Hot Spots in the Weak Detonation Problem and Special Relativity

Problem statement: The initiation of a detonation in an explosive gaseous mixture in the high activation energy regime, in three space dimensions, typically leads to the formation of a singularity at one point, the “hot spot”. It would be suitable to have a description of the physical quantities in a full neighborhood of the hot spot. Results of this paper: (1) To achieve this, it is necessary to replace the blow-up time, or time when the hot spot first occurs, by the blow-up surface in four dimensions, which is the set of all hot spots for a class of observers related to one another by a Lorentz transformation. (2) A local general solution of the nonlinear system of PDE modeling fluid flow and chemistry, with a given blow-up surface, is obtained by the method of Fuchsian reduction. Advantages of this solution: (i) Earlier approximate solutions are contained in it, but the domain of validity of the present solution is larger; (ii) it provides a signature for this type of ignition mechanism; (iii) quantities that remain bounded at the hot spot may be determined, so that, in principle, this model may be tested against measurements; (iv) solutions with any number of hot spots may be constructed. The impact on numerical computation is also discussed.

INTELLIGENT NEIGHBORHOOD EXPLORATION IN LOCAL SEARCH METHOD

Standard tabu search methods are based on the complete exploration of current solution neighborhood. However, for some problems due to the neighborhood size or to the fitness evaluation complexity, the total exploration of the neighborhood is impractical. The main purpose of this paper is to propose a local search method with no enumeration procedure. In other words, a single solution is examined at each iteration and retained for the future iterations. The idea is to randomly select one solution among a sub-set of the neighborhood of the current one. An adaptive exploration of neighborhood, using extension and restriction mechanisms represented by loop detection and tabu list structure, determines this sub-set. This approach is applied to the K-coloring problem and evaluated on standard benchmarks like DIMACS. The objective is to show how a generic method, without full neighborhood exploration, degradation control and problem-oriented operators, provides a very competitive results comparing to the best dedicated algorithms for graph coloring problems published in the literature.