geometric multigrid method for steady buoyancy convection

Contemporary Engineering Sciences, Vol. 9, 2016, no. 2, 47 - 70

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ces.2016.511306

Geometric Multigrid Method for Steady

Buoyancy Convection in Vertical Cylinders

Fedir Pletnyov and Ayodeji A. Jeje

Department of Chemical and Petroleum Engineering, Schulich School of

Engineering, University of Calgary, Calgary, Alberta, T2N 1N4 Canada

Copyright © 2015 Fedir Pletnyov and Ayodeji A. Jeje. This article is distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any

medium, provided the original work is properly cited.

Abstract

An implementation of the Geometric Multi-Grid (GMG) method, with full

approximation scheme/storage (FAS) algorithm, in a numerical study of steady

Buoyancy-driven convection with axis-symmetric flows in vertical cylinders is

presented. The particular system examined is a cylinder with an aspect ratio a

(radius divided by height) of 4. The fluid is heated from below and cooled from

above, and the circular wall of the vessel is insulated. The Rayleigh (Ra) and fluid

Prandtl (Pr) numbers are respectively 56.4 10 and 7. A non-linear system of

equations was formulated in stream function-vorticity-temperature variables and

discretized using a monotonic conservative finite difference scheme of second order

accuracy. The steady state condition was solved for purposes of comparing two

numerical methods: the GMG FAS and the Gauss- Seidel method with

lexicographic ordering (GS-LEX). The GMG FAS method has significantly higher

efficiency in CPU performance compared to the pure GS-LEX method for fine grids

only if the tolerance value for stopping iteration process is chosen not too small. A

procedure for the selection of adjustment parameters for GMG FAS algorithm is

proposed and tested for different grid sizes. Details regarding convergence criteria

are addressed.

Keywords: multigrid method, natural convection, optimization parameters,

convergence criteria, vertical cylinder

Nomenclature

aspect ratio, thermal diffusivity, k/ρCp a /R H

48 Fedir Pletnyov and Ayodeji A. Jeje

acceleration of gravity relative accuracy

Grashof number dependent variable in the

general convection-

diffusion equation ( or )

mesh step in radial direction surface of the wall

mesh step in axial direction part of the boundary

( ) where heat enters

the fluid

height of cylinder part of the boundary

( ) where heat exits

the fluid

radial component of convection-

diffusion operator dimensionless temperature

axial component of convection-

diffusion operator

vorticity

rN number of radial mesh points stream function

zN number of axial mesh points

Nusselt number Subscripts

heat input from the bottom wall cold

heat output through the top wall hot

Prandtl number (= ) mesh point in direction

radial coordinate j mesh point in direction

radius of cylinder iteration number

Rayleigh number (=

)

radial component of vector

Rayleigh number by Leong [7]

(= )

axial component of vector

temperature vorticity transport equation

ΔT temperature difference across bottom

and top surfaces of domain energy transport equation

g

Gr

rh

zh 0Nu

H 0Nu

rL

zL

Nu

Nu c

Nu h

Pr / i r

r z

R n

Ra

4 / , PrT

g R Ra GrH

r

LeongRa3 /g TH

z

T

Geometric multigrid method for steady Buoyancy convection 49

k thermal conductivity of fluid ν kinematic viscosity, μ/ρ

Cp heat capacity of fluid μ viscosity of fluid

dimensionless radial component of

velocity

ρ density of fluid

dimensionless axial component of

velocity

Superscripts

axial coordinate + , - any

variable

coefficient of volumetric expansion _

1. Introduction

Natural convection of fluids within confined spaces routinely occurs in

many engineering systems such as storage tanks for fuels, water and industrial

chemicals. Vertical cylinders are most commonly used for the storage, as reactors

for chemical transformations, and as vessels for precipitation and mixing processes.

Descriptions of thermal convection inside vessels of different geometries and a

comprehensive review of numerical methods for modelling the circulation has been

presented by Lappa [6]. Implementation of the numerical schemes is of current

interest and steady state solutions for the circulation are important for exploring

issues of convergence, stability, uniqueness and accuracy. Steady state solutions are

also the starting point for linear stability analysis. The choice of a schemes for

simulating steady Buoyancy-driven convection is important and the criteria are

primarily the efficiency of the computer processing unit (CPU) and the accuracy of

the solution. Kuzmin [5] has recently undertaken a comprehensive review of

various numerical methods for the equations of transport phenomena. The relative

efficiency and accuracy of this numerical methods are often in conflict, a trade-off

is required. Leong [7] solved the transport equations for steady Rayleigh-Bernard

convection in cylinders by the central difference scheme in three dimensions (3D).

In his approach, unsteady equations for vorticity- vector potential-temperature were

solved until steady state was attained. Neither computational efficiency nor the

effect of variations in grid sizes were explored. The alternating-direction implicit

(ADI) scheme was used along with Fast Fourier Transform (FFT). Preliminary

calculations indicate that methods involving transitioning from unsteady to steady

states are inefficient. The ADI+FFT method with transitioning required much more

CPU time compared to the non-transitioning Gauss-Seidel (GS) method. Liang et

al [8] had earlier successfully applied the GS method for natural convection in cylindrical geometries. The method is efficient for coarse grids. It is slow to converge

r

z

z 1( )

2x x x x

1( )

2x x x


for fine grid meshes and may yield inaccurate results if the convergence factor is

relatively large. An alternate method is the multigrid (MG) approach (Trottenberg

et al. [13], Wesseling [17], Wesseling and Oosterlee [18]) that yields accurate

solutions but it may be slow when small residuals determine convergence. More

commonly applied for fluid dynamic problems is the geometric rather than the

algebraic multigrid method that Muratova and Andreeva [9] have examined.

Competing numerical methods include the Jacobian-free Newton-Krylov scheme

that Wang et al. [15, 16] used with primitive variables to examine linear stability of

natural convection for axially and laterally heated cylinders.

. The geometric multigrid method (GMG) is applied for this study and the

motivation is to optimize the techniques for the non-linear equations that describe

Buoyancy-driven motion in cylindrical coordinates, with the convergence criteria

based on residuals rather than on relative errors for consecutive iterations. Results

are compared with those from using the GS algorithm. The dependent variables are

vorticity, stream function and temperature, and the equations are discretized with a

monotonic conservative finite difference scheme of second order.

2. The multigrid method

Numerical methods involve transforming differential equations, valid for

continuous domains, into algebraic equations for discrete points or elements of

specified networks within a space bounded by surfaces. Algebraic equations

obtained by a finite difference scheme, for example, may be solved by the Jacobi,

Gauss-Seidel or other iterative algorithms. When problems involve spatially

complex regions and a large number of algebraic equations are to be solved,

convergence may be slow or not readily achievable. Typically, solution by

algorithms such as the Gauss-Seidel solver involve considering errors in the

dependent variables and in the residuals for the calculations. In sequential iterations,

errors and residuals associated with large eigenvalues or high frequency are reduced

quickly but the errors at lower frequency slowly or hardly decrease, thus

engendering long computational times.

The multigrid method accelerates the rates of convergence of solutions by

sequentially manipulating the residuals from iterations in two or more grid meshes

of different resolutions. In the simplest application, a differential equation is

discretized for two networks, one fine and the other coarse, for the same domain.

The solution is started from one of the networks, say the one with the fine grid.

After a few iterations, the residual of the equation is projected onto the coarse grid

network with a restriction operator. The residual equation is solved on the coarse

grid for the correction term to the value of the variable in every single point of the

domain. This correction term is interpolated into the finer grid with the prolongation

operator, and is added to the solution earlier obtained for the fine mesh (V-cycle).

The coarse grid is effectively a temporary, computational adjunct [10]. Low

frequency components of errors are quickly removed. Post-smoothing iterations are

usually performed for each of the calculation cycles. The process can be applied on

a recursive basis for coarser grids to accelerate convergence.


The algorithm can be made more efficient by discretizing the governing

equations for the problem on as coarse a mesh as possible. The solution obtained is

used as the initial guess for the next finer mesh in a series, with values for the

unknown intermediate variables interpolated from results for the coarse mesh. The

governing equations are again re-discretized for the new mesh. This is the scheme

for the full multigrid algorithm (FMG) for linear problems and the full

approximation scheme/storage (FAS) for non-linear problems [10]. The approach

used for this study is the geometric multigrid (GMG) that has been well described

and contrasted with the Algebraic Multigrid (AMG) method by Trottenberg and

Oosterlee [13], Chang et al. [3], Shakira [11], St�̈�ben [12] and others.

The framework of the algorithms used are “Mglin” and “Mgfas” routines

[10] for the full multigrid method. They are also known respectively as the nested

iteration method and the full approximation scheme/storage method (FAS). Both

FMG and FAS algorithms were extended in this study from single elliptical partial

differential equations (PDE) to a group of three equations that describe convective

currents in a confined space. Of the three equations, two are non-linear, the vorticity

and energy transport equations, and the third is a linear elliptic equation for stream

function from the definition of vorticity. For smoothing, the Gauss-Seidel iteration

method with a lexicographic ordering of the grid points (GS-LEX) was applied. The

Gauss-Seidel method with red-black ordering was not found to be suitable for

second order boundary conditions, where values for points near boundaries are

estimated before derivatives are calculated. The efficiency of the multigrid method

depends significantly on how its parameters are adjusted - 1) the number of

GS_LEX relaxation iterations (sweeps) ( 1 ) before coarse-grid correction is

computed (pre-smoothing); 2) the number of GS_LEX relaxation iterations

(sweeps) ( 2 ) after coarse-grid correction is computed (post-smoothing; and 3) the

number of V-cycles ( ncycle ) that is used in each grid level. Preliminary estimates

indicated that the FMG algorithm, for any selection of parameters (𝜈1, 𝜈2, ncycle),

does not improve performance time compared to the pure GS-LEX method. A high

number of pre- and post-smoothing iterations are required for convergence of the

non-linear system that the FMG algorithm is inefficient. For the same system, the

GMG-FAS algorithm has lower CPU time than the GS-LEX method when an

optimal set of parameters (𝜈1, 𝜈2, ncycle) is used.

The procedure for calculating the multigrid parameters are demonstrated for

the specific case of thermal convection inside a vertical cylinder with an aspect

ratio a, ratio of container radius to its height, of 4. The fluid is heated from below,

cooled from above and the circular vertical wall is insulated. Results for the axis-

symmetric, 2-dimensional case are similar to those reported by Leong [7] for his

problem analyzed in 3-dimensions. Conditions common for both studies are

aspect ratio a = 4 and 56.4 10Ra . A Rayleigh number of 2,500, as defined by

Leong, has been multiplied by 4a to be consistent with the definition in this study.

The fluid Prandtl number (Pr) is chosen as 7. Conclusions arrived at from the

specific case have been extended to the general case for cylinders with arbitrary

aspect ratios elsewhere.


3. The governing equations

Equations for axisymmetric heat and momentum transfer, by natural

convection in a fluid-filled vertical container that is heated from below and cooled

from above, are written in cylindrical coordinates, in dimensionless and

conservative form as 2

1

0L u fx

, where is the dependent variable, x1

and x2 are spatial coordinates, L1 and L2 are known convection-diffusion

differential operators and is a source function. The Boussinesq approximation

is applied to the equations in primitive variables (Turner [14]), and the equations

transformed into the stream function-vorticity-temperature equations:

1 10, (1)

1 10, (2)

Pr Pr

1 1. (3)

r z

r z

r rr r Gr

r r r z r z r

r rr r z z

r r r z r z

The variables , are the radial and axial components of the velocity vector,

defined as:

, where is the stream function, and

with being the vorticity.

The following dimensional quantities were used as scaling factors:

the internal radius of cylinder, , for length; for velocity, where is the

kinematic viscosity of the fluid and dimensionless temperature , where

subscripts h and c refer to conditions at the lower (hot) and upper (cold) surfaces

of the pool respectively. The product of the dimensionless quantities in the

equations, Grashof number (Gr) and Prandtl number (Pr) is Rayleigh number (Ra).

The spatial convection-diffusion differential operators in equations (1) and

(2) are generalized as

, (4)

u

f

r z

1 1,r z

r z r r

r z

z r

R / R

c

h c

T T

T T

1, ,L a b f r z


For the vorticity transport equation (1), , 1, 1,r a b , f Grr

;

and, for the energy equation (2), , Pr, ,a r b , 0f .

4. The boundary conditions

The boundary conditions are specified, following Berkovskii and Nogotov [1],

as:

a) the stream function :

on the bounding solid surfaces

along the axis of the cylinder where, because of symmetry, both

and . The last term is equivalent to

. (5)

b) vorticity :

along the axis, and (6)

is defined by steam function values at internal points of the domain.

The condition for vorticity, , at the bounding surface of the domain can be

defined by a second order approximation using values of stream function and

vorticity in a single node nearest to the wall. This approach was first derived by

Woods [19] for rectangular coordinate system. The Woods’s formula should be

corrected for the cylindrical coordinate system in terms of a uniform mesh, which

is superimposed on the solution domain consisting of r zN N discrete points in

and directions respectively, as:

0, 0

00, 1, 1, 0 02

1, 2, 2,2

,0 ,1 ,12

, 1 , 2 , 22

0, 0,

3, 0,

2

3 11 , 1,

2

3 1, 0,

2

3 1. .

2

r r r

z z z

j

j j j r

r

N k N k N k r

r

i i i

i z

i N i N i N

i z

at r r

rr h at r r

h

h at rh

at zrh

at z Hrh

, (7)

where 0,..., 1zj N , 0,..., 1ri N ; rh , zh and rN , zN are the mesh sizes and

the number of the mesh points in r and z directions respectively.

0

00

r

00r r

0

0z

rr

0

10

rr r r

00

r

r

z


c) temperature :

1 0,

0 1,

0 0 1.

at z

at z

at r and rr

(8)

The initial guesses for the variables are:

0ij ;

0ij ; for temperature linear temperature profile between bottom and top

of the cylinder has been used:

1 / ( 1) ;ij zj N , /i j , where / refers to

the internal mesh points of the domain. Values were also chosen for the Grashof

number, Gr (= / PrRa ), in Eq. (1) and the aspect ratio a defined the domain

5. The numerical scheme

In designing finite difference schemes that satisfy the maximum principle

for any mesh size , first order derivatives are represented using the

asymmetrical difference expressions. The schemes take into account the sign of the

coefficients preceding these derivatives. The term is approximated with

backwards difference formula if the coefficient is positive, and a forward

difference formula if the coefficient is negative, as suggested by Courant,

Isaacson and Rees [4]. This is the upwind scheme.

The finite difference operators, approximating convection-diffusion

operators (eq. 4) in monotonic conservative form with second order approximation

, are shown below:

1, j , j

1, j

2

1, j

2

, j 1, j

1, j

2

1, j

2

1

12

1

12

i i

ri

rrr

i

i i

irr

ri

L bhh

h b

bhh

h b

(9)

r zh and h

i

i

fb

x

ib

ib

2( )O h max( , )r zh h h


, j 1 , j

1, j

2

1, j

2

, j , j 1

1, j

2

1, j

2

1

12

1

12

i i

zi

zzz

i

i i

izz

zi

L bhh

h b

bhh

h b

(10)

where is coefficient in equation (4).

1 , j 1 , j 1 1, j 1 1, j 1, j

12

2

1 1, j 1, j 1, j 1 1, j 1, j

2

1,

4

1,

4

, 1,..., , 1,..., .2

r i i i ii

zi

z i i i ii

r i

z z

z r z

h r

h r

i N j N

6. Estimation of Nusselt number

Nusselt number (Nu) characterizes the intensity of heat transfer between a

fluid and a bounding surface. Berkovskii and Polevikov [2] defined it as:

, (11)

where θ is dimensionless temperature and variable is the dimensionless normal

to the surface of the wall at .

Heat enters the fluid through a segment of the boundary at ( ) and exits

through a different part of the boundary at ( ). Here describes the

distribution of the dimensionless heat transfer across the entire bounding surface.

The net heat transfer from the wall to the fluid, and from the fluid to the wall

are evaluated from the integrals:

,

2

b bb

b

Nun

n

0Nu 0Nu Nu


,Nu Nud Nu Nud

For the steady rate of energy transfer through a control volume, | |Nu Nu .

Since parts of the boundary and may be unknown in advance, it is

convenient, according to Berkovskii and Polevikov [2], to estimate

from the formulas:

1,

2

1.

2

Nu Nu Nu d

Nu Nu Nu d

(12)

7. Criteria for convergence

a) Relative errors for the dependent variables, from consecutive iterations, are

specified as:

max( , , ) , (13)

where ( ) ( ) 1

, j , j , j , j

, j

, j

1max ; ,

; , , ;

s s

i i i is

s

is

i r zN N

r zN N is the total number of the points in grid, n is the iteration number; is

tolerance for relative error, 3 810 10 .

a) Maximum value of the residual (defect) in the equations (1)-(3) from all

points of the domain.

( )

max, , j 1max , , ,id d

. (14)

Here 1 is tolerance for maximum value of the defect 3

1 0.1 10 .

b) Average value of the residual in the equations (1)-(3) from all points of the

domain.

c)

, j

, 2

, j

; , , ;i

aver

i r z

dd

N N

(15)

,Nu Nu


Calculations show that values for maximum and average residuals for

vorticity transport equation (1) are several orders greater than for residuals in

equations (2) and (3). Thus, it is more important to keep track of the largest values

of the residuals max,d , ,averd in vorticity transport equation (1) for criteria (14) and

(15).

d) Convergence and balance of the heat transfer (Nusselt number) from the

bottom to the top of the cylinder

1

3

1

n n

n

Nu Nu

Nu

. (16)

8. Results and discussion

Results of steady convective patterns from the numerical simulation are

presented in the following with a review of the criteria for convergence and

selection of grid sizes.

8.1. Comparison of GS and GMG methods

The accuracy of the results from solving equations by iterative method such

as the Gauss-Seidel’s is tightly connected to the stopping criterion for the iteration

process. The convergence criteria for this study are equations (13) – (16). The

solution to the discretized form of the PDEs is correct if the values for a dependent

variable at a fixed point within the domain are the same as the mesh size tends to

zero ( 0h ). The relative error criterion, equation (13), with the GS method often

leads to solutions that are not the same for different mesh sizes when the tolerance

is fixed.

The number of steady ring rolls when a relative error of 310 is specified is

shown in the contour map of streamlines in Figure 1 for different mesh sizes (56.4 10Ra ; a= 4). Only the grids 31x17 and 65x65 exhibited three rolls. If the

relative error is reduced to 4 510 10 , all the networks with finer mesh sizes

show three rolls only, thus indicating that this is the correct results. Figure 1 also

includes contours of the residuals for vorticity. In Table 1, the highest values for

each of the residuals d , d and d for equations (1) – (3) are presented for

different grid sizes. These high residual values are at the boundary between adjacent

rings


rotating in opposite directions. Corresponding heat transfer rates entering at

the bottom and exiting at the top of the cylinder are also given in the Table.

Maximum residuals for all the three equations increased with refinement of the

mesh network. An important observation is that the tolerance that produces correct

solutions is not known a priori.

Table 1. Effect of mesh sizes on maximum values for residuals d , d and d ,

and the ultimate number of rings with the relative error 310 , Ra=

( =2500), Pr=7, a=4.

№ Mesh

Size max

d

max

d

max

d

| | Remark

1 31x17 0.21 1.0 1355 29.7 29.8 3 rolls

2 65x65 2.4 12.4 18700 30.8 30.7 3 rolls

3 129x129 15.8 44.4 76860 33.2 33.1 5 rolls

4 169x169 47.0 135.0 106500 32.9 34.4 5 rolls

Application of the relative error criterion (Eq. 13) requires additional

analysis and it is not reliable. Values for the unknowns can be close in successive

iterations and convergence is indicated but the solution may be incorrect. It is

essential to continue to decrease the tolerance 3 4 510 ,10 ,10 ,... for a given

mesh size and only to stop calculation when solutions cease changing.

The criterion for which maximum value of residuals is prescribed (Eq. 14)

is more reliable but it requires a large number of iterations. In the calculations,

stream function contours and temperature isotherms already may be established and

nearly invariant for consecutive iterations, and the heat transfer rates as estimated

with the Nusselt number (Nu) constant, but the iteration continues until the residuals

are reduced below the imposed limit. Further iterations most significantly reduce

residuals at the boundary between rolls than elsewhere.

Instead of requiring that residuals at all points of a grid fall below a limiting

value as the condition for convergence, a criterion that the average residual for all

Nu

Nu

56.4 10

LeongRa

Nu Nu


the grid points of the domain is prescribed (Eq. 15) appears to be more efficient for

stopping iteration. This is less stiff than the criterion that each point must have a

residual less than a value. To save time and effort, even with using an average value

for the residuals in a domain, care is required in selecting the limit. For example,

the same vorticity and temperature contours, as shown in Figure 2, are obtained for

both values of tolerance 2 1 and 2 0.01 , yet setting 2 0.01 consumes much

more CPU time without improving accuracy. The results in Table 2 are for a fine

mesh with 𝜀2 = 0.01. Corresponding values for criteria (13 to 16) at termination are

shown. Specification of average residuals appear to be most efficient as will be

further discussed later.

Table 2. Values of relative error, maximum and average residuals when

calculation was stopped. The inlet and outlet Nusselt numbers are also shown for

a grid mesh 513x129.

How fast convergence to a solution is achieved is compared for the three criteria,

using the GS-LEX method for a 275 x 65 mesh, in Figure 3. The spikes in the curves

corresponds to when flow patterns appear to transition between forms. Changes in

flow structures are reflected in Figure 4 at 1000, 2000, 7000, 15000 and 27000

iterations. The patterns (in section) evolved from stationary fluid to four rolls, five

rolls, four rolls, and finally, the stable three rolls. If iteration is terminated before

the last spike in Figure 3, e.g. at iteration number ~ 24000, the flow pattern obtained

is not the ultimate even though the Nusselt number has attained a steady values.

For this case, the three criteria appear to have worked equally well.

Relative

error Max d Aver d Nu

Nu

71.12 10 25.7 0.00998 30.10 30.10


Figure 1. Contour plots of stream function and residuals of vorticity transport

equation (1) at different mesh sizes. Relative error tolerance 310 (criterion

(13); Ra= , Pr=7, a=4.

56.4 10


Figure 2. Contour plots of the stream function, vorticity, residual for vorticity and

isotherms. The grid is 513x129 and the average residual vorticity2 0.01 .

The effect of grid sizes and the shape of the cells were also examined. In the

classical multigrid approach, with finite difference discretization of two-

dimensional problems, the cell shape is square. This shape allows implementation

of standard restriction and interpolation operators [10, 13]. Since a radial section

through a cylinder with a radius-to-height ratio a equal 4, as considered in this study,

is not square, the grid has more subdivisions in the radial than in the axial directions

to obtain square cells. The coarsest mesh has 17 x 5 points or 16 x 4 cells. Other

grid sizes applied are 33x9, 65x17, 129x33, 257x65 and 513x129.

The results shown in Figure 2 are the same for the GS-LEX and FAS multigrid

methods for a 513x129 mesh size and for the average residual convergence criterion

0.01 . Effects of variations in grid size, number of iterations and the

convergence criteria are shown in Table 3 using pure GS-LEX smoother. For each

mesh, the average residuals 𝜀2 are maintained at 0.01 and 1, and the number of

iterations and convergence criteria ε and 𝜀1 determined. In these cases, 𝜀1 and 𝜀2

are residuals for vorticity.

The number of iterations increased, the relative errors required for convergence

decreased and the tolerance for local maximum residual increased as the mesh

became finer.


Figure 3. The average and maximum residuals for vorticity, relative errors and the

heat transfer rates in relation to the number of iterations using the GS-LEX

method. Grid size is 257x65.

Values of the rates of heat transfer flows into the domain at the bottom ( Nu ) and

out at the top | |Nu ) are presented in Table 4 with respect to grid sizes. Two

considerations are important – closure of the heat balance at steady state and

convergence of the solution to a finite value. Closure is more readily satisfied (as

Nu is always approximately equal | |Nu) than convergence is achieved

especially for coarse grids.


Figure 4. Contour plot of stream function and temperature isotherms at iteration

numbers (1000, 2000, 7000, 15000 and 27000) between spikes of error or residual

(max and average) values in Figure 3. Grid size is 257x65.

Table 3. Values of the relative error, maximum residual for vorticity at two values

of average residual tolerance 2 1 and 0.1.


Table 4. Heat transfer rates at the bottom (Nu+) and to the top (|Nu-|) of the

cylinder with respect to grid sizes.

8.2. Multigrid-FAS algorithm optimization

For the Multigrid method to be preferred over the GS method, optimal

adjustment parameters are to be found (to achieve minimum CPU time). These are

the number of V-cycles and number of pre- and post-smoothing iterations. The

procedure for selection is demonstrated for a grid with 65x17 points. The tolerance

for average residual of vorticity was specified as 2 0.01 . The steps are:

1. Select a small number for pre/post smoothing iterations: iter PRE/POST = 2.

2. Determine the number of V-cycles and the CPU time that give average residual

values nearest 2 0.01 .

3. Increase number of pre/post smoothing iterations and keep track for the required

CPU time. The number of V-cycles and CPU time would decrease.

4. Terminate the iteration when number of V-cycles ncycle = 2.

Results of calculations for the foregoing procedure are presented in Table 5

and Figure 5 (a, b). The GMG FAS algorithm parameters are V-cycles ncycle

equals 2 and pre/post smoothing iterations equal 1350. The GS-LEX method

required 7802 iterations.

N Grid size Nu | |Nu

1 17x5 23.9 23.6

2 33x9 29.7 29.4

3 65x17 30.4 30.0

4 129x33 30.3 30.0

5 257x65 30.2 30.0

6 513x129 30.1 30.1


For finer grids (129x33, 257x65 and 513x129), the best number of V-

cycles ncycle is 2 and the number of PRE/POST smoothing iterations increases

with grid size as shown in Table 6 for 2 1 and 2 0.01 . Fewer PRE/POST

smoothing iterations are required for 2 1 compared to when 2 0.01 . The CPU

times and the ratios of CPU times for the GMG FAS and GS-LEX algorithms in

Table 7 illustrate the higher efficiency of the GMG-FAS method. The application

of the larger average residual 𝜀2 = 1, also yielded faster comparative times than for

the residual at 0.01, especially for the finer meshes. The specification of low

average residuals leads to sharp increases in PRE/POST smoothing iterations and

to decreased efficiency for the GMG FAS algorithm.

Table 5. Finding the optimal number of the iterations for PRE/POST smoothing of

the GMG FAS algorithm - grid size 65x17, 2 0.01 .

N Iter PRE/POST

ncycle Resid, max Resid, aver

CPU, sec

1 2 1400 1.13 0.0076 76.134

2 10 280 1.13 0.0076 23.533

3 20 140 1.13 0.0076 16.692

4 30 95 0.96 0.0065 14.857

5 40 70 1.13 0.0076 13.564

6 50 55 1.31 0.0081 12.73

7 60 46 1.28 0.0086 12.322

8 65 42 1.4 0.0094 12.058

9 70 39 1.4 0.0094 11.9

10 75 37 1.22 0.0082 12.034

11 80 34 1.44 0.0097 11.673

12 85 32 1.44 0.0097 11.615

13 90 31 1.16 0.0077 11.779

14 95 29 1.3 0.0087 11.601

15 100 28 1.13 0.0076 11.697

16 105 26 1.4 0.0094 11.344

17 110 25 1.32 0.0088 11.409

18 115 24 1.27 0.0086 11.38

19 120 23 1.28 0.0086 11.299

20 125 22 1.32 0.0088 11.211

21 130 21 1.4 0.00949 11.129

22 140 20 1.13 0.0076 11.308


Table 5. (Continued): Finding the optimal number of the iterations for PRE/POST

smoothing of the GMG FAS algorithm - grid size 65x17, 2 0.01 .

N Iter PRE/POST ncycle Resid, max Resid, aver

CPU, sec

23 145 19 1.29 0.0087 11.105

24 155 18 1.16 0.0078 11.222

25 160 17 1.45 0.0097 10.878

26 170 16 1.45 0.0097 10.85

27 195 15 1.4 0.0094 10.782

28 210 13 1.4 0.0094 10.72

29 215 13 1.14 0.0077 10.941

30 230 12 1.27 0.0086 10.805

31 250 11 1.32 0.0088 10.719

32 275 10 1.32 0.0088 10.649

33 305 9 1.34 0.00897 10.618

34 340 8 1.45 0.0097 10.494

35 355 8 0.995 0.0067 10.892

36 390 7 1.4 0.0094 10.548

37 550 5 1.32 0.0088 10.477

38 700 4 1.13 0.0076 10.627

39 1350 2 1.48 0.0099 10.176

Table 6. Optimal number of PRE/POST smoothing iterations for lowest GMG

CPU for different grid sizes and average residual tolerances 2 1 and 0.01.

Mesh Size

Optimal number of iter PRE/POST

Tolerance 2 for average residual

1.0 0.01

65x17 615 1350

129x33 680 3110

257x65 1950 12000

513x129 4500 42000


Figure 5. GMG CPU time (a) and number of PRE/POST smoothing iterations (b)

with respect to number of V-cycles. Grid size 65x17, tolerance 2 0.01 for

vorticity.


Table 7. Computer performance time (CPU, min) at different grid sizes for the

GS-LEX and GMG-FAS methods.

Mesh

Tolerance 2 for average residual

1.0 0.01

CPU GMG, min

CPU GS, min

Ratio CPU GS:GMG

CPU GMG, min

CPU GS, min

Ratio CPU GS:GMG

65x17 0.13 0.12 0.92 0.17 0.18 1.05

129x33 0.46 1.43 3.13 1.83 2.3 1.26

257x65 4.85 19.39 4.00 28.26 34.65 1.23

513x129 56 261 4.66 980.50 1066.00 1.09

9. Conclusion

The GMG-FAS algorithm has been used to obtain the solution for steady

axisymmetric natural convection inside a vertical cylinder heated from below,

cooled from above and insulated on the side. Three coupled equations for vorticity,

stream function and temperature were involved. The equations were discretized by

the monotonic conservative finite difference scheme of the second order accuracy.

The method has been shown to be more efficient in CPU performance time

compared to pure GS-LEX method for grid sizes 65x17 and finer only if not too

small tolerance values are selected. A procedure of the selection of the adjustment

parameters for the GMG-FAS algorithm has also been proposed and tested for

different grid sizes. Minimum CPU time is achieved for two V-cycles for all grid

sizes and the number of the PRE- and POST smoothing iterations was equal,

increasing as the grid became finer.

Application of average residual over the domain as the condition for

convergence was more efficient than the use of maximum residual for a point

within the domain. The choice of values for the tolerance for average residuals is

also important. The number does not have to be small for the results to be accurate.

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Received: December 6, 2015; Published: January 15, 2016


http://dx.doi.org/10.1016/s0377-0427%2800%2900517-3

geometric multigrid method for steady buoyancy convection

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