New Fourth Order Explicit Group Method in the Solution of the Helmholtz Equation Norhashidah Hj. Mohd Ali, Teng Wai Ping

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1 World Academy of Science, Engineering and Tecnology International Jornal of Matematical and Comptational Sciences Vol:9, No:, 05 New Fort Order Eplicit Grop Metod in te Soltion of te elmoltz Eqation Norasida. Mod Al Teng Wai Ping International Science Inde, Matematical and Comptational Sciences Vol:9, No:, 05 waset.org/pblication/00005 Abstract In tis paper, te formlation of a new grop eplicit metod wit a fort order accracy is described in solving te two dimensional elmoltz eqation. Te formlation is based on te nine-point fort order compact finite difference approimation formla. Te compleity analysis of te developed sceme is also presented. Several nmerical eperiments were condcted to test te feasibility of te developed sceme. Comparisons wit oter eisting scemes will be reported and discssed. Preliminary reslts indicate tat tis metod is a viable alternative ig accracy solver to te elmoltz eqation. Keywords Eplicit grop metod, finite difference, elmoltz eqation, five-point formla, nine-point formla. W I. INTRODUCTION E consider te two dimensional elmoltz eqation of te form (, ) f y, (, ) Ω yy k were te soltion domain Ω (0,)(0,) y () wit Diriclet conditions defined at te bondary. Te soltion is (,y), k is known as a wave nmber, and te fnction f togeter wit are assmed to be sfficiently smoot and ave necessary continos partial derivatives. Tis eqation governs some problems in pysical penomena sc as water wave propagation and membrane vibration [4]. Te soltion domain is discretized niformly in te and y directions so tat te mes size is /n, were y ( 0,,,, n) i. Te notation i is sed to represent te compted soltion (, y ). Recently, several new point and i grop scemes derived from te standard and rotated fivepoint stencils ave been developed in solving tis type of problem []-[3]. In [], for eample, a alf-sweep point iterative metod is derived in solving model problem () were tis metod is fond to be ave better convergence rates tan te normal fll-sweep iterative sceme de to te lesser compting compleity of te former. Grop iterative scemes were formlated in [] and [3] to solve te same Norasida. Mod Ali is wit te Scool of Matematical Sciences, Universiti Sains Malaysia, 800 Penang, Malaysia (corresponding ator pone: ; fa: ; sida@cs.sm.my). Teng Wai Ping was a gradate stdent of te Scool of Matematical Sciences, Universiti Sains Malaysia, 800 Penang, Malaysia. Se is now wit ELP University, Psat Bandar Damansara, Kala Lmpr ( wp86045@otmail.com). Financial spport provided by te Fndamental Researc Grant Sceme (03/PMATS/673) is grateflly acknowledged. eqation were tese new metods were proven to ave even better convergence rates tan te point sceme in []. owever, all tese developed scemes are of second order accracy. In tis paper, a new grop iterative sceme wit a iger order accracy will be developed in solving (). Tis forpoint eplicit grop metod is formlated by sing te compact nine-point finite difference formla wit fort order accracy. Section II will describe te derivation of te proposed sceme, followed by te compleity analysis of te sceme in Section III. Section IV will report some nmerical eperiments implemented on te proposed sceme. Conclding remarks are given in Section V. II. FORMULATION OF TE GROUP METOD A. Eplicit Grop O( ) Te standard second order central difference can be written as,,,, i i i i,,,,, i i i y i If () is approimated by te standard five-point central difference sceme (), te following is obtained i, i, ( 4) () k f. (3) Tis sceme as a trncation error of O( ). Based on tis approimation, te second order eplicit grop sceme was derived by [] as: p( p ) p p p i, p p( p ) p p i, p p p( p ) p p p p p( p ) i, f i, i, fi, p ( p 4 ) i, i, fi, i, f were p ( 4 k ). Akir et al. [] generate te iterations in grops of for ntil a certain convergence criteria is met. Tey fond tat tis EG O( ) is more sperior in terms of eection timings tan te standard five-point sceme (3) bt wit te same order of accracy. (4) International Scolarly and Scientific Researc & Innovation 9() 05 7 scolar.waset.org/ /00005

2 World Academy of Science, Engineering and Tecnology International Jornal of Matematical and Comptational Sciences Vol:9, No:, 05 International Science Inde, Matematical and Comptational Sciences Vol:9, No:, 05 waset.org/pblication/00005 B. Eplicit Grop O( 4 ) To design a iger order sceme, te Taylor series epansion is sed to obtain te following formlas: O( ) (5) y yy y y O( ) (6) 360 wic can be re-arranged to become [5] (7) yy y y. (8) By employing (7) and (8) into (), and rearranging, we ave te following i, i, i, k 4 ( ) (4k 0) i, i, ( 8 f fi, fi, f f ) wic is a fort order compact approimation to (). Te comptational molecle of tis approimation is of te following form: i-, i-, i-,- - - Fig. Comptational molecle of approimation (9) Te compact nine-point iterative sceme may be formlated based on (9) were iterations are generated on te in te wole soltion domain ntil convergence is acieved. To constrct te proposed eplicit grop sceme, we apply (9) to grops of for in te soltion domain wic will reslt in te following (44) system of eqations: (9) were p q q, q p q i q p q q q p i, i, i, i, F i, i, F q i, i, i, F3 i, i, F4 i, i, i, i, i, i, i, i, i, i, i, i, 3 i, i, 4 i, i, i, i, (0) k p (0 4 k ), and q 4. Te matri at te left and side of te system (0) can be inverted wic transform te system into an eplicit form: i, i, ( 4 ) 8 ( 4 ) 4 p q p q p q were i, i, i, F () b c c d c b d c F c d b c d c c b i, i, i, F i, i, F q i, i, i, i, F3 i,, i F i, i, 4 ( )( ), c q ( p ) d ( p ( q ))( p ). b p p q p and Te trncation error of tis sceme is of O( 4 ). International Scolarly and Scientific Researc & Innovation 9() 05 8 scolar.waset.org/ /00005

3 World Academy of Science, Engineering and Tecnology International Jornal of Matematical and Comptational Sciences Vol:9, No:, 05 Types of Iterative groped Iterative ngroped Total internal TABLE I NUMBER OF POINTS INVOLVED IN TE ITERATIONS EG O( ) EG O( 4 ) Nine-Point Sceme m 4 m m 4 m 4 m m 4 Standard Five Point Sceme m m m m International Science Inde, Matematical and Comptational Sciences Vol:9, No:, 05 waset.org/pblication/00005 Fig. Groping of for te grop metod (n3) All te in te soltion domain are divided in grops of for as sown in Fig. for te case n 3. Using (), iterations on tese grops of are generated ntil a certain convergence criteria is met. In smmary, we describe te fort order eplicit grop metod as follows:. Divide te soltion domain into grops of for as sown in Fig. (for te case n3).. Iterate on te in eac grop sing () wit preferred smooter (e.g. Mltigrid, Sccessive OverRelaation, etc) 3. Ceck te convergence. If te soltions converge, terminate te iterations. Oterwise, repeat step. III. COMPLEXITY ANALYSIS Assme tat te soltion domain is discretized wit integer n, ten te nmber of internal mes is given by m were mn-. If n is even, ten tere will be ngroped near te pper/rigt bondaries. Tese type of will be smooted sing (9). Table I lists te nmber of for te internal mes for te proposed eplicit grop metod as well as for te eisting second order eplicit grop metod de to [], te nine-point compact and te five-point metod. Te comptational compleity of te algoritm is based on te nmber of aritmetic operations performed at eac iteration, i.e. te operation involving addition/sbtraction ( / ) and mltiplication/division ( / ) performed in eac iteration, eclding te convergence test. Te eection time reqired to perform an aritmetic operation is assmed to be te same or almost te same. All te in te soltion domain are involved in te iterative process, i.e.. owever, for te grop metod, it will reqire interior and near te pper/rigt bondary if n is even. Te nmber of aritmetic operations reqired for eac iteration for te point and grop metods in terms of m are recorded in Table II. Note tat, ceiling fnction is defined as min, e.g , 4.9 4, and floor fnction is defined as, e.g , Metod Standard Five Point EG O( ) Nine Point EG O( 4 ) TABLE II ARITMETIC OPERATIONS OF TE METODS UNDER STUDY Per Iteration /- / 4m m m m 4 m 4 m m m 4 5 8m m m m 8 m 4 3 IV. NUMERICAL EXPERIMENTS m m m 4 9 To test te feasibility of te proposed metod, we condct nmerical eperiments in solving te following model problem [4]: ( π ) sin ( π ) sin ( π ) yy k k y () were te soltion domain is Ω (0,)(0,) wit Diriclet conditions defined at te bondary satisfying te eact (, y) sin π sin πy. All te eperiments are soltion ( ) ( ) condcted sing te programming tool C on P Mini wit Windows 7 Starter Edition, processor type is Intel Atom TM CPU wit installed memory (RAM) of GB and 3-bit Operating System type. Trogot tis eperiment, all te algoritms are implemented sing different grid sizes of 8, 6, 3, 64 and 8, wit te vales of k randomly cosen. For te iteration process, te mltigrid tecniqe was sed as te smooter. Several parameters were measred, i.e. te eection times (in seconds), te nmber of iterations (Iter), te maimm absolte errors over te discrete grid, and te estimated order of accracy. Te maimm errors are taken as International Scolarly and Scientific Researc & Innovation 9() 05 9 scolar.waset.org/ /00005

4 World Academy of Science, Engineering and Tecnology International Jornal of Matematical and Comptational Sciences Vol:9, No:, 05 International Science Inde, Matematical and Comptational Sciences Vol:9, No:, 05 waset.org/pblication/00005 ma < ε, were te tolerance ( k ) ( k) ε 3 0. Te estimated order of accracy for eac sceme can be compted for different grid size. Consider two mes sizes and on Ω and Ω respectively. We denote te maimm absolte errors of tese two grids as and respectively. Sppose te order of accracy is A, ten [5] ( ) ( ) A A. (3) So te order of accracy can be estimated as log A. (4) log For comparison prposes, we also rn te programs for te eisting five-point iterative sceme (3), nine-point iterative sceme (9) and te second order eplicit grop iterative sceme de to [] (4). Table III displays te performances of te implemented nmerical scemes. TABLE III PERFORMANCE COMPARISON BETWEEN METODS Order of Accracy (A) e-0 - Metods n Iter Time (secs) Ma error Second Order (Standard Five Point) Second Order EG [] Fort Order (Nine Point) Fort Order EG e e e e e e e e e e e e e e e e e e e Te nmber of iterations for bot five-point and compact nine-point iterative metods are almost te same, bt te eection times for te compact nine-point stencil become longer becase tere are more involved in te comptational molecle, and also fll weigted restriction operator is sed dring te mltigrid smooter process for te tis sceme, compared to te alf weigted restriction operator sed for te standard five-point stencil. owever, te nine-point sceme gives fort order approimation to te soltions compared to five-point sceme. Maimm.80E-0 errors.60e-0.40e-0.0e-0.00e E E E-03.00E E00 EG0 EGO Mes sizes Fig. 3 Maimm errors for EG O( ) and EG O( 4 ) for different mes sizes Amongst te grop metods, te newly developed EG O( 4 ) prodces more accrate soltions compared to te eisting second order accrate soltions of EG O( ) wen te mes is finer as sown in Fig. 3. Bt te compleity of EGO4 is larger tan EG0 sc tat it reqires sligtly more CPU times tan te latter. owever te differences in timings are not very large as te grid sizes increase. It can be observed tat EGO reqires abot 77% of te timings reqired by EGO4. Between te pointwise and te grop metods, te grop metods are more sperior in terms of compting effort were tey reqire only almost alf of te eection times compared to tat of teir point iterative conterparts as sown in Fig Timings (secs) CNP EGO4 SFP EG Mes sizes Fig. 4 Eection timings (in secs) for te compact nine-point (CNP) te eplicit grop O( 4 ) (EGO4), te standard five-point (SFP) and te second order eplicit grop O( ) (EGO) Tables IV and V tablate te total compting efforts between te pointwise and te grop metods in terms of operation conts. Tese vales were compted by combining te nmber of operations in Table II wit te nmber of iterations obtained in te eperiments by eac sceme. From International Scolarly and Scientific Researc & Innovation 9() 05 0 scolar.waset.org/ /00005

5 World Academy of Science, Engineering and Tecnology International Jornal of Matematical and Comptational Sciences Vol:9, No:, 05 te eperimental reslts, we can clearly see tat te eperimental timings of te iterative metods are in agreement wit te compleity analysis. [5] Y. Wang, and J. Zang, Sit order compact sceme combined wit mltigrid metod and etrapolation tecniqe for D Poisson eqation, Jornal of Comptation Pysics, 8, 009, pp International Science Inde, Matematical and Comptational Sciences Vol:9, No:, 05 waset.org/pblication/00005 TABLE IV COMPUTING EFFORTS FOR STANDARD FIVE-POINT AND EXPLICIT GROUP OF ORDER [] n Metods Standard Five Point EG O( ) [] Iter Operation cont Iter Operation cont TABLE V COMPUTING EFFORTS FOR COMPACT NINE-POINT AND EXPLICIT GROUP OF ORDER 4 Metods n Nine Point EG O( 4 ) Iter Operation cont Iter Operation cont V. CONCLUSION In tis work, we ave developed an alternative fort order nmerical sceme sing a specific grop constrction in solving te two dimensional elmoltz eqation. Te originality of tis metod lies in te selection of te groping strategy wic led to an eplicit natre of te approimation formla. Nmerical reslts sow tat te new metod is able to solve te elmoltz eqation wit better accracy compared to te second order grop metod introdced in [] witot adding too mc compting costs. Te new sceme also reqires lesser compting times tan its point iterative conterpart, i.e. te compact nine-point sceme, wic is de to te former's lower comptational compleity. For ftre work, we foresee tat tis grop metod can also be etended to solve time dependent partial differential eqations sc as te D diffsion or convection-diffsion eqations. REFERENCES [] M.K. Akir, M. Otman, J. Slaiman, Z.A. Maid, and M. Sleiman, Nmerical soltions of elmoltz sing a new for-point EGMSOR iterative metod eqations, Appl. Mat. Science, Vol. 80, 0, pp [] M.K. Akir, M. Otman, J. Slaiman, Z.A. Maid, M. Sleiman, alfsweep iterative metod for solving D elmoltz eqations, International Jornal of Applied Matematics & Statistics, Vol. 9, No. 5, 0, pp [3] M.K. Akir, M. Otman, Z. Abdl Maid, and M. Sleiman, For point eplicit decopled grop iterative metod applied to two-dimensional elmoltz eqation. International Jornal of Mat. Analysis, vol. 6, no. 0, 0, pp [4] M. Nabav M.. K. Siddiq J. Darga A new 9-point sit order accrate compact finite difference metod for te elmoltz eqation, Jornal of Sond and Vibration, Vol. 307, 007, pp International Scolarly and Scientific Researc & Innovation 9() 05 scolar.waset.org/ /00005