New A MULTIGRID ACCELERATED HIGH-ORDER COMPACT … · 2018. 9. 14. · acceleration has high resolving efficiency that drastically reduces computational time and high-order accuracy

IJRRAS 5 (3) ● December 2010 San & Kara ● Multigrid Accelerated High-Order Compact Fractional-Step Method

245

A MULTIGRID ACCELERATED HIGH-ORDER COMPACT

FRACTIONAL-STEP METHOD FOR UNSTEADY INCOMPRESSIBLE

VISCOUS FLOWS

Omer San 1,*

& Kursat Kara2

1Detpartment of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University,

Norris Hall, Room 227-A, Blacksburg, VA 24061, USA. 2Department of Aerospace Engineering, Khalifa University of Science, Technology & Research,

P.O.Box 127788, Abu Dhabi, UAE.

*Email: [email protected]

ABSTRACT

The objective of this study is the development of an efficient high-order compact scheme for unsteady

incompressible viscous flows. The scheme is constructed on a staggered Cartesian grid system in order to avoid

spurious oscillations in the pressure field. Navier-Stokes equations are advanced in time with the second order

Adams-Bashford method without considering the pressure terms in the predictor step, the velocity field is then

corrected such that discrete mass continuity equations satisfied through pressure Poisson equation. Since the

efficiency of the fractional step method depends on the Poisson solver, a V-cycle multigrid acceleration with

compact Mehrstellen discretization based iterative method is implemented in the pressure Poisson equation to

enhance the computational efficiency. The efficiency and accuracy of iterative Poisson solvers (pseudo-time, Jacobi,

Gauss-Seidel) are also tested within the multigrid framework. The method is then validated by the simulations of the

Taylor-Green vortex decaying problem. Results show that the fractional-step compact scheme with multigrid

acceleration has high resolving efficiency that drastically reduces computational time and high-order accuracy

making the method applicable for simulation of incompressible viscous flows.

Keywords: Incompressible flow, High-order methods, Compact schemes, Fractional-step procedure, Staggered

grid, V-cycle multigrid acceleration

1. INTRODUCTION

Computational study of incompressible flow problems in both basic research and engineering applications has been

performed for long times. Incompressible flow is an approximation of flow where flow speed is insignificant

compared to the speed of sound [1]. Since the numerical stability of the density-based numerical algorithms (i.e., full

Navier-Stokes algorithms) depends on the speed of sound, the time step is restricted with this high velocity even if

we have low speed flows (i.e. numerical stability criterion is in the form of au

xt

||

). There are mainly two

different approaches for solving viscous incompressible flow problems. The first is based on still compressible flow

formulations where momentum and continuity equations are coupled through the use of density which is called

"density-based preconditioned" approach. The typical idea is to use the pseudo speed of sound to precondition

system such that time integration is performed in physical time, and pseudo compressibility time is used to

precondition the system and obtain the solution at each time level; see e.g., [2, 3, 4]. Incompressibility is recovered

as a limiting case of this formulation. The other alternative way to model low speed flows by using incompressible

flow assumption by changing the character of governing equations from hyperbolic to parabolic. This is based on

satisfying the incompressibility directly through the pressure as a mapping parameter to satisfy this condition. This

class of methods is generally called as the "pressure-based" methods; see e.g., [5, 6, 7, 8]. For a given velocity field,

the pressure can be calculated without knowing history. The pressure equation is elliptic in character. At each time

step (or iterations for steady flows) a pressure Poisson equation has to be solved to obtain the pressure field. Most of

this type of algorithms are decoupled such that momentum equations are solved separately, and pressure equation is

solved to satisfy the mass continuity equation [1].

Many numerical schemes have been developed for solving viscous incompressible flow [9]. Among them several

fractional-step procedures have been introduced as an efficient way within the pressure-based approach especially

for unsteady flows. This procedure was first introduced by Harlow and Welch [10] which was the first primitive

variable methods on the staggered grid using a derived Poisson equation for pressure. This method is known as

"marker and cell" and can be thought one variation of fractional step methods. Fractional step methods are also


246

sometimes called as predictor-corrector, or time-splitting, or projection methods. The common implementation of

this framework is done in two steps: The first step is to solve for intermediate auxiliary velocity field using the

momentum equation, in which the pressure gradient term can be computed from the pressure in previous step or can

be excluded entirely. In the next step, the pressure is computed, which can map the intermediate velocity on the

divergence-free velocity field by satisfying the discrete mass continuity equations to the current step. Fractional-step

methods can be used together with different space discretizations such as finite difference, finite element, and

spectral methods with various order of space and time accuracy [11].

Computational algorithms developed in the past were mainly designed for solving large scale fluid dynamics

problems with the second-order space accuracy. Recently, people have been paying more attention to numerical

simulation of complex flows with multiscale structures such as turbulent flows. Simulation of turbulent or other

convection dominated unsteady flows using direct numerical simulation or large eddy simulation (LES) requires a

numerical method that properly resolves all the multi-structured flow scales. There are two ways to improve the

resolution of the method. One of them is to refine the grid and the other is to construct a high order accurate scheme.

Since high-order accurate computational methods are both desirable and preferred, compact finite difference

methods feature high-order accuracy with smaller stencils and easier implicit application of boundary conditions,

and have been employed as an alternative to spectral methods in simulation of turbulence with their flexibility [12,

13, 14, 15].

The advantage of the finite difference compact methods is to interpolate the variables without taking integration by

parts to keep order of accuracy. Zhang et al. introduced the point/node based compact finite difference scheme on

staggered grids with the classical fractional-step method [13]. In the point/node-based method, all quantities

appeared in a specific governing equation were defined on the same selected nodes and interpolated by unknowns on

staggered or collocated grids by presenting all relevant interpolations. The system was solved using fully explicit

second-order accurate time advancing scheme, where pressure Poisson equation was solved by a pseudo-time

marching procedure. As with other pressure based methods, the efficiency of the fractional step method depends on

the Poisson solver. A multigrid acceleration, which is physically consistent with the elliptic field is one possible

avenue to enhance the computational efficiency.

The multigrid framework is among the most efficient iterative methods to solve linear systems arising from

discretized elliptic differential equations like the pressure Poisson equation. It solves the error correction sub-

problem on coarser grids and interpolates the error correction solution back to fine grids. An important aspect of

multigrid method is that the coarse grid solution can be approximated by recursively using the multigrid idea. Thus,

the method requires a series of different problems to be solved on a hierarchy of grids with different mesh sizes. One

of the efficient multigrid methods is V-cycle that is the process that goes from finest grid down to the coarsest grid

and back from the coarsest up to the finest. Considerable amount of computational time can be saved by doing major

computational work on the coarser grids. For more details on multigrid, readers are referred to Wesseling [16, 17]

and other recent studies on solving higher order Poisson equations with multigrid methods [18, 19, 20].

In this study, the point/node-based high-order compact fractional-step finite difference method introduced by Zhang

et al. [13] are coupled with the V-cycle multigrid acceleration for pressure Poisson equation for simulation of

unsteady incompressible flow. The efficiency of explicit high-order compact scheme and efficient Mehrstellen-

based V-cycle multigrid acceleration for the Poisson equation are combined together to obtain fast and accurate

numerical method for high resolution desiring flows. The paper is organized as follows: The mathematical

formulation of problem with derivation of the fractional-step procedure and implementation issues combined with

compact interpolations are given in Section 2. Several iterative solution methods for pressure Poisson equations in

the multigrid framework are explained and numerically tested in Section 3. The scheme introduced in this study are

validated in Section 4 by simulating Taylor-Green vortex decaying benchmark problem for unsteady incompressible

Navier-Stokes equations. The efficiency and accuracy of the method is also provided in this section which confirm

their theoretical accuracy properties. Final conclusions and some comments about effectiveness of this scheme are

drawn in section 5.

2. MATHEMATICAL MODEL

The governing equations describing incompressible flow in dimensionless conservative form with index notation are

ijj

i

ij

jii fxx

u

Rex

p

x

uu

t

u

21= (1)

0=j

j

x

u

(2)


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where Re is the Reynolds number, iu and if are the velocity and body force components in the ith

direction,

respectively, and p is the pressure. The staggered grid system is used to eliminate well-known pressure-velocity

coupling issues which is shown in Figure 1.

*

x-mom

y-mom

p

Figure 1. The schematics for two-dimensional staggered grid arrangement.. The x-component of the velocity, y-

component of velocity, and pressure are defined on x-mom points, y-mom points, and p point, respectively. The

values at temporary * points are used when nonlinear terms are computed.

2.1. Fractional-step Method

In advancing the momentum equations, pressure can be neglected during the predictor step. For other variants of

fractional-step procedure, see [6, 7, 8]. Therefore the intermediate velocities can be computed by integrating the

momentum equations as

tHuu in

t

nt

nii

1=~ (3)

where iH is the combination of convection, viscous diffusion and body force terms:

ijj

i

j

ji

i fxx

u

Rex

uuH

21

= (4)

With a second-order accuracy, the time integration can be performed with Adams-Bashforth scheme as

)2

1

2

3

2

1

2

3)(

2

1)(

2

3(= 1

12211

ni

ni

jj

ni

jj

ni

j

nji

j

nji

in

t

nt

ffxx

u

Rexx

u

Rex

uu

x

uuttH (5)

Next, at the end of the time step the velocities should be corrected according to

tx

puu

ii

ni

~=1 (6)

Before correction step, the pressure field can be computed such that discrete mass continuity equation is satisfied.

According to Eq. (2) we need to make sure that mass is conserved at the end of the time step as


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0=1

i

ni

x

u

(7)

By substituting Eq. (6) to discretized continuity Eq. (7), the pressure Poisson equation becomes

i

i

ii x

u

txx

p

~1=

2

(8)

Finally, the fractional step algorithm can be summarized as

1. Compute intermediate velocities

)2

1

2

3

2

1

2

3)(

2

1)(

2

3(=~ 1

1221

ni

ni

jj

ni

jj

ni

j

nji

j

njin

ii ffxx

u

Rexx

u

Rex

uu

x

uutuu (9)

2. Solve pressure Poisson equations in order to correct velocity field such that the continuity equation is

satisfied

i

i

ii x

u

txx

p

~1=

2

(10)

3. Correct the intermediate velocities

tx

puu

ii

ni

~=1 (11)

2.1. Compact Interpolations

The basic idea of compact finite difference is rather simple. The first and second order spatial derivatives of a

variable in the governing equations can be obtained from its value implicitly. The conventional method of deriving

compact difference schemes using a truncated Taylor series and determining the coefficients of the interpolations

based on the desired accuracy is straightforward and given by Lele [12] for uniform grid spacing and Shukla et

al.[21] for nonuniform grid spacing. For any scalar pointwise value of u, the derivatives of u is obtained by solving a

tridiagonal matrix system using the well-known Thomas algorithm. For example, the first derivatives can be given at

internal nodes as follows:

h

uub

h

uuauuu iiii'i

'i

'i

42= 221111

(12)

which gives rise to a -family of tridiagonal schemes with 2)(3

2= a , and 1)(4

3

1= b . Here, 0= leads

to the explicit non-compact fourth-order scheme for the first derivative. A classical compact fourt-order scheme,

which is also known as Pade scheme, is obtained by 1/4= . The truncation error in the Eq. (12) is

(5)41)(35!

4uh . Therefore, a sixth-order tridiagonal scheme is also obtained by choosing 1/3= . The compact

interpolations used in this study for computing the spatial derivatives for prediction and correction steps, Eq. (9) and

Eq. (11), are given in the Appendix for completeness. In order to speed-up the simulation, we are introducing the

compact Mehrstellen-based iterative solution procedure for pressure Poisson equation, Eq. (10), with the fourth-

order accuracy. The procedure is explained in the following section within the multigrid framework.

3. MULTIGRID ACCELERATION FOR POISSON EQUATION

The efficiency of fractional step methods depends on the Poisson solver, as with other pressure-based methods for

the computational simulation of incompressible flows. In the numerical simulation, at each time step the pressure


249

Poisson equation need to be solved accurately by taking quite longer CPU time than the outer time integration

procedure.

3.1. The Fourth-order Compact Discretization Scheme for the Poisson Equation

The Poisson equation given in Equation (10) can be written in the form of Sp =£ , where the £ operator is the

discrete Laplacian ii xx

2=£ , and

i

i

x

u

tS

~1=

. For simplicity, the two-dimensional Poisson equation is

Sy

p

x

p=

2

2

2

2

(13)

Then, the general fourth order compact discretization scheme with nine point stencil can be written as [19]

)(8

2=

)()()(

1,1,1,1,,

2

11,11,11,11,1,1,1,1,,

jijijijiji

jijijijijijijijiji

SSSSSx

ppppdppcppbap

(14)

where the coefficients in Equation (14) are )10(1= 2a , 25= b , 15= 2 c , and )/2(1= 2d . Here

is the mesh aspect ratio yx /= . For 1= the scheme is well known and is sometimes called Mehrstellen. The

sparse linear system formed by Eq. (14) at all interior grid points has nine nonzero diagonals. It is usually

advantageous to solve such sparse linear systems using a point iterative method with multigrid acceleration. For

example, Point-Jacobi(PJ) or Gauss-Seidel (GS) relaxation schemes. The GS relaxation scheme for system described

by Eq. (14) is

)]()()(

)(82

[1

=

11,11,11,11,1,1,1,1,

1,1,1,1,,

2

,

jijijijijijijiji

jijijijijiji

ppppdppcppb

SSSSSx

ap

(15)

where we use new values as soon as we get them. Different from GS, PJ iteration takes all the values from previous

iterations. The behaviour of all these point based iterative method in the multigrid framework will be also tested in

this study. Since Mehrstellen type of discretization has compact form of stencils, Drichlet type of boundary

conditions can be directly introduced. For the Neumann type of boundary conditions again forth order accurate

representation for staggered grid configuration shown in Figure 2 can be computed from interior points according to

)22111225(22993

1= 43210 ppppp (16)

p4

p'= 0

p0

p1 p2 p3

h

Figure 2. Neumann boundary conditions schematics for staggered grid system.

Alternatively, we can use pseudo-time marching iterative relaxation procedure as Zhang et al.[13] implemented in

their computations by using compact interpolation. In there, the pseudo-time marching procedure is given as:

)£(=1 Sppp kkk (17)


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where p£ can be computed from compact interpolations. The main disadvantage of this pseudo-time iterations is

numerical stability restriction due to the changing character of partial differential equations from elliptic to parabolic

to obtain steady state solution. The stability criterion for pseudo-time step is given as

2

11122

yx

(18)

The comparisons of pseudo-time iterative scheme and Mehrstellen based relaxation schemes will be shown later.

3.2. V-cycle Multigrid Method

In order to solve Poisson equation effectively, the V-cycle multigrid procedure is constructed with standard mesh

coarsening strategy (the coarser grid mesh size is double that of the finer grid). This methods is an effective

algorithm to accelerate the convergence for relaxation methods [16, 17]. The method effectively reduces the error of

longer wavelength than the grid interval. In this framework, we need to transfer residual from the fine grid to the

coarse grid. The residual pSr £= for the Mehrstellen type of compact discretization can be computed as [19]

)]()()([6

1

)(812

1=

11,11,11,11,1,1,1,1,,2

1,1,1,1,,,

jijijijijijijijiji

jijijijijiji

ppppdppcppbapx

SSSSSr

(19)

The transfer action is done by restriction operators, denoted by hhI2 , i.e., hhhh rrI

22 = . For two dimensional domain,

the simplest restriction operator is direct injection. The coarse grid points take the fine grid point values directly as

h jihji rr ,22

2, = (20)

Another common restriction operator is defined as half weighting by

)4(8

1= ,221,221,221,221,22

2,

hji

hji

hji

hji

hji

hji rrrrrr (21)

Alternatively, the most accurate restriction operator is the full weighting averaging which is given as

]4)2([16

1= ,221,221,221,221,2211,2211,2211,2211,22

2,

hji

hji

hji

hji

hji

hji

hji

hji

hji

hji rrrrrrrrrr (22)

Among those restriction operators the latter one is more appropriate for high-order simulations [19]. The

transformation from the coarse grid to fine is also given by bilinear prolongation operator, hhI2 , as [17]

)(4

1=

)(2

1=

)(2

1=

=

211,

21,

21,

2,11,22

21,

2,1,22

21,

2,1,22

2,,22

hji

hji

hji

hji

hji

hji

hji

hji

hji

hji

hji

hji

hji

uuuuu

uuu

uuu

uu

(23)

Couple of corrections can be done on coarse level before returning the finer level. We may also start with the

coarsest grid in order to provide a good initial guess for finer grids. Such an algorithm is called the full multigrid


251

methods. The V-Cycle multigrid algorithm does one correction on each level. Since computations on coarse levels

are relatively cheap, such actions may pay for the incurred costs. In the V-Cycle algorithm physical boundary values

are only required for the finest grid, they are also zero on the coarser grids. And the initial guesses are zero for all

coarser grids. The V-cycle multigrid algorithm for (n=3) level grid shown in Figure 3 is summarized as:

1. Relax 1 times on hhh fu =£ with initial guess khu

2. Compute restriction )£(== 222 hhhh

hhh ufIrf

3. Relax 1 times on hhh fu 222 =£ with initial guess zero

4. Compute restriction )£(== 2224244 hhh

hhhh ufIrf

5. Solve exactly on hhh fu 444 =£ with initial guess zero;

6. Correct by prolongation )£(= 4442422 hhh

hhhh ufIuu

7. Relax 2 times on hhh fu 222 =£ with initial guess hu2

8. Correct by prolongation )£(= 2222 hhhhhhh ufIuu

9. Relax 2 times on hkhh fu =£

1 with initial guess hu

h

I4h

2h

4h

2h

I2h

h Ih

2h

I2h

4h

Figure 3: Three level ( 3=n ) V-cycle multigrid hierarchy

The number of presmoothing ( 1 ) and postsmoothing ( 2 ) numbers may depend on considered problem. More

relaxation on each level lead to faster convergence with higher cost. In this study, n level V-cycle algorithm with

two relaxation sweeps on each level is considered ( 1== 21 ). In order to characterize of computational cost we

can define a work unit (WU ) which shows the cost of performing one relaxation sweep on the finest grid.

Therefore, the equivalent total fine grid work units can be computed as:

22

21

1=2

)(=

n

n

i

kWU

(24)

where k is the outer iteration number.

3.3. Numerical Experiments for Poisson Equation

Here numerical experiments are conducted to solve a two-dimensional Poisson Equation (13) on the unit square

domain [0,1][0,1] . The source term S and the boundary conditions are prescribed to satisfy the exact solution

))(2)(2(4

1=),(

2ycosxcosyxp

(25)

This solution satisfies the Neumann boundary conditions for all the sides. The initial guess is the zero vector for all

the experiments. All multigrid computations use V-cycle algorithm given in previous section, the coarsest grid is the

one with the one being the coarsest possible (i.e., n=6 for N=64). One presmoothing and one postsmoothing are

applied at each level. The iterative process stops when the subsequent root mean square values on the finest grid is


252

reduced to the specified tiny number, i.e., 10-8

. The root mean square shows measure of averaged 2L norm of the

difference between numerical and exact solution as:

2,,1=1=

)(1

| |=| | exactjiji

jN

j

iN

iji

ppNN

p (26)

First we compare the efficiency of the iterative methods with and without multigrid acceleration. Table 1 compares

the number of work units (WU ) and the CPU time in seconds for these methods with three different finest grid

resolutions. We see that the relaxation schemes with Mehrstellen type of compact discretization is quite faster the

pseudo-type iteration even if we use maximum allowable . We can also see another disadvantage of pseudo-time

iteration that the multigrid framework does not suitable for this type of iteration. However, when the multigrid

method is implemented with Mehrstellen based relaxation schemes, the Poisson equations can be solved more than

100 times faster. It is apparent that the multigrid method is effective for the improvement of the convergence

efficiency of these schemes(i.e., PJ and GS). Moreover, comparisons between pseudo-time and GS-MG shows that

the acceleration rate in term of CPU time is improved 770 times for 64=N and this speed-up is increasing with

higher grid resolutions.

Table 1. Efficiency data for the high-order pressure Poisson iterative methods

32=N 64=N 128=N Method WU CPU ][s WU CPU ][s WU CPU ][s

Pseudo 10935 9.78 41541 144.27 157636 3204.42

Pseudo-MG 6337 4.59 20986 66.33 72680 1731.24

Jacobi 1952 1.08 9731 20.47 46600 570.87

Jacobi-MG 95 0.05 158 0.36 245 3.76

GS 977 0.66 4867 9.91 23301 217.98

GS-MG 63 0.03 108 0.27 170 2.18

k

||r.

m.s

.||

0 10000 20000 3000010

-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Pseudo

Pseudo - MG

Jacobi

Jacobi - MG

GS

GS - MG

Ni= 64, N

j= 64

Figure 4. Comparison of efficiencies of the high-order Poisson schemes


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The convergence history for 64=N is shown in Figure 4 in order to get more accurate insight of convergence rate.

It is again apparent that MG method is very efficient for relaxation schemes. Pseudo-time iteration hits machine zero

earlier. Someone can think that in the numerical simulation of Navier-Stokes equation, usually previous time step

values for pressure are used as initial guess for current step. However if we look the acceleration rate for the

schemes, in the linear relaxation regime, in order to reduce average norm of | || | p to just one order (i.e., from 410

to 510 ), we need 3822 iteration with pseudo-time marching method, just 5 iterations with GS-MG methods. When

the multigird method with a Mehrstellen based relaxation scheme is used, the convergence rate increases order of

1000 times compared with that of the pseudo-type of iterative method. Another issue is the order of accuracy which

can be tested by the following formula

(1/2)

)| || |/| |(| |= 2,2

ln

pplnn NNNN (27)

Table 2 shows the accuracy data and order of discretization n . Results verify that the fourth order accuracy for the

relaxation schemes. Although, pseudo-schemes has the order of accuracy of almost 6, the results in Table 2 show

also that the solution computed from Mehrstellen based relaxation schemes is much more accurate than that of the

pseudo-time marching algorithm.

Table 2. Accuracy data for the high-order pressure Poisson iterative methods

Method

16=N

| || | p

32=N

| || | p

64=N

| || | p

128=N

| || | p )( 32,16nO )( 64,32nO )( 128,64nO

Pseudo 1.15E-05 1.94E-07 3.53E-09 6.51E-10 5.89 5.78 2.45

Jacobi 6.51E-07 3.89E-08 2.39E-09 1.48E-10 4.07 4.02 4.01

GS 7.11E-07 3.96E-08 2.40E-09 1.48E-10 4.17 4.05 4.02

4. NUMERICAL EXPERIMENTS FOR 2D NAVIER-STOKES EQUATION

In this section, the non-dimensional unsteady incompressible Navier-Stokes equations are solved on a two-

dimensional square domain for the Taylor-Green vortex decaying problem. In fluid dynamics, the Taylor-Green

vortex is a two-dimensional, unsteady flow of a decaying vortex, which has the exact closed form solution of

incompressible Navier-Stokes equations in Cartesian coordinates. The exact solution of this vortex flow in

[0,1][0,1] domain is given by

)()()(=),,( tFycosxsintyxu (28)

)()()(=),,( tFysinxcostyxv (29)

where )/2(=)( 2 RetexptF denotes the strength of the vortex, decreasing in time due to viscous dissipation. The

pressure field p can be obtained by substituting the velocity solutions in the momentum equations and is given by

)](2)(2[4

)(=),,(

2

ycosxcostF

tyxp (30)

This problem can be solved using periodic boundary conditions or using no-slip boundary conditions for velocities

and Neumann boundary conditions for pressure. The Reynolds number is set to 10=Re and the final non-

dimensional time is 1=t . The grid system used in the computation includes 88= ji NN , 1616 , 3232 , and

6464 , where iN , and jN represents the number of cells for x and y directions. The variation of velocity component

u at the section of 1/2=x , and v at the section of 1/2=y are compared with exact solutions in Figure 5 for these

grid systems. The pressure field contours with velocity streamlines for 3232 grids are also plotted in Figure 6.

The purpose of these computations is to test the efficiency and accuracy of the scheme introduced in this study.

Since the total CPU times strongly depends on the Poisson solver, Table 3 shows the efficiency data by defining the


254

acceleration parameter which is the ratio of total CPU times with respect to Poisson solvers. In the simulations

the termination criteria for the pressure Poisson equation is set for averaged 2L norm of residual 810| |£| | pS .

Since the fractional-step algorithm depends on the pressure Poisson equation, as shown from the Table 3, the

pseudo-time iterations is not efficient in this framework. The results demonstrate that the fractional-step compact

Navier-Stokes solver with Gauss-Seidel multigrid (GS-MG) acceleration is 218 times faster than that of with

pseudo-time method for 6464= N grid, and increasing with increasing grid resolution. It shows that for large

scale problem, the GS-MG method with compact finite difference fractional step procedure becomes very efficient

algorithm to solve incompressible flows.

Table 3: Efficiency comparisons for different grid numbers

Grid sytem CPU time [s] Speed-up

iN x jN Pseudo( 1t ) GS( 2t ) GS-MG( 3t ) 211/2 /= tt 311/3 /= tt

88 0.77 0.14 0.11 5.42 7.03

1616 35.59 4.67 2.13 7.61 16.71

3232 1695.20 217.89 25.23 7.78 67.10

6464 46159.53 5872.71 211.45 7.86 218.30

The accuracy of methods are also shown Table 4. The root mean square values for u are computed according to Eq.

(26). The order of accuracy can also be tested using one of the variables, say u velocity component, according to

Eq. (27). The results presented here indicate that the same accuracy can be get by using multigrid accelerated

framework. In terms of accuracy and the effective order of accuracy there is no difference between with and without

multigrid acceleration in the Poisson equation. However, in terms of computational efficiency, multigrid

acceleration speed up the simulation drastically, because of fast Poisson solution of O(N) instead of O(N2) without

multigrid acceleration. The results also demonstrate that the effective order of accuracy agree with the theoretical

order of accuracy. The simulations here are performed with using the time step 0.0001=t . Since the scheme is

)( 2tO , it is expected to get less values(i.e., 3.56 for )( 64,32nO ) for bigger grid resolution due to the decreasing

spatial error.

Table 4: The effective accuracy and order of accuracy test

Method 88| || | u 1616| || | u 3232| || | u 6464| || | u )( 16,8nO )( 32,16nO )( 64,32nO

Pseudo 6.77E-04 2.03E-05 7.90E-07 6.57E-08 5.0607 4.6823 3.5883

GS 6.63E-04 1.99E-05 7.69E-07 6.48E-08 5.0561 4.6966 3.5695

GS-MG 6.63E-04 1.99E-05 7.69E-07 6.48E-08 5.0561 4.6966 3.5695


255

y

u

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

8 x 8

16 x 16

32 x 32

Exact

x

v

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

8 x 8

16 x 16

32 x 32

Exact

Figure 5. Comparison of velocity components with exact solution for 10=Re at 1=t ; (a) u component of

velocity along 1/2=x , (b) v component of velocity along 1/2=y .

p

0.3

0.25

0.2

0.15

0.1

0.05

0

-0.05

-0.1

-0.15

-0.2

-0.25

-0.3

Figure 6. Pressure contours and velocity streamlines for 10=Re at 1=t .

5. CONCLUSIONS

An efficient high-order fractional-step compact scheme for unsteady incompressible viscous flows is presented and

tested on staggered grid system in order to avoid the well-known odd-even point decoupling problem on the

pressure, occurring in incompressible flow solver. The numerical implementation procedure for fractional-step

procedure are given by using compact interpolations at the predictor and corrector steps. The underpinning idea of

the a compact scheme is to cancel lower order errors by treating spatial Taylor expansions implicitly by constructing

the tridiagonal set of equations to obtain efficient high-accurate discrete approximations. In the multigrid

framework, the algorithm utilize some relaxation method to dump high frequency errors and make use of coarse grid

correction to remove smooth errors. The combination of the efficiency of compact scheme and multigrid

acceleration for the Poisson equation demonstrates very efficient and accurate scheme which can be used large scale

problems desiring high level resolutions. The scheme is applied to the Taylor-Green vortex decaying problem and

the results show that the success of the method under the light of efficiency and accuracy analysis. The results

indicate that the ratio of computational time of proposed algorithm with Multigrid (GS-MG) acceleration to that of

pseudo-time marching scheme is about 1:218 when we use 64x64 grid spacing and the efficiency increases with

increasing resolution. These huge efficiency ratios are observed such that pseudo-time marching relaxation for

pressure equation takes enormous time for each time steps. The efficiency of Mehrstellen-based GS relaxation


256

procedure with and without multigrid acceleration is also analyzed in order to get an idea of true comparison

between them. In those cases the efficiency ratios for 64x64 grid resolution are 1:27, respectively. It is also

important to notice that those relative efficiencies keep increasing with higher resolutions. Therefore, the present

method can be used as an efficient high-order accurate solver of incompressible flow problems.

6. REFERENCES

[1]. Kwak D., Kiris C., Kim C.S., Computational challenges of viscous incompressible flows. Computers and Fluids 2005; 34:283--299.

[2]. Chorin, A.J., A numerical method for solving incompressible viscous flow problems. Journal of Computational Physics 1967; 2:12--26.

[3]. Soh, WY and Goodrich, J.W., Unsteady solution of incompressible Navier-Stokes equations. Journal of Computational Physics 1988; 79:113--134.

[4]. Mateescu, D. and Paidoussis, MP and Belanger, F., A time-integration method using artificial compressibility for unsteady viscous flows. Journal of Sound and Vibration 1994; 177:197--205.

[5]. Chorin, A.J., Numerical solution of the Navier-Stokes equations. Mathematics of Computation 1968; 22:745--762. [6]. Kim, J. and Moin, P., Application of a fractional-step method to incompressible Navier-Stokes equations. Journal of

Computational Physics 1985; 59:308--323.

[7]. Brown, D.L. and Cortez, R. and Minion, M.L., Accurate projection methods for the incompressible Navier-Stokes equations. Journal of Computational Physics 2001; 168:464--499.

[8]. Guermond, JL and Minev, P. and Shen, J., An overview of projection methods for incompressible flows. Computer Methods in Applied Mechanics and Engineering 2006; 195:6011--6045.

[9]. Hafez M., Numerical Simulation of Incompressible Flows. World Scientific, 2002. [10]. Harlow F.H., Welch J.E., Numerical calculation of time-dependent viscous incompressible flow with free surface. Phys

Fluids 1965; 8(12):2182--9.

[11]. Blasco J., Codina R., Huerta A., A fractional-step method for the incompressible Navier-Stokes Equations related to predictor-corrector algorithm. International Journal for Numerical Methods in Fluids 1998; 28:1391--1419.

[12]. Lele S.K., Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics 1992; 103:16--42.

[13]. Zhang K.K.Q., Shotorban B., Minkowyczand W.J., Mashayek F., A compact fnite difference method on staggered grid for Navier-Stokes flows. International Journal for Numerical Methods in Fluids 2006; 52:867--881.

[14]. Knikker R., Study of a staggered fourth-order compact scheme for unsteady incompressible viscous flows. International Journal for Numerical Methods in Fluids 2009; 59:1063--1092.

[15]. Ferreira V.G., Kurokawa F.A., Queiroz R.A.B., Kaibara M.K., Oishi C.M., Cuminato J.A., Castelo A., Tomé M.F., McKee S., Assessment of a high-order finite difference upwind scheme for the simulation of convection?diffusion

problems. International Journal for Numerical Methods in Fluids 2009; 60:1--26.

[16]. Wesseling P., An Introduction to Multigrid Methods. Wiley: Chichester, 1992. [17]. Moin, P., Fundamentals of Engineering Numerical Analysis. Cambridge University Press, 2001. [18]. Gupta, M.M. and Kouatchou, J. and Zhang, J., Comparison of Second-and Fourth-Order Discretizations for Multigrid

Poisson Solvers. Journal of Computational Physics 1997; 132:226--232.

[19]. Wang Y., Zhang J., Sixth order compact scheme combined with multigrid method and extrapolation technique for 2D poisson equation. Journal of Computational Physics 2009; 228:137--146.

[20]. Sakurai K., Aoki T., Lee W.H., Kato K. Poisson equation solver with fourt-order accuracy by using interpolated differential operator scheme., Computers and Mathematics with Applications 2002; 43:621--630.

[21]. Shukla R. K., Zhong X., Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation. Journal of Computational Physics 2005; 204:404--429.

[22]. Botella O., Peyret R., Benchmark spectral results on the lid-driven cavity flow. Computers and Fluids 1998; 27:421--433.

7. APPENDIX: COMPACT INTERPOLATIONS

7.1. Interpolations from staggered to collocated points for the zeroth-order derivatives:

i+2

N-1

1

2 i-1 i i+1

N

N+1

N

0

0

1

i-1 i i+1 N-1

Figure 7. Interpolations from staggered to collocated points for the zeroth order derivatives

The compact interpolations from staggered grid points to the collocated points as shown in Figure 7 for zeroth order

derivatives (the function itself) by constructing compact tridiagonal system are given as


257

)(22

1

2

3= 420110 hO

(31)

)(210

1

22

3=

10

3

10

3 621111 hO

iiiiiii

(32)

)(22

1

2

3= 4111 hO

NNNNN

(33)

There is no need to specify boundary conditions by constructing the interpolations from staggered grid to the

collocated grids.

7.2. Interpolations from collocated to staggered points for the zeroth-order derivatives:

i+1 N-1 0 1 i-2 i-1 i

N+1

b

N

b

0 2 i-1 i i+1 N-1 1 N

Figure 8. Interpolations from collocated to staggered points for the zeroth-order derivatives

Similarly, schematics for the interpolations from collocated grid to staggered grid points for function itself is shown

in Figure 8. The corresponding tridiagonal system with including the boundary values is established as

)(22

1

2

3= 41010 hO

b

(34)

)(210

1

22

3=

10

3

10

3 612111 hO

iiiiiii

(35)

)(22

1

2

3= 4111 hO

NNNNN

(36)

7.3. Interpolations from staggered to collocated points for the first-order derivatives:

i+2

N-1

1

2 i-1 i i+1

'N

N+1

N

0

'0 '1 'i-1 'i 'i+1 'N-1

Figure 9. Interpolations from staggered to collocated points for the first-order derivatives

Schematics for the interpolations from staggered grid to collocated grid points for the first-order derivatives is

shown in Figure 9. The corresponding tridiagonal system becomes

)(= 3120110 hOhh

''

(37)

)(362

17

62

63=

62

9

62

9 612111 hO

hh

iiii'i

'i

'i

(38)

)(= 3111 hOhh

NNNN'N

'N

(39)

7.4. Interpolations from collocated to staggered points for the first-order derivatives:

i+1 N-1 0 1 i-2 i-1 i

'N+1

b

N

b

'0 '2 'i-1 'i 'i+1 'N-1'1 'N

Figure 10. Interpolations from collocated to staggered points for the first-order derivatives


258

Similarly, schematics for the interpolations from from collocated to staggered points for the first-order derivatives is


)(= 301010 hOhh

b''

(40)

)(362

17

62

63=

62

9

62

9 621111 hO

hh

iiii'i

'i

'i

(41)

)(= 3111 hOhh

NNNN'N

'N

(42)

7.5. Interpolations from collocated to collocated points for the first-order derivatives:

i+2

N-1

0

1

i-2 i-1 i b N

b

'0

'i-1

'i

'i+1

'N-1'1 'N

i+1

Figure 11. Interpolations from collocated to collocated points points for the first-order derivatives

Similarly, schematics for the interpolations from collocated to collocated points for the first-order derivatives is


)(32

1

2

3= 420110 hO

hh

b''

(43)

)(49

1

29

14=

3

1

3

1 6221111 hO

hh

iiii'i

'i

'i

(44)

)(32

1

2

3= 4211 hO

hh

NbNN'N

'N

(45)

7.6. Interpolations from collocated to collocated points for the second-order derivatives:

i+2

N-1

0

1

i-2 i-1 i b N

b

"0

"i-1

"i

"i+1

"N-1"1 "N

i+1

Figure 12. Interpolations from collocated to collocated points for the second-order derivatives

Schematics for the interpolations from staggered to staggered points for the second-order derivatives is shown in

Figure 12. The corresponding tridiagonal system becomes

)(22

= 32

210

2

1010 hO

hh

b''''

(46)

)(2

5

6=

10

1

10

1 42

1111 hO

h

iii''i

''i

''i

(47)

)(4

2

11

32

11

12=

11

2

11

2 62

22

2

1111 hO

hh

iiiiii''i

''i

''i

(48)

)(22

= 32

21

2

11 hO

hh

NNNNNb''N

''N

(49)

7.7. Interpolations from staggered to staggered points for the second-order derivatives:

i+2

N-1

0

1 i-2 i-1 i b N

b

''

0

''

i-1

''

i

''

i+1 ''

N-1''

1

''

N

i+1

h/2 h

Figure 13. Interpolations from staggered to staggered points for the second-order derivatives


259

Similarly, schematics for the interpolations from staggered to staggered points for the second-order derivatives is


)(23

12

23

36=

23

11 32

12

2

0110 hO

hh

''''

(50)

)(2

5

6=

10

1

10

1 42

1111 hO

h

iii''i

''i

''i

(51)

)(4

2

11

32

11

12=

11

2

11

2 62

22

2

1111 hO

hh

iiiiii''i

''i

''i

(52)

)(23

12

23

36=

23

11 32

1

2

11 hO

hh

NNNN''N

''N

(53)

7.8. Interpolations from collocated to staggered points for the second-order derivatives:

i+1 N-1 0 1 i-2 i-1 i

"N+1

b

N

b

"0 "2 "i-1 "i "i+1 "N-1"1 "N

Figure 14. Interpolations from collocated to staggered points for the second-order derivatives

Similarly, schematics for the interpolations from collocated to staggered points for the second-order derivatives is


)(2

2= 22

1010 hO

h

b''''

(54)

)(2

7

62

7

6=

14

5

14

5 42

11

2

1211 hO

hh

iiiiii''i

''i

''i

(55)

)(2

2= 22

11 hO

h

NNb''N

''N

(56)

7.9. Interpolations from staggered to collocated points for the second-order derivatives:

i+2

N-1

1

2 i-1 i i+1

"N

N+1

N

0

"0 "1 "i-1 "i "i+1 "N-1

Figure 15. Interpolations from staggered to collocated points for the second-order derivatives

Similarly, schematics for the interpolations from staggered to collocated points for the second-order derivatives is


)(2

2= 22

21010 hO

h

''''

(57)

)(2

7

62

7

6=

14

5

14

5 42

21

2

1111 hO

hh

iiiiii''i

''i

''i

(58)

)(2

2= 22

111 hO

h

NNN''N

''N

(59)

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New A MULTIGRID ACCELERATED HIGH-ORDER COMPACT … · 2018. 9. 14. · acceleration has high resolving efficiency that drastically reduces computational time and high-order accuracy