One-sided finite-difference approximations suitable for use with Richardson extrapolation

Transcription

1 Journal of Computatonal Physcs 219 (2006) Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng, Indan Insttute of Technology, Kharagpur , West Bengal, Inda Receved 30 March 2006; receved n revsed form 12 May 2006; accepted 29 May 2006 Avalable onlne 10 July Abstract New expressons for one-sded fnte-dfference approxmatons are proposed. In these approxmatons the odd-order error terms are elmnated whle the even-order terms are left to be taken care of by Rchardson extrapolaton. The effectve local truncaton error s shown to be less than for hgher-order one-sded fnte-dfference approxmatons but the solutons for a test problem are shown to have comparable accuracy for both approxmatons. Ó 2006 Elsever Inc. All rghts reserved. Keywords: Fnte-dfference method; One-sded dfference approxmaton; Order of accuracy; Rchardson extrapolaton 1. Introducton A fnte-dfference approxmaton s one of the commonly used methods for numercal soluton of ordnary and partal dfferental equatons. The approxmatons most often used have second-order accuracy. The order of accuracy can be ncreased ether by usng hgher-order fnte dfference approxmatons or by Rchardson extrapolaton [2,3]. For nternal grd ponts, f central dfference approxmatons are used, the truncaton error has the form A 2 h 2 + A 4 h 4 + A 6 h 6 +, where h s the dstance between adjacent grd ponts and A 2,A 4,A 6,... are constants. Dong the computaton wth two or three dfferent grd szes and carryng out one or two extrapolatons to elmnate the error terms of second order or of second and fourth order we obtan a soluton wth fourth- or sxth-order accuracy. Snce each of the computatons uses a second-order-accurate fnte-dfference approxmaton there s no dffculty wth the boundary condtons for a Drchlet problem. However, a problem arses when the boundary condton nvolves a dervatve and we use a one-sded fnte-dfference approxmaton for the dervatve. By usng three grd ponts, ncludng one on the boundary, we can obtan a one-sded approxmaton whch s second-order-accurate [1]. The truncaton error has the form B 2 h 2 + B 3 h 3 + B 4 h 4 +. So f we carry out one extrapolaton to elmnate the second-order error term the result would have thrd-order rather than fourth-order accuracy. One way of obtanng fourth-order accuracy * Correspondng author. Tel.: ; fax: E-mal address: snb@mech.tkgb.ernet.n (S.N. Bhattacharyya) $ - see front matter Ó 2006 Elsever Inc. All rghts reserved. do: j.jcp

2 14 K. Rahul, S.N. Bhattacharyya Journal of Computatonal Physcs 219 (2006) s by usng a one-sded dfference approxmaton whch has a hgher order of accuracy. Such hgher-orderaccurate one-sded fnte-dfference approxmatons are known [1]. To obtan fourth-order accuracy, one could use a one-sded fnte-dfference approxmaton whch has fourth-order accuracy. Ths would nvolve fve grd ponts, ncludng one on the boundary. The addtonal two ponts are requred n order to elmnate the terms of the order of h 2 and h 3 n the truncaton error. However, f we are gong to use Rchardson extrapolaton t s not necessary to elmnate the order h 2 term n the truncaton error for the one-sded fnte-dfference approxmatons. So we can use just one more pont to obtan a one-sded dfference approxmaton whch has a truncaton error of the form B 2 h 2 + B 4 h 4 + B 5 h 5 +. After one extrapolaton, ths would gve a result whch has fourth-order accuracy. Snce ths approxmaton nvolves smaller number of grd ponts compared to the fourth-order approxmaton whch nvolves ponts farther away from the boundary, although both the methods have fourth-order accuracy, we expect the coeffcent of h 4 n the truncaton error to be smaller n the frst case. Contnung n ths manner, f we requre sxth-order accuracy, we can use one more pont to obtan a one-sded fnte-dfference approxmaton whch has a truncaton error of the form of B 2 h 2 + B 4 h 4 + B 6 h 6 + B 7 h 7 +. In ths study, we derve one-sded fnte-dfference approxmatons wth odd-order terms n the truncaton error elmnated and the even-order terms left to be taken care of by Rchardson extrapolaton. We compare the local truncaton error wth that of hgher-order one-sded fnte-dfference approxmatons. We then apply these approxmatons to a test problem and check the accuracy of the solutons obtaned. 2. One-sded fnte-dfference approxmatons Let us consder a functon (x) and defne a set of unformly spaced grd ponts x, =0,1,...,N, wth the grd spacng h = x +1 x. We represent (x )by. Smlarly 0 (x ), 00 (x ),..., are represented by 0 ; 00 ;..., where the prmes denote dfferentaton. We can wrte a Taylor seres expanson for around x k ¼ kh 0 þ k2 h k3 h þ k4 h 4 24 ðvþ k5 h ðvþ þ k6 h ðvþ : ð1þ From Eq. (1) we readly obtan frst-order-accurate one-sded fnte-dfference approxmatons [1] 0 ¼ ð 1 Þ þ OðhÞ; ð2þ h where the upper and lower sgns are to be used to the rght and to the left of x. Wrtng Eq. (1) for k = 1 and 2 and elmnatng 00 between them, we obtan second-order-accurate one-sded fnte-dfference approxmatons [1] 0 ¼ ð3 4 1 þ 2 Þ þ Oðh 2 Þ: ð3þ 2h We now derve an approxmaton for use wth Rchardson extrapolaton whch has truncaton error of the form B 2 h 2 + B 4 h 4 + B 5 h 5 +. Wrtng Eq. (1) for k = 1, 2 and 3 and elmnatng 00 and ðvþ between them we obtan the one-sded approxmatons 0 ¼ ð þ Þ þ h 2 1 6h h ðvþ þ: ð4þ Alternatvely, we can use a fourth-order-accurate one-sded fnte-dfference approxmaton obtaned by wrtng Eq. (1) for k = 1, 2, 3, and 4 and elmnatng 00, 000 and ðvþ, to obtan 0 ¼ ð þ þ 3 4 Þ þ h h 120 ðvþ þ: ð5þ When usng the approxmatons gven by Eq. (4), f we use Rchardson extrapolaton to elmnate the secondorder error term then the leadng order error s due to the fourth-order term. Effectvely the local truncaton error for ths approxmaton becomes ð11=120þh 4 ðvþ. If nstead of Eq. (4) we use Eq. (5) the local truncaton error s ð24=120þh 4 ðvþ. Thus the one-sded fnte-dfference approxmatons we have proposed, gven by Eq. (4), when used together wth Rchardson extrapolaton, effectvely gve a local truncaton error whch s

3 K. Rahul, S.N. Bhattacharyya Journal of Computatonal Physcs 219 (2006) φ =0 2 φ =1 2 φ =0 φ =0 φ =0 ( φ x ) x=1 = r( φ x ) x=1+ φ =0 Fg. 1. The test problem. slghtly less than half the local truncaton error when usng hgher-order one-sded approxmatons gven by Eq. (5). Contnung the logc used n dervng Eq. (4) we can derve one-sded fnte-dfference approxmatons whch have a truncaton error of the form of B 2 h 2 + B 4 h 4 + B 6 h 6 + B 7 h 7 +. These are 0 ¼ ð þ þ 4 Þ : ð6þ 20h When used together wth central dfference approxmatons for nternal grd ponts, f we carry out two Rchardson extrapolatons to elmnate the error terms of order h 2 and h 4, we can obtan a soluton wth sxth-order accuracy. Table 1 Soluton of the test problem usng a frst-order-accurate one-sded fnte-dfference approxmaton at the nterface N N N N=2 N=4 N2 N N=

4 16 K. Rahul, S.N. Bhattacharyya Journal of Computatonal Physcs 219 (2006) Applcaton to a test problem In ths secton, we use the fnte-dfference approxmatons derved n the prevous secton to compute the numercal soluton for a test problem and check the order of accuracy obtaned. The test problem s defned as r 2 ¼ 1000 for 0 < x < 1; 0 < y < 1; r 2 ¼ 0 for 1 < x < 2; 0 < y < 1; o ox x¼1 ¼ r o ox x¼1þ for 0 < y < 1; ¼ 0 for x ¼ 0or2; 0 6 y 6 1; ¼ 0 for 0 6 x 6 2; y ¼ 0or1: Ths s shown n Fg. 1. For most of our numercal calculatons we use r = 2. We defne a set of grd ponts (x j,y k ) where x j = jh for 0 6 x 6 2N, y k = kh for 0 6 x 6 N and h =1N. For the nternal grd ponts, except the grd ponts at the nterface x = 1, we use a second-order central dfference approxmaton ðr 2 Þ j;k ¼ j;k 1 þ j 1;k 4 j;k þ jþ1;k þ j;kþ1 : ð8þ h 2 The truncaton error for ths approxmaton has the form A 2 h 2 + A 4 h 4 + A 6 h 6 +. The boundary condton = 0 on the doman boundary s easly taken nto account. For the condton at the nterface, x = 1, we study the effect of usng the dfferent one-sded fnte-dfference approxmatons derved n the prevous secton. To ð7þ Table 2 Soluton of the test problem usng a second-order-accurate one-sded fnte-dfference approxmaton at the nterface N N N N=2 N=4 N2 N N,N2 N,N2 N=2;N=4 N=4;N=8 N2,N4 N=2 N;N=2 N=2;N=

5 begn wth we use the frst-order-accurate one-sded approxmatons gven by Eq. (2). The nterface condton n fnte-dfference approxmaton becomes N;j N 1;j h K. Rahul, S.N. Bhattacharyya Journal of Computatonal Physcs 219 (2006) ¼ r Nþ1;j N;j : ð9þ h The system of equatons gven by Eqs. (8) and (9) are solved numercally. The solutons at x = 0.25, 0.75, 1.25 and 1.75 and y = for dfferent number of grd ponts are shown n Table 1, where N represents the soluton obtaned usng N grd ntervals along each coordnate axs on each unt square. We observe that at x = 0.25 and y = for small number of grd ponts t almost appears that we have second-order accuracy but as we ncrease the number of grd ponts the order of accuracy s seen to decrease. Near the nterface x = 0.75 and 1.25, y = we observe close to frst-order accuracy. Further at we observe that the accuracy s even worse but, wth ncrease n number of grd ponts, appears to tend to frst order. Ths probably occurs because the soluton n the square on the rght s drven by the source term n the square on the left through the nterface condton. So an naccurate modellng of the nterface condton leads to naccurate soluton n the entre square on the rght, though why the order of accuracy at x = 1.75 seems worse than at x = 1.25 s not clear. Thus we fnd that although the approxmatons for the nternal grd ponts s second-order-accurate, snce we use a frst-order-accurate approxmaton for the condton at the nterface the soluton s only frst-order-accurate. We repeat the calculaton usng second-order-accurate one-sded approxmatons gven by Eq. (3) for the condton at the nterface. The results are shown n Table 2 and we clearly see second-order accuracy. We next carry out Rchardson extrapolaton to elmnate the h 2 term n the error. The extrapolated values are shown n Table 3 Soluton of the test problem usng at the nterface a second-order-accurate one-sded fnte-dfference approxmaton wth the thrd-order error term elmnated N N N,N2 N,N2 N=2;N=4 N=4;N=8 N2,N4 N;N=2 N=2;N=

6 18 K. Rahul, S.N. Bhattacharyya Journal of Computatonal Physcs 219 (2006) Table 4 Soluton of the test problem usng a fourth-order-accurate one-sded fnte-dfference approxmaton at the nterface N N N,N2 N,N2 N=2;N=4 N=4;N=8 N2,N4 N;N=2 N=2;N= the table as N,N2. Here N,N2 denotes the value obtaned by the extrapolaton usng N and N2. The rato of the dfferences appear close to 8 ndcatng thrd-order accuracy. If the nterface condton were not present we should have got fourth-order accuracy after one extrapolaton. In order to obtan fourth-order accuracy after one extrapolaton we use one-sded fnte-dfference approxmatons gven by Eq. (4). The results are shown n Table 3. We observe that after one extrapolaton we clearly obtan fourth-order accuracy. Next we use the approxmatons gven by Eq. (5) and the results are shown n Table 4. Agan after one extrapolaton we obtan fourth-order accuracy. Comparng Tables 3 and 4 we fnd that although the local truncaton error s smaller for the approxmatons n Eq. (4) the accuracy of the soluton for the test problem s smlar to that obtaned usng Eq. (5). However, Eq. (4) stll has the advantage that t s easer to derve. Next we use the approxmaton gven by Eq. (6) and the results are shown n Table 5. Here N,N2,N4 s the value obtaned by usng Rchardson extrapolaton to elmnate the order h 4 error term between N,N2 and N2,N4. After two extrapolatons the dfferences are very small and consequently, t s dffcult to show that we obtan sxth-order accuracy. However, we do observe that agreement to 9 sgnfcant fgures s obtaned wth reasonable number of grd ponts. Alternatvely f we were to use sxth-order-accurate one-sded fnte dfference approxmatons the algebra nvolved n dervng the expressons would be extremely tedous. Thus the advantage of Eq. (6) s that, when used together wth Rchardson extrapolaton, t provdes hgh accuracy wthout requrng very tedous algebra. So far the calculatons have been carred out for r = 2. We now check whether the one-sded approxmatons work well for larger values of r. The results for the test problem wth r = 10 computed usng Eq. (4) are shown n Table 6. Agan we fnd that ths provdes fourth-order accuracy after one Rchardson extrapolaton, as expected. The magntude of the errors are also comparable wth those for r =2.

7 K. Rahul, S.N. Bhattacharyya Journal of Computatonal Physcs 219 (2006) Table 5 Soluton of the test problem usng at the nterface a second-order-accurate one-sded fnte-dfference approxmaton wth the thrd- and ffth-order error terms elmnated N N N,N2 N,N2,N Table 6 Soluton of the test problem usng at the nterface a second-order-accurate one-sded fnte-dfference approxmaton wth the thrd-order error term elmnated and wth r = 10 N N N,N2 N,N2 N=2;N=4 N=4;N=8 N2,N4 N;N=2 N=2;N= (contnued on next page)

8 20 K. Rahul, S.N. Bhattacharyya Journal of Computatonal Physcs 219 (2006) Table 6 (contnued) N N N,N2 N,N2 N=2;N=4 N=4;N=8 N2,N4 N;N=2 N=2;N= Concluson In ths study, we have proposed some one-sded fnte-dfference approxmatons for use wth Rchardson extrapolaton. The essental logc s to use extra grd ponts to elmnate the odd-order terms n the truncaton error but leave the even-order terms to be elmnated by Rchardson extrapolaton. Usng a test problem we have demonstrated that the computed results have the order of accuracy we would expect. These one-sded fnte-dfference approxmatons, when used together wth Rchardson extrapolaton are shown to have smaller local truncaton error than the conventonal hgher-order one-sded fnte-dfference approxmatons. However, for the test problem for both approxmatons the solutons have comparable accuracy. The one-sded dfference approxmatons proposed n ths manuscrpt nvolve smaller number of grd ponts and are easer to derve than the conventonal one-sded approxmatons whch provde same order of accuracy. References [1] J.D. Anderson Jr., Computatonal Flud Dynamcs, McGraw-Hll, New York, [2] R.L. Burden, J.D. Fares, Numercal Analyss, Brooks-Cole, Pacfc Grove, CA, [3] G. Dahlqust, Å. Björck, Numercal Methods (Translated by N. Anderson), Prentce-Hall, Englewood Clffs, NJ, 1974.