Multigrid methods

S.F. McCormick, Editor, Multigrid Methods , SIAM, Philadelphia, 1987... 

In CED, a number of different iterative solvers for linear algebraic systems have been applied. Two of the most successful and most widely used methods are conjugate gradient and multigrid methods. The basic idea of the conjugate gradient method is to transform the linear equation system Eq. (38) into a minimization problem... 

In contrast to the conjugate gradient method, the multigrid method is rather a general framework for iterative solvers than a specific method. The multigrid method exploits the fact that the iteration error... 

A method for smoothing the residual obtained on the fine grid in order to compute the corresponding residual on the coarse grid. In the terminology of the multigrid method, this step is called restriction. 

In one cycle of the multigrid method, first a few iterations are performed on the fine grid in order to obtain a comparatively smooth iteration error. After that the obtained residual is restricted to the coarse grid, where further iterations are performed in order to damp out the long-wave components of the solution error. Subsequently the coarse-grid solution is interpolated to the fine grid and the solution on the fine grid is updated. 

Multigrid methods have proven to be powerful algorithms for the solution of linear algebraic equations. They are to be considered as a combination of different techniques allowing specific weaknesses of iterative solvers to be overcome. For this reason, most state-of-the-art commercial CFD solvers offer the multigrid capability. 

W. Hackbusch, Multigrid Methods and Applications. Springer-Verlag, Berlin, 1985. 

F. Schmidt, et al.. Adaptive multigrid methods for the vectorial Maxwell eigenvalue... 

To be successful in solving applied and mostly differential problems numerically, we must know how to implement our physico-chemical based differential equations models inside standard numerical ODE solvers. The numerical ODE solvers that we use in this book are integrators that work only for first-order differential equations and first-order systems of differential equations. [Other DE solvers, for which we have no need in this book, are discretization methods, finite element methods, multigrid methods etc.]... 

Wesseling P., An Introduction to Multigrid Methods, Wiley, New York (1992)... 

The computational advantages of such multigrid methods arise from two key factors. First, microscopic simulations are carried out over microscopic length scales instead of the entire domain. For example, if the size of fine grid is 1% of the coarse grid in each dimension, the computational cost of the hybrid scheme is reduced by 10 2rf, compared with a microscopic simulation over the entire domain, where d is the dimensionality of the problem. Second, since relaxation of the microscopic model is very fast, QSS can be applied at the microscopic grid while the entire system evolves over macroscopic time scales. In other words, one needs to perform a microscopic simulation at each macroscopic node for a much shorter time than the macroscopic time increment, as was the case for the onion-type hybrid models as well. 

Liu, Z. and Liu, C. (1994). Fourth order finite difference and multigrid methods for modeling instabilities in flat plate boundary layer- 2D and 3D approaches. Computers and Fluids, 23, 955-982. 

The PB equation may be solved numerically for macromolecules (for reviews, see References 36-38. The finite difference, finite element, and multigrid methods are used most commonly to solve the PB equation. Usually, this technique is performed by mapping the molecules onto a three-dimensional cubic grid. To solve the PB equation, a suitable interior relative dielectric constant and definition of the dielectric boundary should be assigned (39, 40). 

Hutchinson, B.R. and Raithby, G.D. (1986), A multigrid method based on additive correction strategy. Numerical Heat Transfer, 9, 511-537. 

Sathyamurthy, PS. and Patankar, S.V. (1994), Block-correction based multigrid method for fluid-flow problems. Numerical Heat Transfer, 25B, 375-394. 

Yet another approach are multigrid methods [28,29]. These methods are based on solving Poisson s equation numerically on a grid. While periodicity is naturally no problem for these methods, the singularity of the Coulomb potential has to be circumvented, which is done by splitting the equation into... 

G. Sutmann and B. Steffen (2005) A particle-particle particle-multigrid method for long-range interactions in molecular simulations. Comp. Phys. Comm. 169, pp. 343-346... 

Hackbusch W (1985) Multigrid Methods and Applications. Springer, Berlin Haltiner GJ, Williams RT (1980) Numerical Prediction and Dynamic Meteorology, 2nd Edition, Wiley, New York... 

Villadsen J, Michelsen ML (1978) Solution of Differential Equations Models by Polynomial Approximation. Prentice-Hall, Englewood Cliffs Warming RF, Hyett BJ (1974) The modified equation approach to the stability and accuracy analysis of finite-difference methods. J Comp Phys 14 159-179 Waterson NP, Deconinck H (2007) Deign principles for bounded higher-order convection schemes-a unified approach. J Comput Phys 224 182-207 Wesseling P (1992) An Introduction to Multigrid Methods. John Wiley Sons, New York... 

---