SOR (Successive Over-Relaxation) MGR (Multigrid Algorithm)(nu) Experiments on the Crystal Multicomputer,

1986 ◽  
Author(s):  
David Kamowitz
Keyword(s):  
Multigrid Algorithm ◽  
Successive Over Relaxation

scholarly journals Augmented Lagrangian preconditioners for the Oseen–Frank model of nematic and cholesteric liquid crystals

2021 ◽  
Author(s):  
Jingmin Xia ◽  
Patrick E. Farrell ◽  
Florian Wechsung
Keyword(s):  
Liquid Crystals ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Mass Matrix ◽  
Unit Length ◽  
Mesh Size ◽  
Schur Complement ◽  
Cholesteric Liquid Crystals ◽  
Multigrid Algorithm ◽  
The Cost

AbstractWe propose a robust and efficient augmented Lagrangian-type preconditioner for solving linearizations of the Oseen–Frank model arising in nematic and cholesteric liquid crystals. By applying the augmented Lagrangian method, the Schur complement of the director block can be better approximated by the weighted mass matrix of the Lagrange multiplier, at the cost of making the augmented director block harder to solve. In order to solve the augmented director block, we develop a robust multigrid algorithm which includes an additive Schwarz relaxation that captures a pointwise version of the kernel of the semi-definite term. Furthermore, we prove that the augmented Lagrangian term improves the discrete enforcement of the unit-length constraint. Numerical experiments verify the efficiency of the algorithm and its robustness with respect to problem-related parameters (Frank constants and cholesteric pitch) and the mesh size.

Increased space-parallelism via time-simultaneous Newton-multigrid methods for nonstationary nonlinear PDE problems

2021 ◽  
Vol 35 (3) ◽  
pp. 211-225
Author(s):  
Jonas Dünnebacke ◽  
Stefan Turek ◽  
Christoph Lohmann ◽  
Andriy Sokolov ◽  
Peter Zajac
Keyword(s):  
Krylov Subspace ◽  
Solution Procedure ◽  
Grid Size ◽  
Waveform Relaxation ◽  
Test Problems ◽  
Subspace Method ◽  
Multigrid Algorithm ◽  
Multigrid Solvers ◽  
Large Systems ◽  
Linear Pdes

We discuss how “parallel-in-space & simultaneous-in-time” Newton-multigrid approaches can be designed which improve the scaling behavior of the spatial parallelism by reducing the latency costs. The idea is to solve many time steps at once and therefore solving fewer but larger systems. These large systems are reordered and interpreted as a space-only problem leading to multigrid algorithm with semi-coarsening in space and line smoothing in time direction. The smoother is further improved by embedding it as a preconditioner in a Krylov subspace method. As a prototypical application, we concentrate on scalar partial differential equations (PDEs) with up to many thousands of time steps which are discretized in time, resp., space by finite difference, resp., finite element methods. For linear PDEs, the resulting method is closely related to multigrid waveform relaxation and its theoretical framework. In our parabolic test problems the numerical behavior of this multigrid approach is robust w.r.t. the spatial and temporal grid size and the number of simultaneously treated time steps. Moreover, we illustrate how corresponding time-simultaneous fixed-point and Newton-type solvers can be derived for nonlinear nonstationary problems that require the described solution of linearized problems in each outer nonlinear step. As the main result, we are able to generate much larger problem sizes to be treated by a large number of cores so that the combination of the robustly scaling multigrid solvers together with a larger degree of parallelism allows a faster solution procedure for nonstationary problems.

scholarly journals The convection of a Bingham fluid in a differentially-heated porous cavity

2016 ◽  
Vol 26 (3/4) ◽  
pp. 879-896 ◽  
Author(s):  
D. Andrew S. Rees
Keyword(s):  
Rayleigh Number ◽  
Yield Stress ◽  
Piecewise Linear ◽  
Rectangular Cavity ◽  
Content Type ◽  
Multigrid Algorithm ◽  
Bingham Number ◽  
Steady Solutions ◽  
Darcy Flux ◽  
Darcy Rayleigh Number

Purpose – The purpose of this paper is to determine the manner in which a yield stress fluid begins convecting when it saturates a porous medium. A sidewall-heated rectangular cavity is selected as the testbed for this pioneering work. Design/methodology/approach – Steady solutions are obtained using a second order accurate finite difference method, line relaxation based on the Gauss-Seidel smoother, a Full Approximation Scheme multigrid algorithm with V-cycling and a regularization of the Darcy-Bingham model to smooth the piecewise linear relation between the Darcy flux and the applied body forces. Findings – While Newtonian fluids always convect whenever the Darcy-Rayleigh number is nonzero, Bingham fluids are found to convect only when the Darcy-Rayleigh number exceeds a value which is linearly dependent on both the Rees-Bingham number and the overall perimeter of the rectangular cavity. Stagnation is always found in the centre of the cavity and in regions close to the four corners. Care must be taken over the selection of the regularization constant. Research limitations/implications – The Darcy-Rayleigh number is restricted to values which are at or below 200. Originality/value – This is the first investigation of the effect of yield stress on nonlinear convection in porous media.

Multigrid Numerical Solutions of Non-Isothermal Laminar Recirculating Flows

1999 ◽  
Author(s):  
Marcelo J. S. de Lemos ◽  
Maximilian S. Mesquita
Keyword(s):  
Numerical Solutions ◽  
Rectangular Domain ◽  
Optimal Number ◽  
Computational Effort ◽  
Temperature Fields ◽  
Two Dimensional ◽  
Discretization Scheme ◽  
Multigrid Algorithm ◽  
Finite Volume Discretization ◽  
Velocity And Temperature Fields

Abstract The present work investigates the efficiency of the multigrid numerical method applied to solve two-dimensional laminar velocity and temperature fields inside a rectangular domain. Numerical analysis is based on the finite volume discretization scheme applied to structured orthogonal regular meshes. Performance of the correction storage (CS) multigrid algorithm is compared for different inlet Reynolds number (Rein) and number of grids. Up to four grids were used for both V- and W-cycles. Simultaneous and uncoupled temperature-velocity solution schemes were also applied. Advantages in using more than one grid is discussed. Results further indicate an increase in the computational effort for higher Rein and an optimal number of relaxation sweeps for both V- and W-cycles.

Implementation of a Semi-Implicit Pressure-Based Multigrid Fluid Flow Algorithm on a Graphics Processing Unit

2009 ◽  
Author(s):  
Aaron F. Shinn ◽  
S. P. Vanka
Keyword(s):  
Stokes Equations ◽  
Graphics Processing Unit ◽  
Navier Stokes ◽  
Processing Unit ◽  
Navier Stokes Equations ◽  
Driven Cavity ◽  
Multigrid Algorithm ◽  
Computational Speed ◽  
Speed Up ◽  
Graphics Processing

A semi-implicit pressure based multigrid algorithm for solving the incompressible Navier-Stokes equations was implemented on a Graphics Processing Unit (GPU) using CUDA (Compute Unified Device Architecture). The multigrid method employed was the Full Approximation Scheme (FAS), which is used for solving nonlinear equations. This algorithm is applied to the 2D driven cavity problem and compared to the CPU version of the code (written in Fortran) to assess computational speed-up.

scholarly journals Solutions of reaction-diffusion equations using similarity reduction and HSSOR iteration

2019 ◽  
Vol 16 (3) ◽  
pp. 1430
Author(s):  
Nur Afza Mat Ali ◽  
Rostang Rahman ◽  
Jumat Sulaiman ◽  
Khadizah Ghazali
Keyword(s):  
Differential Equation ◽  
Linear System ◽  
Iterative Method ◽  
Iterative Methods ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Sparse Linear System ◽  
Successive Over Relaxation ◽  
Number Of Iterations

<p>Similarity method is used in finding the solutions of partial differential equation (PDE) in reduction to the corresponding ordinary differential equation (ODE) which are not easily integrable in terms of elementary or tabulated functions. Then, the Half-Sweep Successive Over-Relaxation (HSSOR) iterative method is applied in solving the sparse linear system which is generated from the discretization process of the corresponding second order ODEs with Dirichlet boundary conditions. Basically, this ODEs has been constructed from one-dimensional reaction-diffusion equations by using wave variable transformation. Having a large-scale and sparse linear system, we conduct the performances analysis of three iterative methods such as Full-sweep Gauss-Seidel (FSGS), Full-sweep Successive Over-Relaxation (FSSOR) and HSSOR iterative methods to examine the effectiveness of their computational cost. Therefore, four examples of these problems were tested to observe the performance of the proposed iterative methods.  Throughout implementation of numerical experiments, three parameters have been considered which are number of iterations, execution time and maximum absolute error. According to the numerical results, the HSSOR method is the most efficient iterative method in solving the proposed problem with the least number of iterations and execution time followed by FSSOR and FSGS iterative methods.</p>

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