Solutions Numerical Simulation - Homework, 4/15/ The DuFort-Frankel method is given by. f n. x 2. (a) Truncation error: Taylor expansion

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1 Solutios Numerical Simulatio - Homework, 4/15/ The DuFort-Frakel method is give by +1 1 = αl = α ( 2 t (a Trucatio error: Taylor epasio i +1 t = m [ m ] m= m! t m. (1 i Sice we cosider the diusio equatio we eed to carry the Taylor epasio i time oly up to hal the order o the spatial derivative. Cosiderig terms up to the 4th or 5th derivative i space it is suiciet to go up to the 2d derivative i time. Note: or the correspodig epasio vor the covectio or trasport equatio we eed to carry the Taylor epasio i time as ar as we carry the epasio i space. Term 1 i the equatio: t = 1 { (1 1 + (1 + 1 t 2 t t = t + t2 3 6 t 3 + O( t 4 + (1 1 t2 2 2 t3 3 } + (1 + 1 t2 6 t Term 2 i the equatio (we separate the temporal ad spatial variatios o the right side: = 1 { 2 ( (1 1 t t = 1 2 Term 3 also o the right side: ( 2 + t 2 2 t 2 + O( t 4 + (1 + 1 t = 1 { 2 ( ( ( ( = 1 ( Combiig the right side terms ad usig 5 + ( } 5 2 t3 3 + (1 1 t2 6 t ( t = α 2 2 ad 2 t 2 = 4 α2 4 1

2 yields α ( = α 2 = α [ ( = α α ] t2 t 2 4 t2 4 α ( s2 4 Compiig all terms yields t α 2 ( α 2 12 s = with the error ( 1 E = α s2 4 (b Ampliicatio actor Re-aragig terms i the algebraic equatio (see problem 31 yields Divisio by with g = +1 / g +1 = 2s ( s + 1 2s 1 2s 1 (ep(ik + ep( ik g = or Solvig this equatio or g: g 2 4scosk g 1 2s = c Stability: g 2 4scosk g + ( 2scosk 2 = 1 2s 2 = ( g 2scosk + ( 2scosk 2 1 ( 1 4s 4 ( 2 + 4s 2 cos 2 k ad g = 2scosk ± 1 1 4s 2 si 2 k α First cosider 4s 2 si 2 k 1 such that the argumet o the square root is positive. Assumig cosk > we ca cosider ust the + sig ad cosider 2scosk s 2 si 2 k 1 2

3 => 2scosk + 1 4s 2 si 2 k This is obvious because 1 4s 2 si 2 k 1 ad 2scosk 2s. Cosiderig cosk ad choosig - sig yields the requiremet i.e., the same coditio as the oe above. 1 2scosk 1 1 4s 2 si 2 k β Cosider 4s 2 si 2 k > 1 reders the argumet o the square root egative sucht that the square root is purely imagiary. Thus the requiremet or stability is g 2 = gg = (2scosk 2 + 4s 2 si 2 k 1 ( 2 = 4s2 1 4s 2 + 4s + 1 < 1 Thus the scheme is ucoditioally stable. 3

4 37. Implemetatio o the DuFort-Frakel ad the Hopscotch (1D schemes i program sim1: First derive the algebraic equatios eeded or the DuFort-Frakel ad the Hopscotch scheme ad eplai what chages you eed to apply to sim1. or the implemetatio. Carry out the implemetatio. Test the schemes irst or a set o stadard parameters used or the FTCS scheme with a grid umber o 21 poits ad compare the result to the FTCS. The use icreasigly larger time steps or the ew schemes ad report your observatios. Pay attetio to the act that the time steps should be chose such that the ial time is i act the same as or the FTCS scheme. DuFort-Frakel - Algebraic equatio: +1 1 = αl = α ( 2 t multiplicatio with 2 t ad deiitio s = α t/ 2 yields => ( +1 ( +1 1 = 2s ( = (1 2s or +1 ad the Hopscotch (1D schemes i program sim1: Hopscotch: 1st stage Equatio to solve: = 1 2s ( t = αl + = eve +1 = (1 2s + s ( d stage: Equatio to solve: +1 t = αl +1, + = odd +1 = 1 + s ( (a Implemetatio o the two schemes: The two schemes are implemeted usig the code modiied or the two level scheme. The ew program is called sim1mul. ad associated chages are i the data ile sim1m.dat ad the iclude ile sim1mi. Chages are: Itroductio o the subrouties: Subroutie itdu (DuFort-Frakel 4

5 subroutie itdu iclude sim1mi iteger i real s,s1 s = (1.-2.*s/(1.+2.*s s1 = 2.*s/(1.+2.*s do 1 i = 1, 1 oold(i = old(i do 2 i = 1, 2 old(i = (i do 3 i = 2,-1 3 (i = s*oold(i + s1*( old(i+1+old(i-1 retur ed Subrouties ithop1, ihop2 (2stages o the hopscotch subroutie ithop1 iclude sim1mi iteger i real s s = 1.-2.*s do 2 i = 2+mod(t,2,-1,2 2 (i = s*(i + s*( (i+1+(i-1 retur ed subroutie ithop2 iclude sim1mi iteger i real s,s1 s = 1./(1.+2.*s s1 = s/(1.+2.*s do 2 i = 2+mod((t+1,2,-1,2 2 (i = s*(i + s1*( (i+1+(i-1 retur ed I additio a ew varible meth is itroduced to select ad keept track o the method used i the scheme. meth is added to sim1m.dat ad sim1mi ad also writte to the biary output ile to prit the method whe a plot is geerated by sim1.pro (with correspodig small chages. The mai program o sim1mul. is maily modiied as ollows (istead o callig ust the FTCS or the two-level schemes: i (meth.eq.2.or. meth.eq.4 the t = t+1 time = time + dt call ittcs call bdco i (mod(t,out.eq. call out 5

6 edi do while( (t.lt. tma.ad. + (time.lt.(tma-.1*dt t = t+1 time = time + dt write(*,* time, time i (meth.eq.1 call ittcs i (meth.eq.2 call itdu i (meth.eq.4 call it2l i (meth.eq.3 the call ithop1 call bdco call ithop2 edi call bdco i (mod(t,out.eq. call out ed do (b Test o schemes ad compariso with FTCS: The ollowig plots show DuFort-Frakel, Hopscotch, ad FTCS results or s = DuFort-Frakel Hopscotch FTCS N = 21 N = 21 N = 21 5 Dt =. s =.3 5 Dt =. s =.3 5 Dt =. s =.3 RMS E =.5355 RMS E =.315 RMS E =.3417 Note that diereces are small sice the rms error is small. Note that the Hopscotch scheme is remarkable better tha the other two. (c Selected results or icreasig itegratio steps: The et plots show results or the DuFort-Frakel, Hopscotch or s = 1. (where the FTCS is already ustable 1 1 DuFort-Frakel hopscotch N = 21 N = 21 5 Dt =. s = 1. 5 Dt =. s = 1. RMS E = RMS E =.5117 s = 2.: 6

7 1 1 DuFort-Frakel hopscotch N = 21 N = 21 5 Dt = 5. s = 2. 5 Dt = 5. s = 2. RMS E = RMS E = s = 4.: DuFort-Frakel N = 21 Dt = 1. s = 4. RMS E = hopscotch N = 21 Dt = 1. s = 4. RMS E = The results show that ideed both the DuFort-Frakel ad the Hopscotch scheme are stable or all chose values o s. However, the error icreases although it is comparable betwee the two methods. Note that the choice s = 4. yields t = 1 such that the result or = 21 used oly 6 itegratio steps which is isuiciet to propagate the diusio across all grid poits. While the DuFort-Frakel scheme teds to develop grid oscillatios or large time steps the Hopscotch is smooth but teds to uderestimate the actual solutio. The lack o temporal resolutio appears to imply that the results or s = 4 is ot really acceptable. However, the ollowig plots show cases or s = 4 ad s = 8 but o a 161 grid: s = 4., = 161: 1 1 DuFort-Frakel Hopscotch N =161 N =161 5 Dt = 15.6 s = 4. 5 Dt = 15.6 s = 4. RMS E =.1438 RMS E = s = 8., = 161: 7

8 1 1 DuFort-Frakel Hopscotch N =161 N =161 5 Dt = 31.3 s = 8. 5 Dt = 31.3 s = 8. RMS E = RMS E = The result or s = 4., = 161 shows that a larger choice o s is possible i the grid is suicietly reied (thus lowerig particularly the error associated with a large t. This implies that the advatage o a larger s value becomes importat or a suicietly small grid spacig. This may be required i boudary layer problems, problems where the diusio coeiciet strogly varies, or equatios where covectio ad diusio are importat. 8