ASSESSMENT OF NUMERICAL UNCERTAINTY FOR THE CALCULATIONS OF TURBULENT FLOW OVER A BACKWARD FACING STEP

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1 Submitted to Worsho on Uncertainty Estimation October -, 004, Lisbon, Portugal ASSESSMENT OF NUMERICAL UNCERTAINTY FOR THE CALCULATIONS OF TURBULENT FLOW OVER A BACKWARD FACING STEP ABSTRACT Ismail B. Celi and Jun Li Mechanical and Aerosace Engineering Deartment West Virginia University, Morgantown, WV 6505 USA Numerical uncertainty analysis has been erformed for the turbulent flow ast a bacward facing ste. The analysis is based on calculations on seven non-rectangular, but structured, grid sets that were rovided by the organizers of the 004 Lisbon Worsho. The calculations were erformed by using a commercial code, namely, FLUENT with the Salart-Allmaras one-equation turbulence model. Since there was no exerimental data made available, the resent study constitutes simly a calculation verification rocess where by a set of artial differential equations are solved on gradually refined grid sets, and then the overall numerical uncertainty in selected quantities is estimated by various methods. Some new ideas are resented for estimating the coefficient of variation, which is related to standard deviation (or standard error of estimate in case of least squares method). The major roblem that stands out again is the case of oscillatory convergence. For such case some alternative methods are roosed, and the results are assessed by comaring different methods with each other. Another issue that can not be ignored is the roblem of interolation to obtain the values of a selected arameter at a given satial location from the results of calculations on different grids. The interolation error could sometime overshadow the discretization error, and the remedy to this roblem is not trivial. BACKGROUND The numerical uncertainty assessment is necessary for CFD (Comutational Fluid Dynamics) to become a reliable design tool. Many of the aroaches roosed in the literature for quantification of the numerical uncertainty are based on grid refinement in conjunction with Richardson extraolation (RE) (Richardson 90, 97; Roache, 998). RE usually uses calculations on 3 sets of grids to determine the extraolated value of a deendent variable to zero grid size, either using the theoretical order of the scheme (on at least two grid levels), or via the aarent or observed order which is calculated as art of the unnowns; in the latter case at least three sets of calculations are needed on significantly different grid levels. The ros and cons of this method has been the toic of many recent ublications (Celi et al. (993), Celi & Zhang (993), Celi & Karatein (997), Roache (998), Stern et al. (00), Cadafalch et al. (00), Eca & Hoestra (00)). In site of being a very useful tool for quantifying discretization errors in CFD, there still remain major roblems that need to be addressed to advance the level of confidence that could be trusted uon this method (Eca & Hoestra 003, Celi et al., 004). Alternative methods are roosed due to the difficulties of Richardson extraolation. For instance, least squares aroach is roosed to avoid the scattering of the data although it needs more than 3 sets of grids. AES (Aroximate Error Sline) method (Celi et al. 004) is roosed to solve the oscillatory convergence roblems. The organizers of the Lisbon worsho (Eca et al., 004) have recognized the need for further refinement and assessment of the methods used for quantifying numerical uncertainty. This aer resents our findings from a careful study of numerical uncertainty for one of the test cases roosed by Eca et al. (004). This is the classical case of a turbulent flow over a bacward facing ste. Our study will address on the following questions: () what is the effect of interolation methods on the extraolation? () Which extraolation method is suitable for oscillatory convergence? (3) Which uncertainty estimation method is a better indicator for grid convergence? (4) How can the error estimates calculated from extraolated values be translated into a quantitative uncertainty?

2 METHODS With more than 3 sets of grids, we can use least squares method for extraolation of comuted quantities (Eca and Hoestra, 003). With 3 sets of grids, the following methods -- ower law method, cubic sline method, olynomial, and Aroximate Error Sline (AES) method are used in this study (See the aendix for the details of these methods; see also Celi et al., 004 for more details). We use nonlinear least squares extraolation (see aendix A) with 4 and 7 sets of grids. In these cases the mean, µ, is simly taen as the extraolated value, φ ext. Once φ ext is nown the numerical uncertainty is calculated using the fine Grid Convergence Index (GCI) roosed by Roache (998) which can be written as φ φ U f.5 ext f φ () φ f An alternative way to estimate the uncertainty is to calculate the coefficient of variation. In this study, the coefficient of variation (CV) for the least squares method are comuted from n r i ext + i i σ [ φ ( φ αh )] (a) σφ / h σr /( n 3) (b) CV σ φ / h (c) Here, µ φext and φext is equivalent to φ 0 in the aendix. σ r is the sum of squares of the errors between the nonlinear regression line and the data oints. σφ / h is the standard error of the fit. The coefficient of variation is the relative error of estimate. For these methods requiring 3 sets of grids, the coefficient of variation is calculated from the samling results with 4 trilets; Four sets of grids are denoted by G, G,G3, &G4, the samle size n4, and the trilets are (G,G,G3); (G,G3,G4); (G,G,G4) &(G,G3,G4). Using these samles the mean µ is given by and the standard deviation is given by The coefficient of variation is comuted from n i ext, i µ µ φ / n (3a) n ( ext, i ) /( n ) i (3b) σ φ µ CV σ (3c) µ Combining different methods of extraolation, larger samles can be obtained to imrove the statistics.

3 CASES AND SPECIFIC ISSUES The case investigated in this study is a D turbulent bacward facing ste flow. The Reynolds number is 50,000 based on the ste height and the maximum inlet velocity. The Exansion ration is 9/8. An examle of seven sets of similar, structured, and non-cartesian grids used in this study is shown in Figure. These grids are refined from 0*0 to 4*4. The averaged grid size is calculated from h A/ nc where A is the area of the domain and n c is the number of the cells. The grid refinement ratios for all these grids are in the range of.~. as listed in the first two columns of Table. These grids are used to estimate the numerical uncertainty with the least squares method. In this study we also selected some trilets to assess those extraolation methods which are alicable for three sets of grids. For these trilets, the refinement ratios are in the range of.9~.43 which is generally believed to be more aroriate in grid convergence study with three sets of grids. Four sets of grids listed in the last two columns of table are also used to estimate the uncertainty with the least squares method and the results are comared to the ones that involved all seven sets of grids. Figure an examle of grids used in this study Table Grid refinement ratios grids ratio grids Ratio Grids Ratio grids ratio 0*0 0*0 4*4 0*0 *.0 4*4.40 8*8.9 4*4.40 4*4.7 0*0.43 4*4.33 8*8.9 6*6.4 4*4.33 8*8.3 0*0. 4*4.0 Sarlart-Allmaras one-equation turbulence model is used to solve for the flow field with the commercial code FLUENT 6.0 (FLUENT CO. 004). The convection terms and the diffusion terms are discretized with the second order uwinding scheme and the central differencing scheme, resectively. The inlet velocity is rovided by Eca (004). At the outlet, the gauge ressure and the derivatives of other quantities are set to be zero. The non-sli wall boundary condition is used at the walls. The shear stress at walls is obtained from laminar stress-strain relationshi u/ u ρ u y/ µ. For ost-rocessing of the data, the bilinear interolation method based on the quantities at four nodes of a cell is used with an order of accuracy in the range <<. Interolation methods with higher order are ossible but they need also the quantities from neighboring cells. As a comarison, the results with quadratic interolation method are also calculated to access the imact of interolation errors. The quadratic interolation method needs 6 nodes, two of which are from the neighboring cells. In this study, τ τ 3

4 for a cell with four nodes -- (x i, y i ), (x i, y i+ ), (x i+, y i ), and (x i+, y i+ ), we add two neighboring nodes (x i, y i+ ) and (x i+, y i ) to build a linear system to solve the interolation coefficients. It is worth to note here that selecting other nodes may result in a singular linear system of equations. For instance, choosing (x i, y i+ ) and (x i+, y i+ ) will lead to singularity. Figure shows us the streamwise velocities interolated with the bilinear and quadratic interolation methods along a vertical line at x/h3.0 (H is the ste height). The difference is not negligible esecially in the recirculation region. Since the node choice for the quadratic interolation method is not unique, we use the bilinear interolation method for the calculations resented later to reserve the consistency. To study the influence of interolation method further, 4 sets of rectangular grids are generated by doubling the grids. The results with these grids did not need any interolation. 8 7 quadratic bilinear 6 y/h U/Umax Figure Interolated streamwise velocity at x3.0 with 0*0 grids Double recision is used for all the calculations in this study. The machine accuracy for double recision is aroximately.e-6 so that the round-off error is exected to be negligible. The iterations were stoed whenever the scaled residual for continuity equation aroached an asymtotic value. The scaled residual is defined as R φ cells nb ( a nb φ + b a φ ) nb a φ cells Here a is the center coefficient of the discretization equation, a nb are the influence coefficients for the neighboring cells, and b is the contribution of the source term. Corresondingly, φ is the value for a general variable at the center cell and φ nb reresent the one at the center of the neighboring cell. In this study, the scaled residual is observed to reach a constant of about e-~ e-5, which varies with the grids. (4) RESULTS AND DISCUSSION () Grid convergence First the convergence atterns are shown (Figure 3) with grid refinement for the calculated velocity at the oints -- (0,.) and (4, 0.). Two tyes of grid convergence are identified namely monotonic and oscillatory. It is also seen that the results are far from being grid indeendent. These figures illustrate that in the assumtion of being in the asymtotic range is still a roblem even with seven set of grids. 4

5 Streamwise Velocity at x/h0 y/h. Wall-normal Velocity at x/h0 y/h. u/uref h/h0 (a) aarent monotonic convergence Streamwise Velocity at x/ H4 y/ H h/ h0 u/uref h/h0 (b) oscillatory convergence Wall-normal Velocity at x/ H4 y/ H h/ h0 (c) oscillatory convergence (d) aarent monotonic convergence Figure 3 Velocities at (0,.) and (4, 0.) calculated with different grids (h0 is the grid length of the coarsest grid) () Least squares extraolation The results in Table &3 are calculated with least squares method using all 7 sets of grids. Uncertainty is estimated by using the Grid Convergence Index from Eq. (). Table local flow quantities extraolated with 7 sets of grids Variable x0, y.h xh, y0.h x4h, y0.h U 6.69E E E-0 Uncertainty U.7E-03.5E-03.E-0 Observed V.08E-0.46E E-03 Uncertainty V.5E-0 7.5E-0 3.E-0 Observed C -.77E-0 -.4E E-0 Uncertainty C.E-0.6E E-03 Observed ν t.439e-03.9e-03.77e-03 Uncertainty ν t 3.9E E E-03 Observed Reattachment oint: 6.03 Uncertainty: 4.6E- Observed : 0.86 Table 3 Integral quantities extraolated with 7 sets of grids Flow quantity Predicted Uncertainty Observed Friction resistance bottom wall(n).640e-0.e Friction resistance to wall(n) 4.88E-0.0E Pressure resistance bottom wall(n).5e-0 5.6E

6 The results in Table 4&5 are calculated using least squares method with 4 sets of grids 0x0, 4x4, 8x8, 4x4. This is done because of two reasons: ) Calculations with seven sets of grids are too many and unrealistic; ) the grid refinement ratio with the set of seven is too small. Here also, uncertainty is estimated by using the Grid Convergence Index. Table 4 local flow quantities extraolated with 4 sets of grids Variable x0,y.h xh,y0.h x4h,y0.h U 6.69E E E-0 Uncertainty U 3.5E E-04.9E-0 Observed V.76E-0.435E E-03 Uncertainty V.3E-0 6.3E-0.7E-0 Observed C -.654E-0 -.4E E-0 Uncertainty C 5.0E-0.3E-03.4E-0 Observed ν t.440e-03.93e-03.77e-03 Uncertainty ν t.e-03.e-03 4.E-03 Observed Reattachment oint: 6.5 Uncertainty: 4.4E- Observed : 0.89 Table 5 Integral quantities extraolated with 4 sets of grids Flow quantity Predicted Uncertainty Observed Friction resistance bottom wall(n).64e-0.0e Friction resistance to wall(n) 4.883E-0.6E Pressure resistance bottom wall(n).5e-0 5.E The extraolated quantities and the uncertainties redicted with 4 sets of grids are very close to the ones redicted with 7 sets of grids. Since 4 sets of grids are the minimum which least squares aroach requires. This seems to be adequate for uncertainty analysis. Of the uncertainties at three different location (0, ), (, 0.), and (4, 0.), the ones at (, 0.) are the largest which may be caused by the raidly changing of the flow field in that region. The observed scatters in the range of 0.86 ~ 3.40 and it is difficult to identify the theoretical order of accuracy directly from these results. Positive observation is that the least squares method does not lead to unrealistically low (i.e. 0.) or high (i.e. 0) observed order of accuracy. (3) Extraolation with methods using trilets The quantities extraolated with ower law, AES, cubic sline, and olynomial method (See aendix for exlanation on these methods) are comared to those calculated with least squares method as shown in Table 6. When oscillatory convergence haens, the extraolated values redicted with the AES method are closest to the results with the least squares method. The GCI and the CV are comared in Table 7. It is seen that the CV redicted with the least squares method is much smaller than the ones calculated using the other methods with only 3 sets of grids. The GCI and CV normalized by the maximum in each row of Table 7 are then lotted in Figure 4. With the least squares method, the uncertainty calculated with the CV shows more consistence than the one with the GCI. Among these methods with trilets, the uncertainty associated with the ower law seems to be much larger than the others. It remains to be seen which of these reresent the unnown reality. 6

7 Table 6 Extraolated quantities with different method Least squares(7 grids) Least squares(4 grids) Power Law AES Cubic sline Polynomial U 6.69E E E E E E-0 x0 V.08E-0.76E-0.3E-0.040E-0.533E E-03 y. C -.77E E E E E E-0 ν t.439e e-03.49e e-03.48e-03.40e-03 U -.95E E E E E E-0 x V.46E-0.435E-0.30E-0.453E-0.977E-0.3E-0 y0. C -.4E-0 -.4E E E E E-0 ν t.9e-03.93e e-03.03e-03.9e-03.74e-03 U -.44E E E E E E-0 x4 V -9.89E E E E E E-0 y0. C -.64E E E E E E-0 ν t.77e-03.77e-03.30e-03.76e-03.63e-03.75e-03 f(south).640e-0.64e-0.750e-0.638e-0.706e-0.704e-0 (south).5e-0.5e E-0.8E-0.044E-0.048E-0 f(north) 4.88E E E E E E-0 reattachment 6.03E E E E E E+00 searation 8.988E E-0.039E E E E-0 Oscillatory convergence haens when refining the grids Table 7 GCI and Coefficient of variation in quantities found by extraolation in Table 6 Power law AES Cubic sline Polynomial Least square GCI GCI (4 CV (7 CV (4 (7grids) grids) grids) grids) CV CV CV CV u.7e E-03.7E E-03.6E-0.E-0 4.3E-03.0E-0 x0 v.5e-0.3e-0 7.9E-0.3E-0 4.E-0 7.5E-0 5.6E-0.4E+00 y. c.e-0 5.0E-0.6E-0 3.9E-0.5E-0 9.8E-0 5.6E-0 5.5E-0 vis 3.9E-04.E-03 5.E E-03.E-0.E-0 3.4E-0 4.8E-0 u.5e E-04 5.E E E-0 4.E-0.E-0 5.7E-0 x v 7.E-0 6.3E-0 5.8E-0.0E-0.E-0.5E-0 5.8E-0 7.9E-0 y0. c.3e-03.3e-03 4.E-03 7.E E-0.E-0 5.4E-03.6E-0 vis 3.3E-03.E E E E-0 4.4E-0 3.E-0 6.4E-0 u.e-0.9e-0 9.E-03.6E-0 4.4E-0.3E-0 4.E-0 5.3E-0 x4 v 3.E-0.7E-0.8E-0 4.7E-0.3E-0 7.5E-0 3.3E-0 4.3E-0 y0. c 9.7E-03.4E-0.6E-0 3.E-0.E-0 3.E-0 5.E-0 3.E-0 vis 4.3E-03 4.E E E E-0.3E-0.E-0.E-0 f(south).e-03.0e-03 3.E E-03.6E-0.4E-0 3.4E-04.4E-0 (south) 5.6E-03 5.E E-03.4E-0 7.6E-0 3.6E-0.0E E-0 f(north).0e-04.6e-04.5e-04.e-04.5e-03.4e-03.9e-03.8e-03 reattachment 4.6E-0 4.4E-0.6E E E-0.E-0.0E-0.6E-0 searation.3e-0.e-0 7.9E-03.4E-0.4E-0 8.6E-0.E-0 5.9E-0 Oscillatory convergence haens when refining the grids 7

8 GCI(7grids) GCI(4 grids) CV(7 grids) Least square CV(4 grids) CV Power law CV AES CV Cubic sline CV Polynomial u at (0,.) v at (0,.) c at (0,.) vis at (0,.) u at (,0.) v at (,0.) c at (,0.) vis at (, 0.) u at (4,0.) v at (4,0.) c at (4,0.) vis at (4,0.) f at south wall at south wall f at north wall Figure 4 Normalized GCI and CV with different methods and different cases When resenting results from a CFD alication it is usually desirable to resent calculated field variables with error bars in terms of rofiles at certain locations in arallel with exerimental results. Figure 5 and 6 deicts the normalized error in the streamwise velocity comonent at x/h as a function of vertical distance y/h. It is seen that extraolation using ower law is roblematic, esecially in the region where oscillatory convergence is resent. When an average value is used for the observed order i.e. ave N, N being the number of data oints, the results obtained from ower law are in concert with the other methods (see Fig. 6) where the calculation of is not an issue. We susect that the truth is somewhere among the four cases shown in Fig. 6. But it remains to be seen which one of these results in Fig. 5 reresents the truth. ave () 0.7 ave ( ).98 Figure 5 Extraolated streamwise velocity rofile using ower law with different ways handling the ower 8

9 Figure 6 Comarison of the extraolated streamwise velocity rofile at x/h with different methods (4) Results with no interolation The searation and reattachment lengths calculated with doubling rectangular grids are shown in Table 8.This grid doubling is done to avoid interolation. Again, the oscillatory convergence is observed for the reattachment length calculated with rectangular grids. The extraolated quantities and coefficients of variation are listed in Table 9. Among the coefficients of variation for all methods, the ones with the least squares method exhibit the smallest variation. Comarison of Table 7 and Table 9 show that the extraolated reattachment length is in general significantly smaller (by about 7%) when the rectangular grids are used with grid doubling. On the other hand, the uncertainty indicated by coefficient of variation is larger when interolation errors are minimized. The streamwise velocity comonent at (.0, 0.5) with different grids are shown in Table 0. The velocities calculated with the rectangular grids and with no interolation are shown on the left two columns. On the right, the velocities are comuted with the non-rectangular grids and with the bilinear interolation method. The extraolations are erformed by using least squares method as shown in Table. It is seen first that the extraolated velocity with rectangular grids are somewhat different from the ones with non-rectangular grid. It is also observed that the numerical uncertainty (GCI and CV) calculated with the rectangular grids are quite smaller than the ones with the non-rectangular grids. The aarent order of accuracy,, with rectangular grids is closer to the theoretical order used in this study. CONCLUSIONS This extensive effort on analysis of grid convergence and the estimation of numerical uncertainty has shown that when calculations are reeated on significantly different set of four grids, the least squares method gave results that seem to be at least consistent among themselves. We obtained very similar results using 7 sets of grids and 4 sets of grids, hence four sets of carefully selected grids should be adequate for uncertainty analysis. The major roblem again, seems to arise from cases that exhibit oscillatory convergence. In fact, it may be erroneous to assume monotonic convergence just by observing the behavior of three or four oints. In this regards, it may be necessary to devise methods which erform well both for monotonic, and oscillatory cases. Some of methods are roosed in this category has shown otential to redict the extraolated values and the variance in that without secifically emloying the observed order. 9

10 In articular, the aroximate error sline method when used with trilets among four sets of grids may be a good choice to estimate the mean extraolated values and the variance in that mean. It remains to be seen if these conclusions will hold even after the conference when all results from different grous are comared with each other. REFERENCES Table 8 Quantities calculated with Cartesian grids and non-interolation 6*6 * 4*4 48*48 Searation oint Reattachment oint Table 9 Extraolated quantities and coefficients of variation (Values in Parenthesis are from Table 6 & 7) Searation oint Reattachment oint Mean CV Mean CV Least squares.9 (0.89) 6.7E-0 (.4E-0) 5.77 (6.).4E-0 (4.6E-03) Power law.39 3.E E-0 AES E E-0 Cubic sline.00.5e E-0 Polynomial.06.7E E-0 Table 0 Streamwise velocity at (.0, 0.5) calculated with different grids Rectangular velocity with no non-rectangular velocity with grids interolation grids interolation 6* E-00* E-0 * -.93E-0* E-0 4* E-04*4 -.90E-0 48* E-06* E-0 8* E-0 0* E-0 4* E-0 Table Extraolated streamwise velocity at (.0, 0.5) with least squares method non-rectangular rectangular 7 sets of grids 4 sets of grids 4 sets of grids U GCI.0E-03.0E-03.0E CV 5.0E E E-03 Cadafalch, J., Perez-Segarra, C.D., Consul, R. and Oliva, A. (00) Verification of Finite Volume Comutations on Steady-State Fluid Flow and Heat Transfer, ASME Journal of Fluids Engineering, Vol. 4,.-. Celi, I., Chen, C.J., Roache, P.J. and Scheurer, G. Editors. (993), Quantification of Uncertainty in Comutational Fluid Dynamics, ASME Publ. No. FED-Vol. 58, ASME Fluids Engineering Division Summer Meeting, Washington, DC, 0-4 June. 0

11 Celi, I. and Zhang, W-M (993) Alicaton of Richardson Extraolation to Some Simle Turbulent Flow Calculations, Proceedings of the Symosium on Quantification of Uncertainty in Comutational Fluid Dynamics, Editors: I. Celi et al. ASME Fluids Engineering Division Sring Meeting, Washington, C.C., June 0-4, Celi, I. and Karatein, O. (997), Numerical Exeriments on Alication of Richardson Extraolation with Nonuniform Grids, ASME Journal of Fluids Engineering, Vol. 9, Celi, I., Li, J., Hu, G., and Shaffer, C. (004) Limitations of Richardson Extraolation and Possible Remedies for Estimation of Discretization Error Proceedings of HT-FED004, 004 ASME Heat Transfer/Fluids Engineering Summer Conference, July -5, 004, Charlotte, North Carolina USA, HT-FED Eca, L. and Hoestra, M. (00), An Evaluation of Verification Procedures for CFD Alications, 4th Symosium on Naval Hydrodynamics, Fuuoa, Jaan, 8-3 July. Eca, L. and Hoestra, M. (003), Uncertainty Estimation :A Grand Challenge for Numerical Shi Hydrodynamics 6th Numerical Towing Tan Symosium, Rome, Setember 003 Eca, L. et al., (004) Worsho on Numerical Uncertainty Estimation Lisbon, Portagul 004 FLUENT comany 004 FLUENT 6.0 User Reference Manual Phillis, G.M. and Taylor, P.J. (973) Theory and Alications of Numerical Analysis Academic Press: London and New Yor,.345 Richardson, L.F. (90) The Aroximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations, with an Alication to the Stresses In a Masonary Dam, Transactions of the Royal Society of London, Ser. A, Vol. 0, Richardson L.F. and Gaunt, J. A. (97) The Deferred Aroach to the Limit, Philos. Trans. R. Soc. London Ser. A, Vol. 6, Roache, P.J.(998) Verification and Validation in Comutational Science and Engineering, Hermosa Publishers, Albuquerque Stern, F., Wilson, R. V., Coleman, H. W., and Paterson, E. G. (00), "Comrehensive Aroach to Verification and Validation of CFD Simulations - Part : Methodology and Procedures," ASME Journal of Fluids Engineering, Vol. 3, , December. APPENDIX () Least squares method With the least squares aroach, we comute 0 φ,α, and by minimizing the following function. n 0 i 0 i (.) i ( φ, α, ) ( φ φ α ) S h where n is the number of grids available. The minimum of (.) is found by setting the derivatives of (.) with resect to φ 0,α, and equal to zero, which leads to a non-linear system of equations. Solving the non-linear system yields values forφ 0,α, and. () Polynomial method This method uses the first few terms in the Taylor exansion of φ ( h) to aroximate φ ( h). For instance, assuming the method is first-order, we can use the first three terms if we have 3 sets of grids. That is φ ( h ) φ(0) + a h + a h (.) If we have 4 sets of grids, we can use 3 φ ( h ) φ(0) + a h + a h + a h (.) If the scheme is higher order ( ), this method will mean essentially a curve fit to the actual error function. For a fourth order method one has to ee at least 4 terms, i.e. five sets of calculations are needed. 3

12 The extraolation to the limit aroach is recommended to solve the equations formed by olynomial (3) method. This aroach uses the following formula to calculate the extraolated solution φ ( h) for 3sets (4) of grids and φ ( h) for 4 sets of grids. φ ( m) φ ( h) ( m ) m ( αh) α φ m α ( m ) ( h) m,, (.3) It s easy to tabulate the sequential stes of the calculation rocedure and to add more oints later. (3) Power law method We use the Power law method roosed by Celi and Karatein(997) for 3 sets of grids. The idea follows φ ( 0) φ( ch (3.) h ) ( 0) ( ε φ φ (3.) h ) 3 sign ch ε φ ( 0) φ( (3.3) h ) 3 ch3 where ε 3 / ε3 ( φ( h3 ) φ( h ))/( φ( h ) φ( h )) the sign of which is ositive for monotonic convergence and negative for oscillatory convergence. There are 3 unnows, φ (0), c, and. We can imlement the same iterative method to solve (3.)-(3.3) as done by Celi and Karatein (997). For 4 sets of grids, we can aly + φ ( h ) φ(0) a h + a h i,,3,4 (3.4) i i i Oscillatory convergence is facilitated if a and a are of oosite sign. It should be noted that for some cases there is no solution to Eq. (3.4). Those cases will be counted as unsuccessful outcomes. (4) Cubic sline method The well nown natural cubic slines curve fitting technique is used to create the cubic slines between three oints or four oints. φ (0) can be found by extraolating the curve for the interval closest to h0. (5) Aroximate error sline method Still using Taylor series exansion for (h) The true error E t is given by φ and substituting αh for h, we have 3 φ ( αh) φ(0) + a αh + a ( αh) + a ( αh + (5.) 3 ) and the aroximate error Ea E ( t α aα h (5.) E a, h) φ( αh) φ(0) ( α, h) φ( αh) φ( h) (5.) where (, h) E t α is the true error and E a ( α, h) is the aroximate error which resents the difference of the subsequent results with the fine grid and the coarse grid. So we have

13 letting Dividing (5.) by (5.3) and moving (, h) E a( α, h) a ( α ) h (5.3) E a α to the right hand side yield Et ( α, h) Ea( α, h) (5.4) ah a α h ah aα h b 0 + bh + bh and exanding the l.h.s. of the above equation and comaring it with the r.h.s. give Now Eq. (5.4) can be rewritten as b b E 0 α α a α a α a b α a 3 a ( α) a (5.5) (5.6) α, h) Ea( α, h) ( b + b h + b h ) (5.7) t ( 0 In order to calculateb 0, b & b, we need to calculate a, a,& a3 first. It is seen from Eq. (5.3) that a ( ) Ea ( α,0),,3!( α ) E () is the th derivative of E. Assuming that we have 3 sets of grids and the solutions as (, ( h )) ( h, φ ( h )) and ( h3, φ ( h3 )) with h3 α h α h. And noting that Ea( α,0) 0 ( h Ea( α, )), ( h Ea ( α, )) and ( 0, ( α,0) ), h, h a (5.8) h φ, leads to 3 oints as E which involves the aroximate error instead of the numerical solution φ ~ itself. Using the information on E a we can interolate with cubic slines using two endsloes given by E a' ( α,0) 0 and Ea '( α, h ) ( Ea ( α, h ) Ea( α, h )) /( h h ). These endsloes are accetable at h0 for any scheme with order larger than. For the first order methods, in general, the sloe at h0 is not zero. We could still obtain excellent results using the zero sloe assumtion for the first order methods as we demonstrate in the assessment art of this aer. Once we ( ) have E a ( α,0), we can calculate a from Eq (5.8). As one might notice, b is singular at h0 if E a' ( α,0) 0. In order to avoid this singularity, Ea' ( α, ε ) can be used to reresent E ' a ( α,0) by using finite differencing at h ε where ε is a small value. Having obtained b 0, b andb, we can calculate φ (0) from Eq. (5.7) together with the definition (5.) and (5.). 3