Multilevel Turbulence Simulations

A novel multigrid method for the accurate and efficient simulation of turbulent flows is described and demonstrated. The method's efficiency relative to direct simulations is of the order of the ratio of required integration time to the smallest-eddy turnover time, potentially resulting in orders-of-magnitude improvement for a large class of turbulence problems.Earth and Planetary Science

Multigrid third-order least-squares solution of Cauchy–Riemann equations on unstructured triangular grids

In this paper, a multigrid algorithm is developed for the third-order accurate solution of Cauchy–Riemann equations discretized in the cell-vertex finite-volume fashion: the solution values stored at vertices and the residuals defined on triangular elements. On triangular grids, this results in a highly overdetermined problem, and therefore we consider its solution that minimizes the residuals in the least-squares norm. The standard second-order least-squares scheme is extended to third-order by adding a high-order correction term in the residual. The resulting high-order method is shown to give sufficiently accurate solutions on relatively coarse grids. Combined with a multigrid technique, the method then becomes a highly accurate and efficient solver. We present some results to demonstrate its accuracy and efficiency, including both structured and unstructured triangular grids. Copyright © 2006 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/55882/1/1287_ftp.pd

Multigrid Methods in Lattice Field Computations

The multigrid methodology is reviewed. By integrating numerical processes at all scales of a problem, it seeks to perform various computational tasks at a cost that rises as slowly as possible as a function of $n$, the number of degrees of freedom in the problem. Current and potential benefits for lattice field computations are outlined. They include: $O(n)$ solution of Dirac equations; just $O(1)$ operations in updating the solution (upon any local change of data, including the gauge field); similar efficiency in gauge fixing and updating; $O(1)$ operations in updating the inverse matrix and in calculating the change in the logarithm of its determinant; $O(n)$ operations per producing each independent configuration in statistical simulations (eliminating CSD), and, more important, effectively just $O(1)$ operations per each independent measurement (eliminating the volume factor as well). These potential capabilities have been demonstrated on simple model problems. Extensions to real life are explored.Comment: 4

Improved finite difference method for long distance propagation of waves

Evaluation of aging aircraft involves inspection of large areas of plates, including lap joints, bonded material and other complex geometries. These inspections need to be done within a reasonable time in order to be economical. Ultrasonic measurements of plate-modes propagating for large distances [1] are a promising method for fast detection of these defects in a commercial environment. There is a need to predict the waveforms of such waves in order to analyze the results of the experimental data. Analytic solution cannot always be found for general cases, therefore, numerical prediction of elastic wave propagation for long distances is desired. However, the large problem size and the long propagation time that is involved require careful attention to the discretization. Small mesh size discretization which is needed for accurate solutions can no longer be practical because of the memory limitations and the high computational costs. On the other hand, a large mesh size can cause severe errors especially when long propagation time is concerned. This paper describes a new method which successfully deals with this issue.</p