive fine level are scaled with the current probability vector iterate at that level. The strength of connection is determined as in the algebraic multigrid method, and the aggregation process is fully adaptive, with optimized aggregates chosen in each step of the iteration and at all recursive levels. The multilevel method is applied to a set of stochastic matrices that provide models for web page ranking. Numerical tests serve to illustrate for which types of stochastic matrices the multilevel adaptive method may provide significant speedup compared to standard iterative methods. The tests also provide more insight into why Googles PageRank model is a successful model for determining a ranking of web pages.

Least-Squares Finite Element Methods and Algebraic Multigrid Solvers for Linear Hyperbolic PDEs

Least-squares finite element methods (LSFEMs) for scalar linear partial differential equations (PDEs) of hyperbolic type are studied. The space of admissible boundary data is identified precisely, and a trace theorem and a Poincare inequality are formulated. The PDE is restated as the minimization of a least-squares functional, and the well-posedness of the associated weak formulation is proved. Finite element convergence is proved for conforming and nonconforming (discontinuous) LSFEMs that are similar to previously proposed methods but for which no rigorous convergence proofs have been given in the literature. Convergence properties and solution quality for discontinuous solutions are investigated in detail for finite elements of increasing polynomial degree on triangular and quadrilateral meshes and for the general case that the discontinuity is not aligned with the computational mesh. Our numerical studies found that higher-order elements yield slightly better convergence properties when measured in terms of the number of degrees of freedom. Standard algebraic multigrid methods that are known to be optimal for large classes of elliptic PDEs are applied without modifications to the linear systems that result from the hyperbolic LSFEM formulations. They are found to yield complexity that grows only slowly relative to the size of the linear systems.

A lightweight Java taskspaces framework for scientific computing on computational grids

A prototype Taskspaces framework for grid computing of scientific computing problems that require intertask communication is presented. The Taskspaces framework is characterized by three major design choices: decentralization provided by an underlying tuple space concept, enhanced direct communication between tasks by means of a communication tuple space distributed over the worker hosts, and object orientation and platform independence realized by implementation in Java. Grid administration tasks, for example resetting worker nodes, are performed by mobile agent objects. We report on large-scale grid computing experiments for iterative linear algebra applications showing that our prototype framework scales well for scientific computing problems that require neighbor-neighbor intertask communication. It is shown in a computational fluid dynamics simulation using a Lattice Boltzmann method that the Taskspaces framework can be used naturally in interactive collaboration mode. The scalable Taskspaces framework runs fully transparently on heterogeneous grids while maintaining a low complexity in terms of installation, maintenance, application programming and grid operation. It thus offers a promising roadway to push scientific grid computing with intertask communication beyond the experimental research setting.

High-order central ENO finite-volume scheme for ideal MHD

A high-order accurate finite-volume scheme for the compressible ideal magnetohydrodynamics (MHD) equations is proposed. The high-order MHD scheme is based on a central essentially non-oscillatory (CENO) method combined with the generalized Lagrange multiplier divergence cleaning method for MHD. The CENO method uses k-exact multidimensional reconstruction together with a monotonicity procedure that switches from a high-order reconstruction to a limited low-order reconstruction in regions of discontinuous or under-resolved solution content. Both reconstructions are performed on central stencils, and the switching procedure is based on a smoothness indicator. The proposed high-order accurate MHD scheme can be used on general polygonal grids. A highly sophisticated parallel implementation of the scheme is described that is fourth-order accurate on two-dimensional dynamically-adaptive body-fitted structured grids. The hierarchical multi-block body-fitted grid permits grid lines to conform to curved boundaries. High-order accuracy is maintained at curved domain boundaries by employing high-order spline representations and constraints at the Gauss quadrature points for flux integration. Detailed numerical results demonstrate high-order convergence for smooth flows and robustness against oscillations for problems with shocks. A new MHD extension of the well-known Shu-Osher test problem is proposed to test the ability of the high-order MHD scheme to resolve small-scale flow features in the presence of shocks. The dynamic mesh adaptation capabilities of the approach are demonstrated using adaptive time-dependent simulations of the Orszag-Tang vortex problem with high-order accuracy and unprecedented effective resolution.

Recursively Accelerated Multilevel Aggregation for Markov Chains

A recursive acceleration method is proposed for multiplicative multilevel aggregation algorithms that calculate the stationary probability vector of large, sparse, and irreducible Markov chains. Pairs of consecutive iterates at all branches and levels of a multigrid W cycle with simple, nonoverlapping aggregation are recombined to produce improved iterates at those levels. This is achieved by solving quadratic programming problems with inequality constraints: the linear combination of the two iterates is sought that has a minimal two-norm residual, under the constraint that all vector components are nonnegative. It is shown how the two-dimensional quadratic programming problems can be solved explicitly in an efficient way. The method is further enhanced by windowed top-level acceleration of the W cycles using the same constrained quadratic programming approach. Recursive acceleration is an attractive alternative to smoothing the restriction and interpolation operators, since the operator complexity is better controlled and the probabilistic interpretation of coarse-level operators is maintained on all levels. Numerical results are presented showing that the resulting recursively accelerated multilevel aggregation cycles for Markov chains, combined with top-level acceleration, converge significantly faster than W cycles and lead to close-to-linear computational complexity for challenging test problems.

Algebraic Multigrid for Markov Chains

An algebraic multigrid (AMG) method is presented for the calculation of the stationary probability vector of an irreducible Markov chain. The method is based on standard AMG for nonsingular linear systems, but in a multiplicative, adaptive setting. A modified AMG interpolation formula is proposed that produces a nonnegative interpolation operator with unit row sums. We show how the adoption of a previously described lumping technique maintains the irreducible singular M-matrix character of the coarse-level operators on all levels. Together, these properties are sufficient to guarantee the well-posedness of the algorithm. Numerical results show how it leads to nearly optimal multigrid efficiency for a representative set of test problems.

Efficiency-based h- and hp-refinement strategies for finite element methods

Two efficiency-based grid refinement strategies are investigated for adaptive finite element solution of partial differential equations. In each refinement step, the elements are ordered in terms of decreasing local error, and the optimal fraction of elements to be refined is determined based on efficiency measures that take both error reduction and work into account. The goal is to reach a pre-specified bound on the global error with minimal amount of work. Two efficiency measures are discussed, ‘work times error’ and ‘accuracy per computational cost’. The resulting refinement strategies are first compared for a one-dimensional model problem that may have a singularity. Modified versions of the efficiency strategies are proposed for the singular case, and the resulting adaptive methods are compared with a threshold-based refinement strategy. Next, the efficiency strategies are applied to the case of hprefinement for the one-dimensional model problem. The use of the efficiency-based refinement strategies is then explored for problems with spatial dimension greater than one. The ‘work times error’ strategy is inefficient when the spatial dimension, d, is larger than the finite element order, p, but the ‘accuracy per computational cost’ strategy provides an efficient refinement mechanism for any combination of d and p. Copyright c

The Arab Spring: A simple compartmental model for the dynamics of a revolution

The self-immolation of Mohamed Bouazizi on December 17, 2011 in the small Tunisian city of Sidi Bouzid, set off a sequence of events culminating in the revolutions of the Arab Spring. It is widely believed that the Internet and social media played a critical role in the growth and success of protests that led to the downfall of the regimes in Egypt and Tunisia. However, the precise mechanisms by which these new media affected the course of events remain unclear. We introduce a simple compartmental model for the dynamics of a revolution in a dictatorial regime such as Tunisia or Egypt which takes into account the role of the Internet and social media. An elementary mathematical analysis of the model identifies four main parameter regions: stable police state, meta-stable police state, unstable police state, and failed state. We illustrate how these regions capture, at least qualitatively, a wide range of scenarios observed in the context of revolutionary movements by considering the revolutions in Tunisia and Egypt, as well as the situation in Iran, China, and Somalia, as case studies. We pose four questions about the dynamics of the Arab Spring revolutions and formulate answers informed by the model. We conclude with some possible directions for future work.