Two Full Multigrid Algorithms in Cartesian and Polar Coordinate Systems to Solve an Elliptic Partial Differential Equation with a Mixed Derivative Term in a Quarter of a unit Circle

8/10/2019 Two Full Multigrid Algorithms in Cartesian and Polar Coordinate Systems to Solve an Elliptic Partial Differential Equ

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International Journal of Mathematics andComputer Applications Research (IJMCAR)ISSN(P): 2249-6955; ISSN(E): 2249-8060Vol. 4, Issue 6, Dec 2014, 75-86 TJPRC Pvt. Ltd.

TWO FULL MULTIGRID ALGORITHMS IN CARTESIAN AND POLAR

COORDINATE SYSTEMS TO SOLVE AN ELLIPTIC PARTIAL DIFFERENTIAL

EQUATION WITH A MIXED DERIVATIVE TERM IN A QUARTER OF A UNIT CIRCLE

OSAMA EL-GIAR

Department of Basic Science, Modern Academy for Engineering and Technology in Maadi, Cairo, Egypt

ABSTRACT

In this paper we introduce two full multigrid algorithms in a Cartesian and a Polar coordinate system that have

been modified to treat the elliptic partial differential equations with a mixed derivative term, in the two dimensional space

in a quarter of a unit circle, special modification for the points in the neighborhood of the curved boundary has been done.

An advantage of the two algorithms is that the extension from two to three space dimensions is straightforward. numerical

examples have been given.

KEYWORDS: Multigrid Method, Numerical Analysis, Elliptic PDE, Cartesian and Polar Co-Ordinate Systems

1 FULL MULTIGRID ALGORITHMS IN CARTESIAN COORDINATES

1.1 INTRODUCTION

Consider the general linear elliptic partial differential equation:

LU = AU xx + 2BU xy + CU yy + F = 0 R 2 (1.1a)

with Dirichlet boundary conditions

U = F R 2 (1.1b)

Where is a quarter of a unit circle. A difference approximation L h of L in equation 1.1a

is said to be of order p if L h(U) = O(hp), where h is the mesh size. In our case p = 2, so we

shall use the 6-point difference equation to the mixed derivative term U xy to avoid the absent

points in our grids because of the curved boundary see [3] and [4]

1.2 Finite Difference Discretizaton

For each grid we replace each term in equation 1.1a by a second order finite difference approx-

imation for two kinds of points as follows:

1.Normal Points: These points are interior points are not laying near the curved boundary:

for a grid with mesh size is h, we have the following discretizaton:

U xx = h2 (u i+1,j + u i1,j 2u i,j )

U yy = h2 (u i,j+1 + u i,j1 2u i,j ) (1.2)

U xy = h 2 (u i,j u i1,j u i,j1 + u i1,j1 )


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Impact Factor (JCC): 4.2949 Index Copernicus Value (ICV): 3. 0

Substitute these values in equation 1.1a we get the following system of linear equations for normal

points:

(1.3)

2.Curved Boundary Points: These points are points that laying near the curved boundary:

see [4] and Figure 1 When the boundary is curved and intersects the rectangular grid at

points that are not grid points, so we shall

Figure 1: Curved Mesh

Give the finite difference approximation to the derivatives at the points in the neighborhood of the

curved boundary thus:

( )

++

+= +

ji ji j j xx uuuhU ,,1,1

2 2)1(

21

2

( )

++

+= +

ji ji ji yy uuuhU ,1,1,

2 21

2)1(

2

(1.4)

Uxy = h 2 (u i,j u i1,j u i,j1 + u i1,j1 )

Substituting 1.4 into 1.1a we get a system of linear equations for the points in the neighborhood

of the curved boundary given by:

( ) ( ) 1,,

,1,,

,1,

12

12

12

++

++

++

+ ji ji

ji ji ji

ji jai ucubau


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Two Full Multigrid Algorithms in Cartesian and Polar Coordinate Systems to Solve an Elliptic 77 Partial Differential Equation with a Mixed Derivative Term in a Quarter of a unit Circle


ji ji ji ji ji

ji ji ji ji

ji f hububca

ubc

,2

1,1,,,,,

1,,, 22

12 =+

+

+

+ (1.5)

Where 1 i,j N 1, ,1

1

+=

N h N is the number of interior points in each direction h, h

are the distances from the point in the neighborhood of the boundary and the curve boundary in

direction of the x-coordinate and the y-coordinate respectively.

1.3 Multigrid Method

A sequence of 5 grids kh have been used with mesh sizes kh, k = 2n , n = 0, 1, 4,

where h = 64. The elliptic equation have been solved using full multigrid V ( 1 , 2)-cycle

method with the coarse grid correction (CGC), see Figure 2 and has the following components:

1. Coarse to Fine Interpolation with Operator I h

h2 : Within the V-cycle, we will use correction

interpolation where corrections are to be interp olated from coarser grids and added to finer grids and the full

multigrid interpolation where current values of the approximate solution on the coarser grid are interpolated as the first

approximation to the solution (on the first visit to the grid) on the finer grids, see [2], we choose for both cases of

interpolations the bilinear interpolation. Now we have two kinds of points see figure 3:

Figure 2: FMG V (2, 1)

(a) Normal Points:

uu h

ji

h

ji

2

,2,2 = for common points in the two grids, (point 13)

( )21 2

,1

2

,2,12 uuu h

ji

h

ji

h

ji ++ += for odd points in the x direction, (point 12)

( )21 2

1,

2

,12,2 uuu h

ji

h

ji

h

ji ++ += for odd points in the y direction, (point 8)

( )41 2

1,1

2

1,

2

,1

2

,12,12 uuuuu h

ji

h

ji

h

ji

h

ji

h

ji ++++++ +++= for remaining points in, (point 7) (1.6)

(B) Curved Boundary Points


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78 Osama El-Giar


Since we have used red- black gauss-Seidel iteration, then we need interpolation only for odd points in

the x and y directions respectively defined by

for odd points in the x direction, (point 14)

+

+=

++ 1 2

1,

2

,12,2 huuu h

ji

h

ji

h

ji h for odd points in the direction, (point 19) (1.7)

Figure 3: Numbering Two Grids

2. Fi ne to Coarse Restriction Operator: The restriction operator denoted by I h

h

2 takes the residual equation

computed on the fine-grid and transfer it to the coarse-grid according to the rule: hhhh R R I 22 = where the

components of hij R2 of h R 2 are given by the following two schemes- see [2] and [3]:

(a) Half-Weight: If B = 0 in equation 1.1a, we use the red-black Gauss-Seidel relaxation it is better to use thehalf-weight operator since the residual at the black (odd) points must vanish and the stencil of the half-weight operator

takes the form:

h

h

h

h I

2

2

010

141

010

81

= (1.8)

(b) Full-Weight: If B = 0 in equation 1.1a, we use the red-black Gauss-Seidel relaxation, so we may use either the

half-weight operator or the full-weight operator, or it is better to use the half-weight operator for normal points and

full-weight operator for curved boundary points. The stencil of the full-weight operator takes the form:

h

h

h

h I

2

2

121

242

121

161= (1.9)

1.4 Test Problems, Results and Performance

In this section we report the results and performance of some test problems All our implemen-tations in

Fortran were executed on a pentium 4 PC using FORTRAN90 workstation compiler.

The accuracy of the method is measured by both L 2 norm of both defect and error and the

max norm of both defect and the error. The general formula for the test problem is given by :


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21

21

AU xx + 2BU xy + CU yy + F = 0 R2

Consider the following test problems:

1. Test Problem 1: Let A = C = 1, B = 0 using half-weight restriction:

Uexact = e 2xy , F (x,y) = 4(x2 + y 2)e 2xy (1.10a)

Uexact = sin(3x + y), F (x,y) = 10 sin(3x + y) (1.10b)

2. Test Problem 2: Let A = C = B = 1, using half-weight restriction:

Uexact = e 2xy , F (x,y) = 4(x2 + y 2 + xy + )e 2xy (1.11a)

Uexact = sin(3x + y), F (x, y) = 13 sin(3x + y) (1.11b)

3. Test Problem 3: Let A = C = B = 1, using full-weight restriction:

Uexact = e 2xy , F(x,y)=4(x 2 +y 2 +xy+ )e 2xy (1.12a)

Uexact = sin(3x + y), F (x, y) = 13 sin(3x + y) (1.12b)

Test Problem 1: Equation 1.10a and Equation 1.10b

Table 1: Fortran Implementation of Test Problem 1 Using F M V (2, 1) on PC




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2 Full Multigrid Algorithms in Polar Coordinates

2.1 Introduction

A multigrid algorithm has been modified to treat the elliptic boundary value problem for partial differential

equations with a mixed derivative term in two dimensional space in polar coordinates. A full multigrid algorithm in which

W ( 1, 2) with a line relaxation scheme are used to deal with the finite difference approximation of the mixed derivative

term in polar coordinates. Full weight restriction of residuals and linear interpolation operators are used. Numerical

examples are given.

Consider the elliptic partial differential equation with a mixed derivative term U r in polar coordinates which is


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equivalent to equation 1.1 with A = C = 1, B =21

:

(1 + sin cos ) U rr + r 1

(1 sin cos ) U r + 21r (1 sin cos ) U

( ) ( ) =++ ),,(sincos1 cossin1 22222 r F U r U r r (2.1)

where ( ) =4

00,10,, r r , subject to the dirchlit boundary conditions:

= gU (2.2)

2.2 Finite Difference Discretizaton

For each grid we replace each term in equation 2.1 by a second order finite approximation as follows

see Figure 4

U rr = h2 (u i+1,j + u i1,j 2 ui,j )

U r = (2h)1 (u i+1,j u i1,j )

U = ( ) 2 (u i,j+1 + u i,j1 2 ui,j ) (2.3)

U = (2 ) 1 (u i,j+1 u i,j1 ) and

U r = (4h )1

(u i+1,j+1 + u i1,j1 u i+1,j1 u i1,j+1 )

Figure 4 : Descritization of a Quarter of a Unit Circle

substitute these values in equation 2.1 we get the following system of linear equations:

=

1

2

1

1

2

1

11

111

11

111

11

0...0

...0

0............

010

0...

0...0

N N b

b

b

u

u

u

DC

B DC

B DC

B DC

B D

M

M

M

M (2.4)


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82 Osama El-Giar


where 1 I, j N 1, h1

1+ N

, =( )12

1+ N

, are the mesh size and

D 1 = 2( )

++ 2sin2

1

1

1

2sin2

1

1 2i

B 1=( )

22 22cos

2sin21

11

ii

(2.5)

C 1=( )

22 2

2cos2sin

21

11

ii+

where N is the number of interior points in each direction

2.3 Multigrid Method

A sequence of 5 grids kh have been used with mesh sizes kh, k = 2 n , n = 0, 1, 4,

Where h =641

. The elliptic equation have been solved using full multigrid W-cycle method with the coarse

grid correction (CGC), see Figure 5 and has the following components:

Figure 5: FMG W-Cycle

Line Relaxation: Since the finite difference approximation of the mixed derivative term contains the four corner

points, then a scheme of line-relaxation should be used, (first we may first relax the solution at even (odd) lines, then relax

at the odd (even) lines, this gives the tridiagonal system of linear equation 2.4.

Interpolation Operator I h

h2: It takes coarse-grid vectors u 2h defined on G 2h and produces fine grid vectors u h

defined on G h using I h

h2 hu 2 = hu , the components of u h are divided into 4 groups as follows- see figure 4:

(a) Common points of the two grids

h ji

h ji uu

2,2,2 = (point 5) (2.6)

(b) Radial points

( )uuu h jih jih ji 2 ,12,122,12 21 ++ += (points 4, 6) ` (2.7)


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(c) Angle points

( )uuuuu h jih jih jih jih ji 22,1222,122,122,1212,2 41

+++++ +++= (points 2, 8) (2.8)

where the right hand side is obtained from 2.7.

(d) Intermediate points: They are two kinds:

i. Points are not in the neighborhood of the center of the circle

( )uuuuu h jih jih jih jih ji 2 1,12 1,2 ,12,12,12 41

++++++ +++= (points 3, 9) (2.9)

ii. Points are in the neighborhood of the center of the circle

( )uuuu h

ji

h

ji

h

ji

h

ji

2

1,1

2

,1

2

,12,12 31

+++++ ++= (points 1, 7) (2.10)

Restriction Operator hh I 2 : It takes fine grid vectors R h defined on G h and produces coarse grid vectors R 2h

defined on G 2h using hh I 2 h R = h R 2 since we use the line relaxation for every grid it is better to use the full-weight

restriction operator whose stencil takes the form:

( )( )

( )

( )

( )

( )

h

h

hh i

i

i

i

is

i

iii

I

2

2

2

2

2

2

2

121

12

1

2

112

121

12

1

18

1

+

+

=

(2.11)

2. 4 Test Problems, Results and Performance

In this section we report the results and performance of some test problems All our implementations in Fortran

were executed on a pentium 4 PC using FORTRAN90 workstation compiler. The accuracy of the method is measured by

both L 2 norm of both defect and error and the max norm of both defect and the error. Consider the following test

problems: Consider the following test problems:

1. Test Problem 1:

Uexact = 3e 0.5r2 sin cos + 2r 2 sin cos (2.12)

2. Test Problem 2:

Uexact =24 )cos(sin r e (2.13)

3. Test Problem 3:

Uexact = sin (r2 sin cos + 1 ) (2.14)

4. Test Problem 4:

Uexact = r 6 (sin cos )3 + 3r 4 (sin cos )2 +6r 2 (sin cos ) + 2 (2.15)


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84 Osama El-Giar


5. Test Problem 5:

Uexact =2r e (sin

cos ) (2.16)

Test Problem 1: Equation 2.12Table 4: Fortran Implementation of Test Problem 1 Using F M W (2, 1) on PC

Test Problem 2: Equation 2.13Table 5: Fortran Implementation of Test Problem 2 Using F M W (2, 1) on PC

Test Problem 3: Equation 2.14

Table 6: Fortran Implementation of Test Problem 3 Using F M W (2, 1) on PC





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3. CONCLUSIONS

The implementation of an elliptic partial differential equation in the 2-dimensional space in a quarter of a unit

circle have been shown to be an efficient method. The convergence rate of the method in Cartesian coordinates has been

improved as we increase the number of relaxation steps before coarse grid correction, in the polar coordinates we should

use the W-cycle instead of V-cycle to achieve almost the same or a little bit better convergence rate than that of the method

in Cartesian coordinates. The other advantage of the two algorithms in Cartesian and in Polar coordinates is that the

extension from two to three space dimensions is straightforward.

REFERENCES

1. Brandt A., Multi-level adaptive technique (MLAT) for fast numerical solution to bound-ary value problem,

Springer, Berlin - 1973.

2.

Briggs L., A Multigrid Tutorial, Society for Industrial and applied mathematics - 2000.

3. Hackbucsh W., Multigrid Methods and applications, Springer Berlin Heidelberg Berlin,- 2003.

4. G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods., Oxford

Applied Mathematics & Computing Science Series, - January 16, 1986.

5. McCormick S. F., Mulitigrid Methods, SIAM, Phildelphia-Pennsylvania - 1987

6. Briggs W. L, Mulitigrid tutorial, SIAM, Phildelphia-Pennsylvania - 1987

7. John D. Anderson, Jr. Computational Fluid Dynamics: The Basics with Applications, McGraw-Hill Inc, - 1995.

8. J.E. Dennis and R.B. Schnabel, Numerical Methods for Unconstrained Optimization

and Nonlinear Equations, SIAM Books, Philadelphia, - 1996.

9. J.C. Tannehill, D.A. Anderson, Computational Fluid Mechanics and Heat Transfer,

Second Edition , Series in Computational and Physical Processes in Mechanics and Thermal Sciences., - April 1,

1997.

10. W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, Second Edition SIAM Books,

Philadelphia - 2000

11. Bobby Philip, An Introduction to Multigrid Techniques The Institute for Mathematics and Its Applications,

Minnesota - January 11, 2008.


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Documents

Two Full Multigrid Algorithms in Cartesian and Polar Coordinate Systems to Solve an Elliptic Partial Differential Equation with a Mixed Derivative Term in a Quarter of a unit Circle