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Application of Numerical Methods to Selected Model Equations
Published in Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar, Computational Fluid Mechanics and Heat Transfer, 2020
Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar
The multigrid method is one of the most efficient general iterative methods known today. The key word here is “general.” More efficient schemes can be found for certain problems or certain choices of grids, but it is difficult to find a method more efficient than multigrid for the general case. The multigrid technique can be applied using any of the iterative schemes discussed in this chapter as the “smoother,” although the Gauss–Seidel procedure will be used to illustrate the main points of this technique in the introductory material presented here. The objective of the multigrid technique is to accelerate the convergence of an iterative scheme.
Recent Developments to Improve the Numerical Accuracy
Published in James Fern, Alexander Rohe, Kenichi Soga, Eduardo Alonso, The Material Point Method for Geotechnical Engineering, 2019
James Fern, Alexander Rohe, Kenichi Soga, Eduardo Alonso
Meanwhile, p-multigrid can be applied to solve linear systems resulting from IgA discretisations to ensure an efficient computation with a consistent mass matrix. Compared to the standard techniques, such as the Conjugate Gradient method, p-multigrid requires a considerably lower number of iterations. In addition, p-multigrid can be used to solve the momentum balance equation within BSMPM. Based on the research of Love and Sulsky [176], it is expected that a consistent mass matrix will conserve energy and angular momentum.
Numerical Methods for ODEs
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
Multigrid methods are iterative methods for solving systems of linear equations arising from differential equations. Generally, different grid sizes are used with only a few iterations per grid. The last approximation on one grid becomes the first approximation on the next grid.
Newton-type multilevel optimization method
Published in Optimization Methods and Software, 2022
Chin Pang Ho, Michal Kočvara, Panos Parpas
Multigrid methods are a well-known and established method for solving differential equations [3,11,13,23,24,26]. When solving a differential equation using numerical methods, an approximation of the solution is obtained on a mesh via discretization. The computational cost of solving the discretized problem, however, varies and it depends on the choice of the mesh size used. Therefore, by considering different mesh sizes, a hierarchy of discretized models can be defined. In general, a more accurate solution can be obtained when a smaller mesh size is chosen, which results in a discretized problem in higher dimensions. We shall follow the traditional terminology in the multigrid literature and call a fine model to be the discretization in which its solution is sufficiently close to the solution of the original differential equation; otherwise we call it a coarse model [3]. The main idea of multigrid methods is to make use of the geometric similarity between different discretizations. In particular, during the iterative process of computing the solution of the fine model, one replaces part of the computations with the information from coarse models. The advantages of using multigrid methods are twofold. Firstly, coarse models are in lower dimensions compared to the fine model, and so the computational cost is reduced. Secondly and interestingly, the corrections generated by the coarse model and fine model are in fact complementary. It has been shown that using the fine model is effective in reducing the high frequency components of the residual (error) but ineffective in reducing the low frequency component of the error. Those low frequency components of the error, however, will become high frequency errors in the coarse model. Thus, they could be eliminated effectively using coarse models [3,23].
A Perspective on Data-Driven Coarse Grid Modeling for System Level Thermal Hydraulics
Published in Nuclear Science and Engineering, 2022
Arsen S. Iskhakov, Cheng-Kai Tai, Igor A. Bolotnov, Nam T. Dinh
Other methodologies such as mesh and algorithm (model) refinement (coarsening)16 and multigrid methods17 allow one to algorithmically reduce the computational cost thereby enabling multiscale treatment. The idea of adaptive mesh and algorithm refinement is to resolve microscale effects only in regions of interest; its main disadvantage is in a lack of scalability. Multigrid methods accelerate the convergence of iterative methods; they are also hardly scalable as they work best for elliptic problems such as steady-state flows.
Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines
Published in Nuclear Science and Engineering, 2018
R. N. Slaybaugh, M. Ramirez-Zweiger, Tara Pandya, Steven Hamilton, T. M. Evans
The idea of multigrid methods is to take advantage of the smoothing effects of iterative methods by making smooth errors look oscillatory and thus easier to remove. Errors that are low-frequency on a fine grid can be mapped onto a coarser grid where they are high frequency. A relaxer is applied on the coarser grid to remove the now oscillatory error components. The remaining error is mapped to a still coarser grid and smoothed again. The problem is restricted to coarser and coarser grids until the coarsest grid is reached.