SOLUTIONS OF THE 2D LAPLACE EQUATION WITH TRIANGULAR GRIDS AND MULTIPLE RICHARDSON EXTRAPOLATIONS

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1 SOLUTIONS OF THE 2D LAPLACE EQUATION WITH TRIANGULAR GRIDS AND MULTIPLE RICHARDSON EXTRAPOLATIONS Carlos Henrique Marci, Luciano Kiyosi Araki, Department of Mecanical Engineering (DEMEC) - Federal University of Paraná (UFPR), Curitiba - PR, Brazil Arileide Cristina Alves, aalves@up.edu.br Postgraduate Course in Mecanical Engineering (PGMEC) - Federal University of Paraná (UFPR), Curitiba - PR, Brazil Sector of Exact Sciences and Tecnology - Positivo University (UP), Curitiba - PR, Brazil Roberta Suero, roberta.suero@ifpr.edu.br Federal Institute of Paraná (IFPR) - Campus of Paranaguá, Paranaguá - PR, Brazil Simone de Fátima Tomazzoni Gonçalves, simone-tg@otmail.com Postgraduate Course in Mecanical Engineering (PGMEC) - Federal University of Paraná (UFPR), Curitiba - PR, Brazil Marcio Augusto Villela Pinto, marcio_villela@ufpr.br Department of Mecanical Engineering (DEMEC) - Federal University of Paraná (UFPR), Curitiba - PR, Brazil Abstract. Numerical solutions for te two-dimensional Laplace equation using isosceles rigt triangular cell-centred and square control volumes are compared in tis work. Te metodology employed for te square grids is te one related to unstructured grids, wile for te square volumes te discretization process is based on te structured grids metodology. For bot geometries, te number of control volumes varies from 4 up to 16,777,216, wit a refinement ratio of 2. In order to speed up te obtainment of te numerical results, te algebraic multigrid metod was employed for te triangular grids and te geometric multigrid metod for te square ones. An unexpected observation was tat, for a grid wit te same number of control volumes, te triangular grid as a worse performance tan te square one, exibiting a numerical error greater. Wit te intention of reducing te discretization error, providing more accurate results, a strategy based on Ricardson Extrapolations was employed, called Multiple Ricardson Extrapolations (MRE). According to previous works, tis metodology was successfully used in square grids, but its use is not a common practice for triangular ones. It was verified tat MRE efficiently reduces te discretization error in triangular grids, altoug error magnitudes are considerably iger tan te ones acieved for a square grid wit te same number of control volumes and MRE. Te main results of tis work can be summarized as follows: (1) altoug triangular grids are more adaptable tan square ones, te last ones sould be used as frequent as possible due to te lower discretization errors involved; (2) MRE is efficient and can be used for te reduction of discretization errors in triangular grids; (3) for te same approximation sceme, numerical errors wit MRE can not be lessening under tan a limit value: tere is always a dependence on te performance of te original results. Keywords: Finite Volume Metod, Multiple Ricardson Extrapolations, discretization error reduction 1. INTRODUCTION Te continuous improvement of te computer resources leads to te opportunity of describing natural penomena at previously unimaginable scales. Tis access and tis opportunity ave served as strong drivers for computational sciences and engineering, especially in te last 20 years (Ganem, 2009). In order to acieve accurate results, owever, verification procedures are required. Te numerical error E () related to te numerical solution can be evaluated by te following expression: E (1) in wic is te exact analytical solution. Numerical errors are composed by four elements: truncation, iteration, round-off and programming errors (Marci and Silva, 2002). Wen te numerical error consists on te contribution of none but te truncation one, it is also called discretization error (Tanneill et al., 1997). Since te beginnings of te 20t Century, procedures for discretization error estimates were proposed by Ricardson (Ricardson, 1910; Ricardson and Gaunt, 1927). Besides te common use of Ricardson extrapolations only as uncertainty estimator, te tecnique provided by Ricardson can also be used to reduce te discretization error, as made by im for te two-dimensional eat diffusion problem (Ricardson and Gaunt, 1927). In tis case, Ricardson extrapolations were employed recursively for two grid levels, providing more accurate results. Oter autors (Benjamin and Denny, 1979; Screiber and Keller, 1983, Erturk et al., 2005) also employed Ricardson extrapolations recursively for a iger number of grid levels (but four at maximum), intending te reduction of te discretization errors in CFD

2 problems. More usual, neverteless, is te use of only one Ricardson extrapolation for te reduction of te discretization error, as made by Wang and Zang (2009a, 2009b) and Ma and Ge (2010). Marci et al. (2008) and Marci and Germer (2009), owever, employed Ricardson extrapolations recursively for several grid levels, in a process named Multiple Ricardson Extrapolations (MRE), for te two-dimensional Laplace equation and one-dimensional advection-diffusion equation, respectively, using structured grids. For bot cases, discretization errors were substantially reduced. According to bot works, MRE sould be used to: (1) for a given discretization error magnitude, reduce te computational requirements by te use of coarser grids; or (2) for a given grid, considerably reduce te magnitude of te discretization errors in order to obtain bencmark results. Te aim of tis work is to investigate te use of MRE in te reduction of discretization errors of te twodimensional Laplace equation, in a unitary square domain (Fig. 1), discretized wit isosceles rigt triangular cellcentred grids; te results migt be compared to te ones obtained in square volumes grids. Triangular volumes are related to unstructured grids, wic are te most general form of grid arrangement for more complex geometries (Versteeg and Malalasekera, 2007). Te Laplace equation was cosen in tis work by its simplicity and te fact tat, for suc problem, tere is an analytical solution, wic allows numerical verification. Te use of Ricardson extrapolations to reduce te discretization error in triangular grids, neverteless, is not a common practice. Works like te one of Jyotsna and Vanka (1995), in wic te Ricardson extrapolation was employed to obtain more accurate results for te velocity pattern (and consequently, for te flow field) in triangular grids are yet exceptions. 2. MATHEMATICAL MODEL Figure 1. Pysical domain and boundary conditions. Te matematical model considered in tis work is related to te two-dimensional Laplace equation wit Diriclet boundary conditions, scematically given in Fig. 1: 2 2 T T 0; 2 2 x y T ( x,1) sin x ; 0 x, y 1 T (0, y) T (1, y) T( x,0) 0 (2) were: x and y are te spatial coordinates and T is te temperature. Tis equation can be pysically related to te eat diffusion on a two-dimensional plate in steady state wit constant termal properties and absence of eat generation (Incropera et al., 2008), wose analytical solution is given by T ( x, y) sinxsin y sin. Te variables of interest in tis problem includes: (1) te temperature at te domain centre (Tc), in oter words, te temperature at position x = 1/2 and y = 1/2; (2) te average temperature (Tm) of te wole domain; and te eat transfer rates at te four boundaries, namely: (3) y = 1 (Qn), (4) y = 0 (Qs), (5) x = 1 (Qe), and (6) x = 0 (Qw). Te variables Tm, Qs and Qe (Qn and Qw are defined in an analogous way) are defined by te following expressions: Tm T 1 T T( x, y) dx dy, Qs k dx, Qe k dy 0 y 0 x y 0 x1 (3) were k is te termal conductivity, wic is assumed to ave unitary value in tis work. 3. NUMERICAL MODEL 3.1. Numerical solutions witout MRE Te unitary side square domain is discretized using te finite volume metod (Maliska, 2004; Versteeg and Malalasekera, 2007), using bot isosceles rigt triangular cell-centred and square grids, as sown at Fig. 2. Wile te metodology applied to square grids is te one related to structured grids, for triangular grids te metodology was te

3 same employed to unstructured ones. For bot cases, second order approximation scemes (central differencing sceme, CDS) were used. In order to speed up te convergence of te numerical codes, two different multigrid metods were employed: for triangular grids, an algebraic multigrid (AMG) algoritm, adapted from Ruge and Stüben (1986); and for square grids, a geometric multigrid (GMG) algoritm (Briggs et al., 2000; Trottemberg et al., 2001). (a) (b) Figure 2. (a) Square and (b) isosceles rigt triangular grids, bot wit 16 real control volumes. Te AMG features employed to acieve te numerical results are: correction sceme (CS) (Brandt, 1986; Ruge and Stüben, 1986; Stüben, 2001); V-cycle; parameter of te strengt of connection (θ) equals to 0.25; parameter of te strong dependence on te coarser grid (ε) equals to Oterwise, for GMG, te main features used are: full approximation sceme (FAS) (Briggs et al., 2000; Trottemberg et al., 2001); V-cycle; and grid-size ratio of 2. For bot multigrid metods: lexicograpic Gauss-Seidel (Burden and Faires, 2008) was employed as smooter (wit one internal iteration); te number of cycles was ig enoug to acieve te macine round-off error; double precision operations were used for all te calculations; and null temperature was employed for te wole domain as initial guess. In order to evaluate numerically te integrals related to te average temperature of te wole domain and te eat transfer rates on te four boundaries, rectangle rule (Kreyszig, 1999) was employed. Moreover, in te evaluation of te derivatives related to te eat transfer rates, upstream differencing sceme (UDS) or downstream differencing sceme (DDS) (Tanneill et al., 1997) was adopted, depending on te case. Oterwise, te temperature at te domain centre is evaluated by te aritmetical mean of te temperatures of te volumes wit one of te vertices at te coordinates x = 1/2 and y = 1/2. Tis procedure is needed once neiter in triangular cell-centred nor in square grids, tere is a nodal point wic is placed exactly on te domain geometric centre Numerical solutions wit MRE Once te numerical solutions are obtained, Ricardson extrapolations can be used for reducing te numerical errors associated to te discretization process according to te following expression: g, m1 g 1, m1 g, m g, m1 (4) pm1 r 1 were: is te numerical solution of a given variable of interest; te index g refers to te grid in wic te numerical solution is evaluated; te index m is te number of Ricardson extrapolations; r is te refinement ratio (r= g-1 / g ); and p m are te true orders of te discretization error (Marci and Silva, 2002). Equation (4) is valid for g [ 2, G] and m [ 1, g 1], in wic g=1 refers to te coarsest grid, g=g is te most refined grid, m = 0 refers to te numerical solution witout any extrapolation and m = 1 is related to te standard Ricardson extrapolation. For eac value of in Eq. (4), numerical solutions of in two different grids (g and g 1) in te m 1 extrapolation are needed. For a given value of g, Eq. (4) can be used recursively g 1 times, providing m Ricardson extrapolations. In tis work, Multiple Ricardson Extrapolations results are obtained wen m 1. Te values of te true orders (p m ) are related to te exponents of te truncated terms of te Taylor series employed in te approximation scemes for te derivatives. More details about Eq. (4) and/or MRE teory can be found in Marci et al. (2008). 4. NUMERICAL RESULTS Twelve different grids are employed in tis work for bot triangular cell-centred and square grids: from grids wit only 4 real volumes (2 2 ) up to 16,777,216 real volumes (2 24 ), respecting a (two-dimensional) refinement ratio of 2. g, m

4 Double precision are used for all te operations and te number of multigrid cycles for bot grid geometries were ig enoug to minimize te iteration error. Numerical results for te six variables of interest are compared to te values of te analytical solutions, wit 30 significant figures, obtained in Maple. Tese comparisons allow te evaluation of te real numerical error in order to study te efficiency of MRE in reducing numerical errors. Te consistency of bot triangular and square volumes can be observed in Fig. 3: as expected, for bot cases, te mean l 1 -norm of te numerical error of te temperature decreases wit grid refinement ( represents te twodimensional grid spacing). Curiously, considering te same number of volumes for bot triangular and square grids, triangular norm is always iger tan te corresponding square one, by a factor of about (except by te two coarsest grids). Tis result indicates tat triangular cell-centred grids present iger discretization errors tan teir square counterparts. Based on tis, despite te adaptability of triangular grids for complex geometries, square structured grids sould be employed as frequent as possible, once te discretization errors are smaller for suc grids mean l 1 - norm [ E ( T ) ] Triangular grid Square grid Figure 3. Mean l 1 -norm of te numerical error E(T). Te use of several grid sizes allows te evaluation of apparent orders ( ) for all te variables of interest. Apparent orders (De Val Davis, 1983) sould be used for a posteriori verification of te values obtained a priori for te true orders of numerical error. Details about te evaluation of te true orders and te use of tem in MRE are explained in Marci et al. (2008). Results of apparent orders ( ) for te eat transfer rate at x = 1 (Qe) for bot grid volume geometries are presented in Fig. 4. Wen m = 0, results are related to te asymptotic error order, wile for m =1 and m = 2, results are related to te second and tird true error orders, respectively. Apparent orders tend to values 2, 4 and 6, being in concordance wit te results of Giacomini and Marci (2009) for te first order approximations (UDS) of derivatives. Tis result, owever, comes from a type of order degeneration, once te UDS presents as asymptotic error order te unitary value and not te value of 2. Neverteless, as te apparent orders tend to 2, 4, 6 and so on, tese values were employed as true orders for MRE. Similar beaviour was seen for te oter eat transfer rates (Qn, Qs, Qw) and also for te oter variables of interest (Tc, Tm), altoug for tese two last ones, te expected true order values were te ones found. 6 5 (m=0) Triangular (m=1) Triangular (m=2) Triangular (m=0) Square (m=1) Square (m=2) Square Figure 4. Apparent orders for Qe.

5 Te discretization error results for te variables of interest are presented in Fig. 5; numerical results for te eat transfer rates at y = 1 (Qn) and x = 0 (Qw) are similar to te ones of te eat transfer rate at x = 1 (Qe) and are terefore omitted. In all te cases, bot results for triangular cell-centred and square grids are presented, were E is te numerical error witout MRE and Emre is te numerical error wit MRE. As anticipated by te l 1 -norm (Fig. 3), in all te cases te numerical error observed, witout MRE, in square grids are smaller tan te counterparts for triangular grids. Even for Qe, Fig. 5(c), square grid results are a little better to te ones of triangular grids, altoug te curves for bot cases are almost coincident. E - Triangular E - Square E - Triangular E - Square E ( Tc ) E ( Tm ) (a) (b) E - Triangular E - Square E - Triangular E - Square E ( Qe ) E ( Qs ) (c) (d) Figure 5. Modulus of te numerical error wit (Emre) and witout (E) MRE for: (a) Tc, (b) Tm, (c) Qe and (d) Qs. As can be seen for all te variables presented in Fig. 5, MRE substantially reduces te numerical error for bot te triangular cell-centred and te square grids. Wile te numerical error is smaller tan for all te variables in a grid wit 2 14 = 16,384 volumes ( 8x10-3 ) for bot geometries using MRE, even in te finest grid, wit 2 24 = 16,777,216 volumes ( 2x ), te numerical error witout MRE is greater tan (it is, mostly, about ). Te use of MRE for te problem presented, owever, does not reduce te numerical error for grids wit more tan 2 14 volumes: as seen for all te variables, numerical error acieves a minimum for suc grid. In fact, tis is an effect of te round-off errors: as te discretization errors associated to MRE reduces muc faster tan te ones obtained by simple grid refinement, te use of a small amount of grids is enoug to acieve te round-off error related to double precision calculations. Because of tis, in spite of te numerical error keep on lessening wit te grid refinement, it grows as a consequence of te increasing of te round-off error. Suc situation can be avoided by te use of te quadruple precision, wose round-off error is about , as presented by Marci et al. (2008). Considering again te results for te grid wit 2 14 = 16,384 volumes, for triangular cell-centred grids, it can be seen in Fig. 5 tat numerical errors are under, for all te variables, using MRE. In comparison, taking te same grid, but not employing MRE, numerical errors are about or In tis case, te use of MRE could reduce numerical errors of about 7 or 8 orders of magnitude, proving te efficiency of MRE in lessening te numerical error for triangular grids. Tis effect is similar to te one observed to square grids: taking te same grid (wit 16,384 volumes), te numerical error witout MRE is about to, wile te use of MRE provides numerical errors of about or Comparing bot results, te use of MRE could reduce numerical errors of about 8 to 10 orders of magnitude, as previously observed by Marci et al. (2008) and Marci and Germer (2009).

6 One of te possible uses of MRE is illustrated in Tab. 1: te reduction of te grid refinement necessary to acieve a given error magnitude. In tis case, numerical error was taken constant and equal to for tree different variables of interest (Tc, Tm and Qe). As can be seen, taking te same geometric volume sape, te ratio of te number of volumes needed to acieve te error magnitude witout and wit MRE is at least equal to 64 (for Tc, using square grids). Tis ratio, neverteless, can be as ig as 1,024, as seen for Qe. Hence, as a consequence of te use of coarser grids to acieve a given error magnitude, te need for CPU time and RAM associated to refined grids are significantly reduced by te employment of MRE. Observing bot results presented in Tab. 1 and Fig. 5, te effect of te use of MRE on te numerical errors for bot triangular cell-centred and square grids is clear. Similarly to previous results for square grids (Marci et al., 2008; Marci and Germer, 2009), MRE application for triangular grids is very effective for te reduction of numerical errors. It must be noted, neverteless, tat numerical errors associated to triangular volumes are greater to te counterparts wit square grids, wit or witout te use of MRE. As te numerical approximations adopted for triangular and square grids are te same (second order, Diriclet boundary conditions applied wit gost-cells), te beaviour of numerical error is similar in bot cases. Te use of MRE is not effective to reduce numerical errors associated to triangular grids to values smaller tan te ones of square grids wit MRE. So, te dependency of MRE on te obtained numerical solutions is obvious: MRE procedure efficiently reduces numerical errors but tis reduction as a limit if a discretization tecnique provides better numerical results witout MRE tan a second one, te results for tis first tecnique wit MRE keeps on being better tan te second one wit MRE. It is clearly represented by te comparison between triangular cell-centred and square grids. Table 1. Quantity of volumes needed to acieve a given numerical error magnitude. Volume geometry Tc, E Variable and Error magnitude Tm, E Qe, E Triangular, witout MRE 2 18 = 262, = 262, = 1,048,576 Triangular, wit MRE 2 10 = 1, = 1, = 1,024 Square, witout MRE 2 14 = 16, = 65, = 1,048,576 Square, wit MRE 2 8 = = = 1, CONCLUSION Isosceles rigt triangular cell-centred and square grids were employed in te discretization of te two-dimensional Laplace equation wit Diriclet boundary conditions by te finite volume metod in order to study te efficiency of MRE. Te numerical model implemented includes: second order approximation sceme (CDS); boundary conditions applied wit gost-cells; te discretization wit triangular grids made according to te metodology for unstructured grids; te discretization wit square grids made according to te procedures for structured grids; algebraic multigrid for triangular grids and geometric multigrid for square ones, in order to speed up te numerical convergence; Gauss-Seidel smooter; number of multigrid cycles ig enoug to acieve te macine round-off error; double precision calculations. Te main results of tis work are: 1. Despite te versatility of te triangular grids, te use of square grids is recommended (if te geometry of te domain allows its employment) by te smaller discretization errors associated to tis geometry. 2. MRE is efficient to reduce numerical errors in triangular grids. 3. Considering different grid types but te same approximation sceme, MRE results depend on te original numerical errors: even if numerical errors can be reduced wit te use of MRE, tey can not be lessening more tan a limit value. In tis work, as triangular volume grids present iger levels of discretization error tan te square ones witout MRE, so te results wit MRE present te same tendency. 6. ACKNOWLEDGEMENTS Te autors would acknowledge te financial support provided by CNPq (Conselo Nacional de Desenvolvimento Científico e Tecnológico), AEB (Agência Espacial Brasileira, by te Uniespaço Program) and Fundação Araucária (Paraná), and te Federal University of Paraná (UFPR) and te Department of Mecanical Engineering (DEMEC) by pysical support given for tis work. Te first autor is scolarsip of CNPq (Conselo Nacional de Desenvolvimento Científico e Tecnológico) Brazil. Te fourt and fift autors are grateful by financial support given by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) by teir scolarsips. Te autors would also acknowledge Dr. Stüben, by te providing of is algebraic multigrid (AMG) code, wic was used as basis for te AMG employed in tis work.

7 7. REFERENCES Benjamin, A.S. and Denny, V.E., 1979, "On te Convergence of Numerical Solutions for 2-D Flows in a Cavity at Large Re", Journal of Computational Pysics, Vol.33, pp Brandt, A., 1986, "Algebraic Multigrid Teory: Te Symmetric Case", Applied Matematics and Computation, Vol.19, pp Briggs, W.L., Henson, V.E. and McCormick, S.F., 2000, "A Multigrid Tutorial", 2 ed., SIAM, Piladelpia, 193 p. Burden, R.L. and Faires, J.D., 2008, "Análise Numérica", Cengale Learning, São Paulo, 721 p. De Val Davis, G., 1983, "Natural Convection of Air in a Square Cavity: A Benc Mark Solution", International Journal for Numerical Metods in Fluids, Vol.3, pp Erturk, E., Corke, T.C. and Gökçöl, C., 2005, "Numerical Solutions of 2-D Steady Incompressible Driven Cavity Flow at Hig Reynolds Numbers", International Journal for Numerical Metods in Fluids, Vol.48, pp Ganem, R.G., 2009, "Uncertainty Quantification in Computational and Prediction Science", International Journal for Numerical Metods in Engineering, Vol.80, pp Giacomini, F.F. and Marci, C.H., 2009, "Verificação da Forma de Aplicar Condições de Contorno em Problemas Unidimensionais com o Método de Volumes Finitos", Proceedings of te 30t Iberian Latin American Congress on Computational Metods in Engineering, Armação de Búzios, Brazil, pp Incropera, F.P., DeWitt, D.P., Bergman, T.L. and Lavine, A.S., 2008, "Fundamentos de Transferência de Calor e Massa", 6 ed., LTC Editora, Rio de Janeiro, 644 p. Jyotsna, R. and Vanka, S.P., 1995, "Multigrid Calculation of Steady, Viscous Flow in a Triangular Cavity", Journal of Computational Pysics, Vol.122, pp Kreyszig, E., 1999, "Advanced Engineering Matematics", 8 ed., Jon Wiley & Sons, New York, 1274 p. Ma, Y. and Ge, Y., 2010, "A Hig Order Finite Difference Metod wit Ricardson Extrapolation for 3D Convection Diffusion Equation", Applied Matematics and Computation, Vol.215, pp Maliska, C.R., 2004, "Transferência de Calor e Mecânica dos Fluidos Computacional", 2 ed., LTC Editora, Rio de Janeiro, 453 p. Marci, C.H. and Germer, E.M., 2009, "Verificação de Esquemas Advectivo-Difusivos 1D com e sem Múltiplas Extrapolações de Ricardson", Proceedings of te 30t Iberian Latin American Congress on Computational Metods in Engineering, Armação de Búzios, Brazil, pp Marci, C.H. and Silva, A.F.C, 2002, "Unidimensional Numerical Solution Error Estimation for Convergent Apparent Order", Numerical Heat Transfer, Part B, Vol.42, pp Marci, C.H., Novak, L.A. and Santiago, C.D., 2008, "Múltiplas Extrapolações de Ricardson para Reduzir e Estimar o Erro de Discretização da Equação de Laplace 2D", Proceedings of te 29t Iberian Latin American Congress on Computational Metods in Engineering, Maceió, Brazil, pp Ricardson, L.F., 1910, "Te Approximate Numerical Solution by Finite Differences of Pysical Problems involving Differential Equations, wit an Application to te Stresses in a Masonry Dam", Pilosopical Transactions of te Royal Society of London, Series A, Vol.210, pp Ricardson, L.F. and Gaunt, J.A., 1927, "Te Deferred Approac to te Limit", Pilosopical Transactions of te Royal Society of London, Series A, Vol.227, pp Ruge, G. and Stüben, K., 1986, "Algebraic Multigrid (AMG), in S. F. McCormick, Multigrid Metods, Vol. 5, Frontiers of Applied Matematics, SCIAM, Piladelpia, 57p. Screiber, R. and Keller, H.B., 1983, "Driven Cavity Flows by Efficient Numerical Tecniques", Journal of Computational Pysics, Vol.49, pp Stüben, K., 2001,"A review of algebraic multigrid", Journal of Computational and Applied Matematics, Vol.128, pp Tanneill, J.C., Anderson, D.A. and Pletcer, R.H., 1997, "Computational Fluid Mecanics and Heat Transfer", 2 ed., Taylor & Francis, Piladelpia, 789 p. Trottemberg, U., Oosterlee, C. and Scüller, A., 2001, "Multigrid", Academic Press, San Diego, 631 p. Versteeg, H.K. and Malalasekera, W., 2007, "An introduction to Computational Fluid Dynamics: Te Finite Volume Metod", 2 ed., Pearson Education Limited, Harlow, United Kingdom, 503 p. Wang, Y. and Zang, J., 2009a, "Sixt Order Compact Sceme Combined wit Multigrid Metod and Extrapolation Tecnique for 2D Poisson Equation", Journal of Computational Pysics, Vol.228, pp Wang, Y.-M. and Zang, H.-B., 2009b, "Higer-order Compact Finite Difference Metod for Systems of Reaction- Diffusion Equations", Journal of Computational and Applied Matematics, Vol.233, pp RESPONSIBILITY NOTICE Te autors are te only responsible for te printed material included in tis paper.