Practical Ship Hydrodynamics

Introduction 31
ž LSOR (line successive overrelaxation), ADI (alternating direction implicit)
(line iterative)
ž ILU (incomplete lower upper) decomposition, e.g. SIP (strong implicit
procedure)
ž CG (conjugate gradient) method
The various iterative methods differ in their prerequisite (dominant main diagonal,
symmetry etc.), convergence properties, and numerical effort per iteration.
Strongly implicit schemes such as SIP feature high convergence rates.
The convergence is especially high for multigrid acceleration which today is
almost a standard choice.
1.5.5 Multigrid methods
Multigrid methods use several grids of different grid size covering the same
computational fluid domain. Iterative solvers determine in each iteration (relaxation)
a better approximation to the exact solution. The difference between the
exact solution and the approximation is called residual (error). If the residuals
are plotted versus the line number of the system of equations, a more or less
wavy curve appears for each iterative step. A Fourier analysis of this curve
then yields high-frequency and low-frequency components. High-frequency
components of the residual are quickly reduced in all solvers, but the lowfrequency
components are reduced only slowly. As the frequency is defined
relative to the number of unknowns, respectively the grid fineness, a given
residual function is highly frequent on a coarse grid, and low frequent on a
fine grid. Multigrid methods use this to accelerate overall convergence by the
following general procedure:
1. Iteration (relaxation) of the initial system of equations until the residual is
a smooth function, i.e. only low-frequent components are left.
2. ‘Restriction’: transforming the residuals to a coarser grid (e.g. double the
grid space).
3. Solution of the residual equation on the coarse grid. Since this grid contains
for three-dimensional flow and grid space halving only 1/8 of the unknowns
and the residual is relatively high frequent now, only a fraction of the
computational time is needed because a further iteration on the original
grid would have been necessary for the same accuracy.
4. ‘Prolongation’: interpolation of the residuals from the coarse grid to the
fine grid.
5. Addition of the interpolated residual function to the fine-grid solution.
This procedure describes a simple two-grid method and is recursively repeated
to form a multigrid method. If the multigrid method restricts (stepwise) from
the finest grid to the coarsest grid and afterwards back to the finest grid, a
V-cycle is formed. If the prolongation is only performed to an intermediate
level, again before restriction is used, this forms a W-cycle (Fig. 1.5).
The multigrid method accelerates the overall solutions considerably,
especially for grids with many unknowns. Multigrid algorithms obtain
computational times which are almost proportional to the number of cells,
while single-grid solvers yield computational times proportional approximately
to the square of the number of cells. Multigrid methods are relatively easy
to combine with all major iterative solvers. The considerable speed-up of