Multilevel Adaptive Method Research Articles

Implicit adaptive mesh refinement for 2D reduced resistive magnetohydrodynamics

An implicit structured adaptive mesh refinement (SAMR) solver for 2D reduced magnetohydrodynamics (MHD) is described. The time-implicit discretization is able to step over fast normal modes, while the spatial adaptivity resolves thin, dynamically evolving features. A Jacobian-free Newton–Krylov method is used for the nonlinear solver engine. For preconditioning, we have extended the optimal “physics-based” approach developed in [L. Chacón, D.A. Knoll, J.M. Finn, An implicit, nonlinear reduced resistive MHD solver, J. Comput. Phys. 178 (2002) 15–36] (which employed multigrid solver technology in the preconditioner for scalability) to SAMR grids using the well-known Fast Adaptive Composite grid (FAC) method [S. McCormick, Multilevel Adaptive Methods for Partial Differential Equations, SIAM, Philadelphia, PA, 1989]. A grid convergence study demonstrates that the solver performance is independent of the number of grid levels and only depends on the finest resolution considered, and that it scales well with grid refinement. The study of error generation and propagation in our SAMR implementation demonstrates that high-order (cubic) interpolation during regridding, combined with a robustly damping second-order temporal scheme such as BDF2, is required to minimize impact of grid errors at coarse-fine interfaces on the overall error of the computation for this MHD application. We also demonstrate that our implementation features the desired property that the overall numerical error is dependent only on the finest resolution level considered, and not on the base-grid resolution or on the number of refinement levels present during the simulation. We demonstrate the effectiveness of the tool on several challenging problems.

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 Google's PageRank model is a successful model for determining a ranking of web pages.

Multilevel Adaptive Methods for Semi-Implicit Solution of Shallow-Water Equations on a Sphere

Abstract A multigrid algorithm for local refinement in time and space for an Eulerian formulation of the shallow-water equations on a sphere is presented. It is shown that the accuracy obtainable on a full global grid can be reached using local patches in time and space. Results are presented using a model problem with a large-scale disturbance and a problem with topography and realistic initial conditions.

Multilevel Adaptive Methods for Elliptic Eigenproblems: A Two-Level Convergence Theory

The fast adaptive composite grid method (FAC) is a multilevel scheme that attempts to provide accurate and optimal computation of solutions of partial differential equations on refined grids. The central feature of FAC that ensures its efficiency is its domain-decomposition-like use of uniform grids that cover different regions with different meshsizes, but that fully overlap. This full overlap is geometric in the sense that the regions covered by the finer grids are also covered by the coarser ones, but only at the coarser grid scales. FAC’s full overlap yields fast convergence rates, while its limited scales in the overlap region assure optimal complexity. This paper develops a two-level convergence theory for FAG applied to elliptic eigenproblems. The FAG method is realized by constructing a refined grid version of RQMG, which is a multilevel Rayleigh quotient minimization scheme developed in an earlier paper. The theory shows that this form of FAG converges independent of the meshsizes of either the global or local grids. As such, this is the first theory established for multilevel adaptive methods that applies to other than linear equations.

Multilevel adaptive methods for laminar diffusion flames

A multilevel adaptive method is applied to the numerical simulation of laminar diffusion flames. A local fine grid is embedded near the jet inlet of the simulation to provide increased resolution and accuracy. Computational results confirm that the multilevel local refinement process substantially increases local and global accuracy with little added cost.

Multilevel Adaptive Methods for Partial Differential Equations.

1. Introduction 2. The Finite Volume Element Method (FVE) 3. Multigrid Methods (MG) 4. The Fast Adaptive Composite Grid Method (FAC) 5. The Asynchronous Fast Adaptive Composite Grid Method (AFAC) Appendix References Index.

Multilevel adaptive methods for incompressible flow in grooved channels

Incompressible moderate-Reynolds-number flow in a periodically grooved channel is investigated by direct numerical simulation using the finite-volume method on a staggered grid. A second-order, fully-implicit time-marching scheme is used together with a multigrid full approximation scheme (FAS) to accelerate the convergence process. Convergence factors of about 0.15 for each V(2,2) cycle are observed. The computational results for steady flow show good agreement with that of Ghaddar (1986). A local fine grid is placed about the cavity to achieve better accuracy, without the need for a global fine grid. Both FAC (see McCormick (1989)) and MLAT (see Brandt (1984)) adaptive techniques show optimal efficiency as solvers for the resulting composite grid problem.

Multilevel Adaptive Methods for Partial Differential Equations (Stephen F. McCormick)

Previous article Next article Multilevel Adaptive Methods for Partial Differential Equations (Stephen F. McCormick)Alan WeiserAlan Weiserhttps://doi.org/10.1137/1033165PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"Multilevel Adaptive Methods for Partial Differential Equations (Stephen F. McCormick)." SIAM Review, 33(4), pp. 680–681 Previous article Next article FiguresRelatedReferencesCited byDetails Volume 33, Issue 4| 1991SIAM Review History Published online:18 July 2006 InformationCopyright © 1991 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1033165Article page range:pp. 680-681ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

Computational complexity of the fast adaptive composite grid (FAC) method

This paper develops bounds on the computational complexity of the FAC algorithm which is a multi-level adaptive method for discretization and solution of partial differential equations. The estimates include the cost of the complete scheme, including the dynamic grid refinement process. They show that, for many applications, FAC has computational complexity of the order of the number of points in the final adapted grid. This is significant because adaptive methods that are not based on multi-level techniques generally have much higher complexity.

Asynchronous multilevel adaptive methods for solving partial differential equations on multiprocessors: Performance results

The fast adaptive composite grid method (FAC) is an algorithm that uses various levels of uniform grids (global and local) to provide adaptive resolution and fast solution of PDEs. Like all such methods, it offers parallelism by using possibly many disconnected patches per level, but is hindered by the need to handle these levels sequentially. The finest levels must therefore wait for processing to be essentially completed on all the coarser ones. A recently developed asynchronous version of FAC, called AFAC, completely eliminates this bottleneck to parallelism. This paper describes timing results for AFAC, coupled with a simple load balancing scheme, applied to the solution of elliptic PDEs on an Intel iPSC hypercube. These tests include performance of certain processes necessary in adaptive methods, including moving grids and changing refinement. A companion paper reports on numerical and analytical results for estimating convergence factors of AFAC applied to very large scale examples.

Asynchronous multilevel adaptive methods for solving partial differential equations on multiprocessors: Basic ideas

Several mesh refinement methods exists for solving partial differential equations that make efficient use of local grids on scalar computers. On distributed memory multiprocessors, such methods benefit from their tendency to create multiple refinement regions, yet they suffer from the sequential way that the levels of refinement are treated. The asynchronous fast adaptive composite grid method (AFAC) is developed here as a method that can process refinement levels in parallel while maintaining full multilevel convergence speeds. In the present paper, we develop a simple two-level AFAC theory and provide estimates of its asymptotic convergence factors as it applies to very large scale examples. In a companion paper, we report on extensive timing results for AFAC, implemented on an Intel iPSC hypercube.