Adaptive Wavelet Representation And Differentiation On Block-structured Grids

This paper considers a new adaptive wavelet solver for two-dimensional systems based on an adaptive block refinement (ABR) method that takes advantage of the quadtree structure of dyadic blocks in rectangular regions of the plane. The computational domain is formed by non-overlapping blocks. Each block is a uniform grid, but the step size may change from one block to another. The blocks are not predetermined, but they are dynamically constructed according to the refinement needs of the numerical solution. The decision over whether a block should be refined or unrefined is taken by looking at the magnitude of wavelet coefficients of the numerical solution on such block. The wavelet coefficients are defined as differences between values interpolated from a coarser level and known function values at the finer level. The main objective of this paper is to establish a general framework for the construction and operation on such adaptive block-grids in 2D. The algorithms and data structure are formulated by using abstract concepts borrowed from quaternary trees. This procedure helps in the understanding of the method and simplifies its computational implementation. The ability of the method is demonstrated by solving some typical test problems. © 2003 IMACS. Published by Elsevier B.V. All rights reserved.4703/04/15421437Cohen, A., Wavelet methods in numerical analysis (2000) Handbook of Numerical Analysis, 7. , P.G. Ciarlet, LionsJ.L. Amsterdam: ElsevierHolmström, M., (1997) Wavelet Based Methods for Time Dependent PDEs, , Ph.D. Thesis, Uppsala UniversityWalden, J., A general adaptive solver for hyperbolic PDEs based on filter bank subdivisions (2000) Appl. Numer. Math., 33 (1-4), pp. 317-325Vasilyev, O.V., Browman, C., Second generation wavelet collocation method for the solution of partial differential equations (2000) J. Comput. Phys., 165, pp. 660-693Knuth, D.E., (1997) The Art of Programming, , Reading, MA: Addison-WesleyHunter, G.M., Steiglitz, K., Operations on images using quad trees (1979) IEEE Trans. Pattern Anal. Mach. Intell., PAMI-1 (2), pp. 145-153Tromper, R.A., Verwer, J.G., Runge-Kutta methods and local uniform grid refinement (1993) Math. Comput., 60 (202), pp. 591-616Bacry, E., Mallat, S., Papanicolau, G., A wavelet based space-time adaptive numerical method for partial equations (1992) Math. Model. Numer. Anal., 26 (7), pp. 793-834Lötstedt, P., Söderberg, S., Ramage, A., Hemmingsson-Frändén, L., Implicit solution of hyperbolic equations with space-time adaptivity (2002) BIT, 42, pp. 134-158Glowinski, R., Pan, T.-W., Périaux, J., A fictitious domain method for Dirichlet problem and application (1994) Comput. Methods Appl. Mech. Engrg., 111, pp. 283-303Koshigoe, H., Kitahara, K., Finite difference method with fictitious domain applied to a Dirichlet problem (2001) 12th Conference on Domain Decomposition Methods, pp. 151-163. , T. Chan, T. Kako, H. Kawarada, & O. Pironneau. DDM.orgKunoth, A., Wavelet techniques for the fictitious domain - Lagrange multiplier approach (2001) Numer. Algorithms, 27, pp. 291-316Rieder, A., Embedding and a priori wavelet-adaptivity for Dirichlet problems (2000) Modél. Math. Anal. Numér., 34 (6), pp. 1189-120

CMS Physics Technical Design Report: Addendum on High Density QCD with Heavy Ions

This report presents the capabilities of the CMS experiment to explore the rich heavy-ion physics programme offered by the CERN Large Hadron Collider (LHC). The collisions of lead nuclei at energies $\sqrt{s_{NN}}= 5.5\,{\rm TeV}$ , will probe quark and gluon matter at unprecedented values of energy density. The prime goal of this research is to study the fundamental theory of the strong interaction \u2014 Quantum Chromodynamics (QCD) \u2014 in extreme conditions of temperature, density and parton momentum fraction (low- x ). This report covers in detail the potential of CMS to carry out a series of representative Pb-Pb measurements. These include "bulk" observables, (charged hadron multiplicity, low p T inclusive hadron identified spectra and elliptic flow) which provide information on the collective properties of the system, as well as perturbative probes such as quarkonia, heavy-quarks, jets and high p T hadrons which yield "tomographic" information of the hottest and densest phases of the reaction