Sixth and Fourth Order Compact Finite Difference Schemes for Two and Three Dimension Poisson Equation with Two Methods to derive These Schemes

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1 Basra Journal of Scienec (A) Vol.(),-0, 00 Sit and Fourt Order Compact Finite Difference Scemes for Two and Tree Dimension Poisson Equation wit Two Metods to derive Tese Scemes Akil J. Harfas Huda A. Jalob Department of Matematics, College of Science, Universit of Basra, Abstract In tis paper we improve te accurac of te numerical approimation used to solve te two and tree dimension Poisson equation. We get tis improvement b using te compact finite difference scemes. Tese scemes give ig order accurac; terefore we derive five scemes to solve two and tree dimension Poisson equation: central finite difference, fourt and sit order compact finite difference b using Talor epansion sceme, fourt and sit order compact finite difference b using compact operators sceme. Numerical eperiments are conducted to test accurac of tese scemes. Keword: compact finite difference, Poisson equation, fourt and sit order. الفروقات المحددة المضغوطة من الرتبة الرابعة والسادسة المستخدمة لحل معادلة بواسون ذات البعدين والثلاث ابعاد باستخدام طريقتين لاشتقاقهما عقيل جاسم حرفش هدى عبد الجبار جلوب قسم الرياضيات, كلية العلوم, جامعة البصرة. الخلاصة في هذا البحث قمنا بتحسين دقة التقريبات العددية المستخدمة لحل معادلة بواسون ذات البعدين والثلاث ابعاد. وقد استخدمنا اساليب الفروقات المحددة المضغوطة للحصول على هذه التحسينات. حيث ان هذه الاساليب تعطي دقة عالية. لذلك قمنا باشتقاق خمس اساليب لحل معادلة بواسون ثناي ية وثلاثية البعد: الفروقات المحددة المركزية, الفروقات المحددة المضغوطة من الرتبة الرابعة والسادسة باستخدام اسلوب مفكوك متسلسلة تايلر,, الفروقات المحددة المضغوطة من الرتبة الرابعة والسادسة باستخدام اسلوب المو ثرات المضغوطة. وقد قمنا بتطبيق هذه الطرق على عدة امثلة لاختبار دقتها. Introduction Te Poisson equation is one of te fundamental equations in matematical psics. It occurs in a broad range of applications including acoustics, electromagnetism and fluid mecanics. Tere are man different approaces to tis problem in literature. (Marcuk et al, 983) give a finite difference solution to 3D Poisson equation as onl second

2 Akil J. Harfas Sit and Fourt... order of accurac. (Koromsk, 980) solve te D Poisson equation wit fourt order accurac. Also, (Samarsk, 98) derive oter fourt order difference sceme. Te compact finite difference metod (CFD) one of te metods used to increase te accurac of te numerical solutions of te partial differential equation. Terefore, we can use man metods to derive te compact finite difference sceme for an problem. (Spot, 995) use te Talor series to drive a fourt and sit compact finite difference for D and 3D Poisson equation. (Zang, 00) derive a fourt order compact finite difference for D Poisson equation b using compact operators metod. In tis paper we ave done te following: - Use te compact operators metod to get te sit order compact finite difference for D Poisson equation. - Develop te compact operators metod to be used it for 3D Poisson equation. 3- Obtain a fourt and sit compact finite difference for D and 3D Poisson equation b using te compact operators metod - Testing te accurac of all scemes wic are derived in tis paper. Compact Sceme b Using Talor Epansion Spot (995) derive te compact sceme to solve Poisson equation using tis metod as follows:. Two Dimension Poisson equation of te form f (, ), (, ) Ω () Were Ω is a rectangular domain, or a union of rectangular domains, wit suitable boundar condition defined on Ω. Te solution (, ) and te forcing function f(,) are assumed to be sufficientl smoot and ave required continuous partial derivatives. For convenience, let us consider a rectangular domain Ω [ 0, L] [ 0, L]. Here subscripts are obviousl not derivatives. We discretie Ω wit uniform mes sies and respectivel in te and coordinate directions. Denote N L/ and N L/ te numbers of uniform intervals along te and coordinate directions, respectivel. Te mes point are ( i, j) wit i i and j j, 0 i N, 0 j N. In te sequel, we ma also use te inde pair (i,j) to represent te mes point ( i, j). In tis paper we take.also N N N. Te standard second order central difference operators define at grid point (i,j) can be written as i j δ () δ

3 Basra Journal of Scienec (A) Vol.(),-0, 00 Te derivatives in Eq.() can be approimated to second order accurac as δ O( ) 30 (3) δ O( ) 30 Using te finite difference operators in () and (3), Eq.() can be discretied at a given grid point ( i, i ) as δ δ τ f. () Were te truncation error is τ [ ] [ ] O( ) (5) 30 Equation () is called a ig Central Difference Sceme (CDS). We ave include bot O( ) and O( ) term in Eq.(5) because we wis to approimate all of tem in order to construct an O( ) sceme. To obtain compact approimation to te O( ) terms in Eq.(5), we simpl take te appropriate derivatives of Eq.(), f () f (7) Substituting Eq.() and Eq.(7) into Eq.(5) ields f f τ [ ] [ ] O( ) (8) 30 Note tat all term on te rigt and side of Eq.(8) ave compact O( ) approimations at noted, and te approimation of tese terms as te following form: δ δ O( ) (9) Also, 3

4 Akil J. Harfas Sit and Fourt... [ ( ] i j i j (0) We can easil get an O( ) metod in te same wa b substituting difference i j ) epressions for te O( ) term in Eq.(8) and including tese in te finite difference approimation (). Te resulting iger-order sceme follows from δ δ δ δ τ f ( δ f δ f) () Equation () is called a ig order compact difference sceme of order (HOC-). We can, owever, obtain an O( ) sceme for tis governing equation. Note tat te approimation to te cross derivative of in Eq.(8) introduces additional O( ) term tat we must be careful to include in our derivation of an O( ) sceme. More specificall, substituting te finite difference epression for te cross derivative of and its truncation error terms into Eq.(8) give us f f τ [ ] [ 5 5 ] O( ) 30 () Clearl, to get a compact O( ) approimation, we require compact epressions for te four derivative of order in Eq.(). Tis can actuall be done b furter differentiating (). Te required epressions are f (3) f And () f (5) Te ke ere is tat we can use Eqs. (3) and () to algebraicall eliminate all te derivatives of. Te two dimensions, O( ) compact approimation to () is terefore δ δ δ δ τ f ( δ f δ f ) ( δ f δ f ) δ δ f () Were τ O( ) Equation () is called a ig order compact difference sceme of order (HOC-).

5 Basra Journal of Scienec (A) Vol.(),-0, 00. Tree Dimension Now, we are interested in te ig accurac numerical solution of tree dimensional (3D) Poisson equation of te form f (, ), (,,) Ω (7) Were Ω is a rectangular domain, or a union of rectangular domains, wit suitable boundar condition defined on Ω. Te solution (,,) and te forcing function f(,, ) are assumed to be sufficientl smoot and ave required continuous partial derivatives. For convenience, let us consider a rectangular domain Ω [ 0, L] [ 0, L] [ 0, L]. Here subscripts are obviousl not derivatives. We discretie Ω wit uniform mes sies, and, respectivel in te, and coordinate directions. Denote N L/, N L/ and N L/ te numbers of uniform intervals along te, and coordinate directions,respectivel. Te mes point are ( i, j, k ) wit i i and j j, k k, 0 i N, 0 j N, 0 k N. In te sequel, we ma also use te inde pair (i, j, k) to represent te mes point ( i, i, i ).In tis paper we take.also N NN N. Te standard second order central difference operators define at grid point (i, j, ) can be written as i jk k δ k k δ (8) δ Te derivatives in Eq.(7) can be approimated to second order accurac as δ O( ) 30 δ O( ) (9) 30 δ O( ) 30 Te relative simplicit of te Poisson equation makes it a good candidate for our first attempt at an HOC sceme in tree dimensions. Te central difference sceme for () in tree dimensions is 5

6 Akil J. Harfas Sit and Fourt... f. τ δ δ δ (0) Were te truncation error as te form ) O( ] [ 30 ] [ τ () Equation (0) is called a Central Difference Sceme (CDS). We ave include bot ) ( O and ) ( O term in Eq.() because we wis to approimate all of tem in order to ield an ) ( O sceme. To obtain compact approimation to te ) ( O terms in Eq.(), we take te appropriate derivatives of Eq.(7) to write f () f (3) f () Substituting Eq.(), Eq.(3) and Eq.() into Eq.() we obtain ] [ 30 ] f f f [ τ (5) Note tat all term on te rigt and side of Eq.(5) ave compact O( ) approimations at noted, and te approimation of tese term as te following form: ) O( δ δ () ) O( δ δ (7) ) O( δ δ (8) Also,

7 Basra Journal of Scienec (A) Vol.(),-0, 00 [ ( ] k i jk k k ) k i jk i j k k [ ( ) k k k k k k ]...(9) (30) [ ( k i jk k ] i jk ) k i jk (3) We can easil get an O( ) metod in te same wa as before b substituting difference epressions for te O( ) term in Eq.(5) and including tese in te finite difference approimation (). Te resulting iger-order sceme follows from [ δ δ δ ] [ δ δ δ δ δ δ ] τ f ( δ δ δ ) f (3) Equation (3) is called a ig order compact difference sceme of order (HOC-). Note tat Eq.(3) corresponds to a 9-point stencil, encompassing all te adjacent nodes of te mes located on te tree grid planes tat intersect te node, but not te corner points of te surrounding cube. We can, owever, obtain a O( ) sceme for tis governing equation. Note tat te approimation to te cross derivative of in Eq.(5) introduces additional O( ) term tat we must be careful to include in our derivation of an ( ) sceme. More specificall, substituting te finite difference epression for te cross derivative of and its truncation error terms into Eq.(5) give us τ [ 7 ] O( ) O ] [ 30...(33) Clearl, to get a compact O( ) approimation, we require compact epressions for te four derivative of order si in Eq.(33). Tis can actuall be done b furter differentiating (7). Te required epressions are f (3) f (35) 7

8 Akil J. Harfas Sit and Fourt... f (3) And f (37) f (38) f (39) Te ke ere is tat we can use Eqs. (3) - (39) to algebraicall eliminate all te derivatives of epect, wic as a compact approimation as follows δ δ δ [ 8 ( i jk i jk k k ) ( ) ( k k k k i j k k k k k i jk k i jk i jk ) i jk k k i j k i jk i j k (0) Tis operator brings te eigt corner points into our O( ) approimation, wic follows from [ δ δ δ ] [ δ δ δ δ δ δ ] δ δ δ τ f [ δ 30 δ δ ]f [ δ δ δ ]f [ δ δ δ δ δ δ ]f () Were τ O( ) Equation () is called a ig order compact difference sceme of order (HOC-). - Compact Operator Sceme ] first For an illustration purpose, we consider te analogous one dimensional problem d f () d 8

9 Basra Journal of Scienec (A) Vol.(),-0, 00 Te second derivative operator as d at a grid i can be approimated using central difference d d Φ d Φi d Φi δ O( ) (3) d d 30 i d Dropping te last two terms in te rigt and side of Eq.(3) ields te standard second order difference sceme. Te idea beind te ig order compact approimation d Φ sceme is to approimate te term i to second order accurac and to acieve an d overall truncation accurac of fourt order in (3). To tis end, we double differentiate Eq.() to ave d Φ d f () d d Substituting () into (3) ields a fourt compact approimation for te second derivative d Φ d fi d Φi δ i O( ) (5) d d 30 i d Hence te fourt order compact approimation of te one dimension Poisson () is d f δ i i f i () d We note tat tis fourt order compact approimation is different from te second order approimation onl in te approimation of te rigt and side function f. We can rewrite () as d δ i ( ) f i (7) d Also, b using te same process we get d Φ d f (8) d d Substituting (8) into (3) ields a sit compact approimation for te second derivative d Φ d f d f δ i i i O( ) (9) d d 30 i d Hence te sit order compact approimation of te one dimension Poisson () is d d δ i ( ) f i (50) d 30 d We denote to te fourt and sit approimation as A L f (5) i i 9

10 Akil J. Harfas Sit and Fourt... Were A d δ in fourt and sit order and L ( ) in fourt order d d d and L ( ) in sit order. d 30 d Sstem (5) can be rewrite as L A f i i L (5) Here te operator as smbolic meaning onl. In application, te fourt and sit order compact difference sceme is given b Eq.(5), not b Eq.(5). Analogous smbolic fourt and sit order compact approimation operator can be obtained for te variable. For two dimension, we can appl te smbolic fourt order compact d d approimation operators to te second derivative and in Eq.(), respectivel. d d Tis ields smbolicall L A L A f Appling to bot sides of Eq.(53) wit te operator L L, we obtain (53) L A LA LL f (5) Appling te smbolic operators for te fourt order compact approimation, we ave ( ) δ ( ) δ ( )( )f (55) Simplifing (55) we get δ δ δ f ( δ f δ f) δ δ f δ (5) We called Eq.(5) a modified ig order compact difference sceme of order (MHOC- ). Appling te smbolic operators for te sit order compact approimation, we ave ( ) δ ( ) ( δ )( 30 ) f...(57) Simplifing (57) and use Eq.(5) we get δ δ δ δ f ( δ f δ f) ( δ f δ f) 30 8 δ δ f ( δ δ f δ δ f) δ δ f (58) We called Eq.(58) a modified ig order compact difference sceme of order (MHOC-). 0

11 Basra Journal of Scienec (A) Vol.(),-0, 00 For tree dimension, we can appl te smbolic fourt order compact approimation operators to te second derivative respectivel. Tis ields smbolicall d d d, and in Eq.(5), d d d L A L A L A f Appling to bot sides of Eq.(59) wit te operator L L, we obtain (59) L LA LLA LL A LL Lf (0) Appling te smbolic operators for te fourt order compact approimation, we ave ( ( )( ) δ Simplifing () we get ) δ ( ( )( )( )( ) δ )f ( ) () ( δ δ δ ) ( δ δ δ δ δ δ ) δ δ δ f 3 ( δ δ δ )f ( δ δ δ δ δ δ )f δ δ δf 78 () We called Eq.() a modified ig order compact difference sceme of order (MHOC- ). Appling te smbolic operators for te sit order compact approimation, we ave ( )( ) δ ( )( ) δ ( )( ) δ ( )( )( )f (3) Simplifing (3) and removing te term wic include te derivatives of ecept te derivatives δ, δ, δ, δ δ, δ δ, δ δ and δ δ δ we get

12 Akil J. Harfas Sit and Fourt... ( δ δ δ ) ( δ δ δ δ δ δ ) δ δ δ f ( δ 8 δ δ )f ( δ δ δ )f ( δ δ δ δ δ δ )f ( δ δ δ δ δ δ δ δ δ δ δ δ )f δ δ δf ( δ δ δ δ δ δ δ δ δ )f ( δ δ δ δ δ δ )f ( δ δ δ δ δ δ δ δ δ )f δ δ δf 5000 () We called Eq.() a modified ig order compact difference sceme of order (MHOC-). 3. Numerical Solver we will now sow te procedure and algoritm tat is used to solve te general linear sstem of te sit order compact finite difference sceme as eample, were tis sstem can be written in te following general form : [ δ δ ] [ δ δ δ δ δ δ ] δ δ δ F 30 δ (5) And to solve te linear sstem (5), we use te Gauss-Seidel iterative metod.also to appl tis iterative metod to te above linear sstem, we must arrange tis sstem. Firstl tis sstem can be written as follows:- Substituting te value of,,,,, and, in Eq(5) ield i jk k k k [ ] [ ] [ ] [{ ( i j ) i j i j } { ( k k ) k k k k } { ( k i jk ) k i jk i jk k }] [ 8 ( k i jk k k ) ( k k 30 k i jk k i jk i j k ) ( k k k k i jk ) k i jk k i j k k i jk k i j k ] F...( ) Simplifing Eq. () and appling te Gauss Seidel on it we get te following iterative formula

13 Basra Journal of Scienec (A) Vol.(),-0, 00 (n ) 5 (n ) (n ) (n ) [[ ] [ ] [ ] i jk k k k (n ) (n ) (n ) [{ ( ) } k i jk k k k i jk i j k k (n ) (n ) (n ) { ( ) } k (n ) { ( k k k (n ) (n ) ) }] i jk k i jk i jk k (n ) (n ) (n ) (n ) [( ) ( 30 k i jk k k k k (n ) (n ) ) ( k i jk k i jk i j k k k k (n ) ) k i jk k i jk k i j k k ] F ] i jk k i j k k k k (7) Were represents te value of te function at te point ( i, j, k ) and at iterative step. Firstl we give an initial condition to te function at ever node as follows (0), i, j,k 0()N (8) And ten calculate te value of te function at te step () from Eq.(7) and ceck te convergence criteria NN N (n ) < γ k j i (9) were γ 0 if te are satisfied stop te computation and out put te results. Oterwise, use te newl obtained results as initial guess and repeat te computation. Te absolute error evaluated b using te following equation Error Were N N N e k i j 3 (N ) represents te approimate value and e represents te eact value. (70) 3

14 Akil J. Harfas Sit and Fourt.... Numerical Results. Two Dimension We now consider two dimension model problems to test te ig order compact formulation for Poisson equation. Test : Te first test problem (,0) e, (,) e, 0, (7) (0, ) e, (, ) e, 0, Combined wit te following forcing function, f e Te resulting eact solution is e e (7) Surface of te eact solution are sown in Figure. Figure sow surface for CDS, HOC-, MHOC-, HOC- and MHOC- solution respectivel, to test problem wit /. Table sow te value of te absolute error computed for different value of N (N,,, 8, 0, ) Figure : Surface of te eact solution to D test problem for /.

15 Basra Journal of Scienec (A) Vol.(),-0, 00 Figure : Surface of te CDS, HOC-, MHOC-, HOC- and MHOC- solutions to D test problem for /. 5

16 Akil J. Harfas Sit and Fourt... Table : Te absolute error computed b CDS, HOC-, MHOC-, HOC- and MHOC- for test problem. N N N N8 N0 N CDS 7.38E-03.50E-03.05E-0 3.5E-0.00E-0.35E-0 HOC-.38E-0.989E E-0.08E-0.038E-07.89E-07 MHOC-.58E-0.E E-0.003E E-07.0E-07 HOC- 7.09E-0 8.8E-08.30E-09.0E E E- MHOC-.07E-0.85E E E-0.7E E- Test : Te second test problem (,0) e, (,) e e, 0, (73) (0, ) e, (, ) e e, 0 combined wit te following forcing function, f e e ( 7) Te resulting eact solution is e e e (75) Surface of te eact solution are sown in Figure 3. Figure sow surface for CDS, HOC-, MHOC-, HOC- and MHOC- solution respectivel, to test problem wit /. Table sows te value of te absolute error computed for different value of N (N,,, 8, 0, ). Figure 3: Surface of te eact solution to D test problem for /.

17 Basra Journal of Scienec (A) Vol.(),-0, 00 Figure : Surface of te CDS, HOC-, MHOC-, HOC- and MHOC- solutions to D test problem for /. Table : Te absolute error computed b CDS, HOC-, MHOC-, HOC- and MHOC- for test problem. N N N N8 N0 N CDS.330E E-0 3.9E-0.9E-0.8E E-05 HOC-.33E-05.97E-0 3.9E-07.0E E-08.83E-08 MHOC-.33E-05.97E-0 3.9E-07.0E E-08.83E-08 HOC-.9E-07.8E-09.3E-0.787E- 3.E- 7.3E- MHOC-.9E-07.8E-09.3E-0.787E- 3.E- 7.3E- In tis eample we note te HOC- and HOC- ave te same results of MHOC- and MHOC- respectivel tis occur because te addition terms in te MHOC finised because of te derivatives of its equal to ero. 7

18 Akil J. Harfas Sit and Fourt.... Tree Dimension We now consider tree Test problems to test te ig order compact formulation for Poisson equation. Test : Te first test problem (,,0) e, (,,) e, 0,, (0,,) e, (,,) e, 0,, (7) (,0, ) e, (,, ) e, 0,. combined wit te following forcing function, f 3 e (77) Te resulting eact solution is e e (78) Table 3 sow te value of te absolute error computed b use CDS, HOC-, MHOC-, HOC- and MHOC- scemes for different value of N (N,,, 8, 0, ) Table 3: Te absolute error computed b CDS, HOC-, MHOC-, HOC- and MHOC- for test problem. N N N N8 CDS.77E-0.507E E E-0 HOC-.33E E-05.0E E-0 MHOC E E-05.07E-0.89E-0 HOC-.7E-05.8E-07 3.E-08.07E-09 MHOC- 3.79E-05 3.E-07.80E E-09 Test : Te second test problem (,,0) (,,) (0,,) (,,) (,0, ) (,, ) 0, 0,,, (79) Combined wit te following forcing function, f 3π sin( π) sin( π) sin( π) (80) Te resulting eact solution is sin( π) sin( π) sin( π) (8) e Table sow te value of te absolute error computed b use CDS, HOC-, MHOC-, HOC- and MHOC- scemes for different value of N (N,,, 8, 0, ) 8

19 Basra Journal of Scienec (A) Vol.(),-0, 00 Table : Te absolute error computed b CDS, HOC-, MHOC-, HOC- and MHOC- for test problem. N N N N8 CDS.337E-0.7E-0 9.3E E-03 HOC-.90E-0.775E-03.00E-03.78E-0 MHOC-.E E-03.00E-0.8E-0 HOC-.08E-0.8E-0.0E-05.85E-0 MHOC- 5.73E E-0.7E-05.09E-0 Test 3: Te tird test problem (,,0) cos( ), (0,, ) cos( ), (,0, ) cos( ), (,,) cos( ), (,, ) cos( ), (,, ) cos( ), 0 0 0,,,,,. (8) combined wit te following forcing function, f 3 cos( ) (83) Te resulting eact solution is cos( ) (8) e Table 5 sow te value of te absolute error computed b use CDS, HOC-, MHOC-, HOC- and MHOC- scemes for different value of N (N,,, 8, 0, ) Table 5 : Te absolute error computed b CDS, HOC-, MHOC-, HOC- and MHOC- for test problem 3. N N N N8 CDS.87E-0 9.9E E-05.0E-05 HOC-.89E-05.0E-0.53E-07.93E-07 MHOC-.7E-05.70E-0.5E E-08 HOC- 9.3E E-08.9E-08.50E-08 MHOC E E E-08.50E-08 Conclusions In tis paper, te ig order compact finite difference scemes are derived to solve te Poisson equation subject to Diriclet boundar condition on a bounded two and tree dimension region. We use two metods two drive te fourt and sit order finite difference scemes, and tese metods give four scemes. We use two problems to test te accurac of tese scemes in two dimension case and tree problem to test te accurac of tese scemes in tree dimension case. We note tat te accurac of te 9

20 Akil J. Harfas Sit and Fourt... sit order scemes is ver ig compare wit te fourt order scemes; also te accurac of te fourt order scemes is ver ig compare wit te central finite difference sceme. Moreover, te scemes wic derived b use te compact operators metod ave more accurac compare wit sceme drive b use Talor series epansion metod. Te linear sstem resulted from tese scemes is solve b use te Gauss-Seidel iterative metod. Reference Koromsk B. N., 980, Metod of increasing accurac of discrete solution of boundar value problems wit te operator invariant wit respect to turning of coordinate sstem. Institute of Nuclear resources, Dubna, No.p Marcuk G. I., Sairdurov V.V., 983, Difference metods and teir Etrapolations. N.Y. Springer. Samarsk A.A., 98, Teorie der differenenverfaren. Leipiing. Spot W. F., 995, Hig-order compact finite difference scemes for computational mecanics, PHD tesis, Universit of Teas at Austin, Austin, T,. Zang J., 00, Multigrid Metod and fourt order compact difference sceme for D Poisson equation wit unequal Mes sie Discretiation, Tecnical Report No.3-0, Department of computer Science, Universit of Kentuck, Leington, KY. 0