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