Multigrid Iterative Methods Research Articles

Numerical research on flow and thermal transport in cooling pool of electrical power station using three depth-averaged turbulence models

This paper describes a numerical simulation of thermal discharge in the cooling pool of an electrical power station, aiming to develop general-purpose computational programs for grid generation and flow/pollutant transport in the complex domains of natural and artificial waterways. Three depth-averaged two-equation closure turbulence models, {m1}, {m2}, and {m3}, were used to close the quasi three-dimensional hydrodynamic model. The {m3} model was recently established by the authors and is still in the testing process. The general-purpose computational programs and turbulence models will be involved in a software that is under development. The SIMPLE (Semi-Implicit Method for Pressure-Linked Equation) algorithm and multi-grid iterative method are used to solve the hydrodynamic fundamental governing equations, which are discretized on non-orthogonal boundary-fitted grids with a variable collocated arrangement. The results calculated with the three turbulence models were compared with one another. In addition to the steady flow and thermal transport simulation, the unsteady process of waste heat inpouring and development in the cooling pool was also investigated.

Flow and transport simulation of Madeira River using three depth-averaged two-equation turbulence closure models

This paper describes a numerical simulation in the Amazon water system, aiming to develop a quasi-three-dimensional numerical tool for refined modeling of turbulent flow and passive transport of mass in natural waters. Three depth-averaged two-equation turbulence closure models, k˜−ε˜,k˜−w˜, and k˜−ω˜ , were used to close the non-simplified quasi-three dimensional hydrodynamic fundamental governing equations. The discretized equations were solved with the advanced multi-grid iterative method using non-orthogonal body-fitted coarse and fine grids with collocated variable arrangement. Except for steady flow computation, the processes of contaminant inpouring and plume development at the beginning of discharge, caused by a side-discharge of a tributary, have also been numerically investigated. The three depth-averaged two-equation closure models are all suitable for modeling strong mixing turbulence. The newly established turbulence models such as the k˜−ω˜ model, with a higher order of magnitude of the turbulence parameter, provide a possibility for improving computational precision.

PORTA: A three-dimensional multilevel radiative transfer code for modeling the intensity and polarization of spectral lines with massively parallel computers

The interpretation of the intensity and polarization of the spectral line radiation produced in the atmosphere of the Sun and of other stars requires solving a radiative transfer problem that can be very complex, especially when the main interest lies in modeling the spectral line polarization produced by scattering processes and the Hanle and Zeeman effects. One of the difficulties is that the plasma of a stellar atmosphere can be highly inhomogeneous and dynamic, which implies the need to solve the non-equilibrium problem of the generation and transfer of polarized radiation in realistic three-dimensional (3D) stellar atmospheric models. Here we present PORTA, an efficient multilevel radiative transfer code we have developed for the simulation of the spectral line polarization caused by scattering processes and the Hanle and Zeeman effects in 3D models of stellar atmospheres. The numerical method of solution is based on the non-linear multigrid iterative method and on a novel short-characteristics formal solver of the Stokes-vector transfer equation which uses monotonic B\'ezier interpolation. Therefore, with PORTA the computing time needed to obtain at each spatial grid point the self-consistent values of the atomic density matrix (which quantifies the excitation state of the atomic system) scales linearly with the total number of grid points. Another crucial feature of PORTA is its parallelization strategy, which allows us to speed up the numerical solution of complicated 3D problems by several orders of magnitude with respect to sequential radiative transfer approaches, given its excellent linear scaling with the number of available processors. The PORTA code can also be conveniently applied to solve the simpler 3D radiative transfer problem of unpolarized radiation in multilevel systems.

Flow and contaminant transport simulations of the Solimes River using three depth-averaged two-equation closure turbulence models

This paper reports a numerical simulation in the Amazon water system, aiming to develop a quasi-3D numerical tool for refinedly modeling turbulent flows and passive mass transport phenomena in natural waters. Three depth-averaged two-equation closure turbulence models, - , - , and - , were used to close the non-simplified quasi three-dimensional hydrodynamic fundamental governing equations. The discretized equations were solved by advanced multi-grid iterative method under non-orthogonal body-fitted coarse and fine grids with collocated variable arrangement. Except for steady flow computation, the processes of contaminant inpouring and plume development at the beginning of discharge, caused by a side-discharge of a tributary, also have been numerically investigated. The used three depth-averaged two-equation closure models are all suitable for modeling strong mixing turbulence. The newly established turbulence model, such as - model with higher order of magnitude of transported variable , provides a possibility to crease the computational precision. Key words: River modeling, numerical modeling, contaminant transport, depth-averaged turbulence models, multi-grid iterative method.

Electromagnetic scattering for dielectrics: iterative methods for solving boundary integral equations

We consider multigrid iterative methods for the solution of electromagnetic scattering for dielectric materials. We show convergence of the iteration using coarse grids which are two to four times coarser in each dimension than the fine grid. These results allow a significant increase in problem size and solution speed, for a given hardware configuration. We report in particular on the solution of scattering problems which require the solution of 31,000 equations on the fine grid, using the direct solution of 3,500 double precision equations on the coarse grid, and project the ability to solve significantly larger systems using larger machines or an out-of-core capability.

On higher order multigrid methods with application to a geothermal reservoir model

AbstractThis paper considers the multigrid iterative method applied to the solution of finite difference approximations to a linear second‐order self‐adjoint elliptic equation. It represents an extension of work by Dinar and Brandt. We compare two methods to obtain fourth‐order convergence. The first is local error extrapolation developed by Brandt, the second is iterative improvement developed by Lindberg. This work considers non‐separable problems, but only on a rectangular domain with Dirichlet boundary conditions. We consider test cases with non‐smooth (i.e. discontinuous second derivatives) as well as smooth solutions. We also apply the multigrid method to an elliptic equation with non‐separable coefficients which occurs in a geothermal model. In this case an analysis of the error fails to show any advantage in a fourth‐order difference scheme over a second‐order scheme. However, we do demonstrate that the multigrid iteration performs well on this problem. Also, this example shows that the multigrid iteration can be combined with iterative improvement to create an efficient fourth‐order method for a non‐separable elliptic equation which is coupled with a marching equation. Other work has found an advantage in this fourth‐order scheme for a similar geothermal model.