Comparison of Second Order Numerical Techniques for Linear and Non-Linear ODEs

Transcription

1 Comparson of Second Order Numercal Technques for Lnear and Non-Lnear ODEs Davd Keffer Department of Chemcal Engneerng Unverst of Tennessee, Knoxvlle date begun: September 999 updated: October, 5 updated agan: Februar 8, 4 Table of Contents I. Purpose... II. Sngle Ordnar Dfferental Equaton... II.A. Second Order ethod for Generc ODE... 3 II.B. Second Order Implct ethod for Lnear ODE... 3 Opton. Solve for sequentall... 4 Opton. Solve for all smultaneousl... 4 II.C. Comparson of Second Order Explct and Implct ethods... 6 III. Sstems of Ordnar Dfferental Equatons... 7 III.A. Second Order ethod for a Sstem of Generc ODEs... 7 III.B. Second Order Implct ethod for a Sstem of Lnear ODE... 7 Opton. Solve for sequentall... 8 Opton. Solve for all (k ) smultaneousl... 8 Appx. atlab Subroutnes... Code A.. Heun s ethod ODE (heun_short)... Code A.. Implct ethod, Sequental Soluton ODE (lnode_o_seq short)... Code A.3. Implct ethod, Smultaneous Soluton ODE (lnode_o_sm short)... Code A.4. Heun s ethod n ODEs (heunn_short)... 3 Code A.5. Implct ethod, Sequental Soluton n ODEs (lnode_o_seq_n_short)... 4

2 I. Purpose The purpose of ths lecture package s not to ntroduce a new second order numercal soluton technque for solvng ODEs when we alread have a ver excellent fourth-second order numercal soluton technque for solvng ODEs. Rather, the purpose s to ntroduce these second order technques because the are used to solve PDEs. B frst applng the technques to ODEs, we shall see how the ft nto the faml of methods that nclude the Euler method and the Runge-Kutta method. When we then see the technque used for the soluton of PDEs, t won t seem so foregn. Addtonall, n ths document we compare mplct and explct second order algorthms. The mplct algorthms work onl for lnear ODEs. The explct algorthms are appled to an (nonlnear or lnear) ODEs. Agan, these mplct technques are not tpcall used to solve ODEs because f we have a lnear ODE we wll solve t analtcall. However, the lnear technques are used to solve lnear PDEs. We ntroduce them here, where the applcaton to ODEs s famlar and we can clearl compare the advantages and dsadvantages of mplct and explct technques. II. Sngle Ordnar Dfferental Equaton Consder the sngle frst-order ODE (ether lnear or nonlnear) d f (, t) () dt wth the ntal condton ( t o ) o () We can agan solve ths numercall, usng an Euler-lke formula ~ ( t t ) f (3) where ~ f s an approxmaton to the dervatve over the nterval from t to t. For a second order method, the approxmaton to the dervatve s smpl an average of the values at t and t. ~ f f, t, f t (4)

3 The problem s that we don t know the value of that appears on the rght hand sde of equaton (4). II.A. Second Order ethod for Generc ODE In the general case (meanng that equaton () s ether lnear or nonlnear n ), we can use the Euler method to approxmate for the purposes of equaton (4). ( t t ) f t, (5) We then substtute ths value nto equaton (4) and substtute the resultng value of f ~ nto equaton (3), eldng a second order method: ) ~ t, f t, ( t t ) f t ( t t ) f ( t t ) f, (6 Ths mplementaton of the second order method s a member of a class of methods known as predctor-corrector methods, because ou use Euler s method to predct and ou use equaton (6) to correct the value. Specfcall, equaton (6) s called Heun s method. A code that mplements equaton (6) s provded at the of these notes n appx A.. II.B. Second Order Implct ethod for Lnear ODE In obtanng equaton (6), we were forced to make an addtonal approxmaton, namel we had to use the Euler method for a prelmnar estmate of. If the ODE (eqn. ()) s lnear, then we do not have to make ths approxmaton n order to solve the problem. Therefore, an mplct method that does not make ths approxmaton wll be more accurate than Heun s method. When we appl ths technque to PDEs we wll be seekng to gan advantages n accurac wherever we can fnd them. We now derve the mplct method. Consder a lnear ODE of the form d dt f (, t) a( t) ( t) b( t) (7) wth the ntal condton of equaton (). We substtute ths lnear ODE nto equaton (4) obtanng ~ f a b a b (8) 3

4 where we have used the shorthand notaton that a a( t ), as we have done for and b as well. We substtute equaton (8) nto equaton (3) ( t t ) a b a b (9) We solve for the unknown,, obtanng A, A, B () where A, j s the coeffcent (n general a functon of the ndepent varable t) n front of the j th varable at the th tme, t, and has the form t t - a j f j A, j () t t -- a j f j ( t t ) B b b () Ths method s classfed as an mplct method because the value of the unknown appears on both the left hand sde and rght hand sde of equaton (9). Opton. Solve for sequentall We could proceed as we dd n the Euler method, where we evaluate at t, then use that result to generate at t, etc. through repeated applcatons of equaton (). Ths s totall legtmate and wll lead to the correct approxmaton of the soluton. A code that mplements the sequental soluton of ths mplct method s gven n appx A.. Opton. Solve for all smultaneousl Consder that we dvde the ndepent varable, t, nto n ntervals, each of sze ( t f to ) t (3) n where s the ntal tme from the ntal condton n equaton () and t f s the fnal tme, beond whch we are no longer nterested n the soluton of the ODE. Our approxmate soluton wll be evaluated at n+ ponts, the ntal condton and the n subsequent values of t. If we desgnate 4

5 the soluton of the ODE at each of these ponts as for = to n+, then we can wrte the followng set equatons: o for = A A B,, for = to n+ (4) Ths s a sstem of lnear algebrac equatons. It can be wrtten n matrx form as: A B (5) where the vector B s prncpall defned b equaton (), except for the frst entr whch s the ntal condton, B B Bn o (6) and where the matrx A s prncpall defned b equaton (), except for the frst row whch s the ntal condton, A A, A, A n A, n n, n (7) Thus, we have transformed the numercal soluton of a lnear ODE nto the soluton of a sstem of lnear algebrac equatons, whch we know how to solve. One nverson of ths matrx effectvel solve the ODE problem. The transformaton of ODEs to lnear algebrac equatons through the dscretzaton of the ndepent and depent varables s a commonl encountered transformaton. It s one that s used ubqutousl through-out the soluton of PDEs and Integral Equatons. Therefore, t was educatonal to ntroduce the concept here, even though we would lkel never use ths methodolog to solve a sngle lnear ODE as we dd here. Note that the sequental and smultaneous solutons of the mplct method eld (to wthn machne precson) the same result. The mplct method wll eld a more accurate soluton than Heun s method. 5

6 A code that mplements the smultaneous soluton of ths mplct method s gven n appx A.3. II.C. Comparson of Second Order Explct and Implct ethods Consder the ntal value problem: d a( x) b( x) dx where we have an ntal condton of the form: ( x x o ) o wth the specfc values gven b: a ( x), b( x) xsn(3x), ( x ) Ths IVP has the analtcal soluton nh 57 x e cos(3 ) x x 6x 5sn(3x) 39x We now ths ODE usng the three second-order methods descrbed above, Heun s method, the mplct sequental method and the mplct smultaneous method. For all methods, we set x f = 4 and use n = ntervals. Plots of the soluton and absolute error are shown below..5 x absolute error.5 Heun lnear, sequental lnear,smultaneous - analtcal Heun lnear, sequental lnear,smultaneous x x From the plot of the solutons, t s dffcult to see the dfference between the methods for ths tme step. From the plot of the error, one sees that the errors of the two mplct methods are 6

7 exactl the same and both are better than Heun s method. The wavness of the absolute error s due to the non-monotonc nature of the soluton. III. Sstems of Ordnar Dfferental Equatons A general sstem of frst order ODEs can be expressed as d dt f (, t) (8) wth the ntal condtons ( t o ) o (9) III.A. Second Order ethod for a Sstem of Generc ODEs In the general case (the equatons are ether lnear or nonlnear n, we can wrte the straghtforward mult-equaton analog of Heun s method. We agan use Euler s method to predct the value of the functon, p ~ t t ) f ( t t ) f (, t ) () ( We super-scrpted ths wth a p to dentf ths as the predcton step. The predcton step s followed b the correcton step ~ ( t t ) f ( t ) (, ) t f t p f (, t ) () Therefore, these technques are sometmes called predctor-corrector methods. As we dd n the sngle equaton case, we sequentall solve for each based on. A code that mplements equatons () and () s provded at the of these notes n appx A.4. Ths s essentall Heun s method exted to a sstem of ODEs. III.B. Second Order Implct ethod for a Sstem of Lnear ODE We can also repeat the dervaton for a more accurate second order method f ever equaton n the sstem of ODEs s lnear. In fact, f the equatons are lnear, we don t even need to make the approxmaton that we can solate the dervatves on the left hand sde of the equaton, as we dd above n the general soluton. If the equatons are lnear, the general sstem can be wrtten as C(t) d A(t)(t) B(t) () dt 7

8 Equaton () would be dentcal to equaton (9) f C (t) were the dentt matrx. In solvng a sstem of mass and energ balances, ths s frequentl the case. The dervaton of the method, now becomes more complcated. We need three ndces on the elements of A (t). The frst ndex ndcates the equaton. The second ndex ndcates the varable. The thrd ndex ndcates the tme step, once we have performed the dscretzaton of ( k ) tme necessar to obtan the numercal soluton. We wll use the notaton A, j to desgnate the coeffcent (the tme functonalt s now mplct) of the j th varable n the th equaton at (k ) dscretzed tme t k. Smlarl, A represents the entre matrx of coeffcents at dscretzed tme t k. If C s a constant matrx, we can dscretze equaton () as C ( k ) t k t ( k) k ( k) ( k) ( k) ( k ) ( k ) ( k ) A B A B (3) Once agan, ou can proceed to solve ths problem sequentall or smultaneousl. Opton. Solve for sequentall We can rearrange equaton (3) to solate our vector of unknowns t ( k ) ( k ) t ( k ) ( k ) t ( k ) ( k ) C A C A B B (4) Everthng on the rght-hand sde of equaton (4) s known. You nvert and solve for (k ) (k ). A code that mplements the sequental soluton of ths mplct method s gven n appx A.5. Opton. Solve for all (k ) smultaneousl We could wrte equaton (4) for all values of k from to n+ (for a dscretzaton of t nvolvng n ntervals). We then have a sstem of m ODEs, we now have a sstem of m(n+) lnear algebrac equatons. We could nvert ths larger matrx f we so chose. In solvng all of the equatons smultaneousl, we would have to create a sngle matrx of unknowns. We have some freedom as to how we choose to order our unknowns. Two possble arrangements for a sstem wth m equatons and n tme ntervals are shown below. 8

9 Y ( n ( n m m ( n m () () ) () () ) () () ) or () () Y (5) ( n) ( n ) In the frst arrangement, the unknowns are grouped b varable. In the second arrangement, the unknowns are grouped b tme. The latter form s preferable for two reasons. Frst, t reduces the bandwdth of the resultng matrx, whch equates to a qucker computaton. Second, t also facltates the mathematcal descrpton of the sstem. Here we assume that the vector of unknowns takes the form of the second arrangement. We then can wrte a sstem of m(n+) lnear algebrac equatons of the form A Y B (6) where, n order to present a compact descrpton of the matrx and vector n equaton (6), we frst rewrte equaton (4) as ( k ) ( k ) B k, k k, k k (7) where the matrx, k, k, and the vector, B k, are defned b comparson to equaton (4). The matrx and vector used n equaton (6) are related to the smaller matrces and vectors used n equaton (7) b the followng equatons: 9

10 ,, 3,3 3,,, n n n n I A (8) and 3 n o B B B B (9) Thus, we have shown how we can transform the numercal soluton of a sstem of lnear frstorder ODEs nto the soluton of a sstem of lnear algebrac equatons. A code that mplements the smultaneous soluton of ths mplct method s left as an exercse for the curous reader.

11 Appx. atlab Subroutnes In ths secton, we provde routnes for mplementng the varous optmzaton methods descrbed above that are not translated from Numercal Recpes. Note that these codes correspond to the theor and notaton exactl as lad out n ths book. These codes do not contan extensve error checkng, whch would complcate the codng and defeat ther purpose as learnng tools. That sad, these codes work and can be used to solve problems. As before, on the course webste, two entrel equvalent versons of ths code are provded and are ttled code.m and code_short.m. The short verson s presented here. The longer verson, contanng nstructons and servng more as a learnng tool, s not presented here. The numercal mechancs of the two versons of the code are dentcal. Code A.. Heun s ethod ODE (heun_short) functon [x,]=heun(n,xo,xf,o); dx = (xf-xo)/n; x = zeros(n+,); for = ::n+ x() = xo + (-)dx; = zeros(n+,); () = o; for = ::n x = x(); = (); k = funkeval(x,); x = x() + dx; = () + dxk; k = funkeval(x,); ddx =./.(k + k); (+) = () + dxddx; close all; plot = ; f (plot == ) plot (x,,'k-o'), xlabel( 'x' ), label ( '' ); fd = fopen('heun_out.txt','w'); fprntf(fd,'x \n'); fprntf(fd,'%3.5e %3.5e \n', [x,]'); fclose(fd); functon ddx = funkeval(x,); ddx = -.^; An example of usng heun_short s gven below.» [x,]=heun_short(,,,);

12 Ths program generates outputs n three forms. Frst, the x and vectors are stored n memor and can be drectl accessed. Second, the program generates a plot of vs. x. Thrd, the program generates an output fle, heun_out.txt, that contans x and vectors n tabulated form. Code A.. Implct ethod, Sequental Soluton ODE (lnode_o_seq short) functon [x,]=lnode_o_seq short(n,xo,xf,o); dx = (xf-xo)/n; x = zeros(n+,); for = ::n+ x() = xo + (-)dx; = zeros(n+,); () = o; for = ::n+ A =. -.5dxfunkeval_a(x()); Aj = dxfunkeval_a(x(-)); B =.5dx(funkeval_b(x()) + funkeval_b(x(-))); () =./A(B - Aj(-)); close all; plot = ; f (plot == ) plot (x,,'k-o'), xlabel( 'x' ), label ( '' ); % % wrte result to fle 'lnode_o_seq out.txt' % fd = fopen('lnode_o_seq out.txt','w'); fprntf(fd,'x \n'); fprntf(fd,'%3.5e %3.5e \n', [x,]'); fclose(fd); % ddx = a(x)(x) + b(x); functon a = funkeval_a(x); a = -x; functon b = funkeval_b(x); b = sn(x); An example of usng lnode_o_seq short s gven below. >> [x,]=lnode_o_seq short(,,,); Code A.3. Implct ethod, Smultaneous Soluton ODE (lnode_o_sm short) functon [x,]=lnode_o_sm short(n,xo,xf,o); dx = (xf-xo)/n; x = zeros(n+,); for = ::n+ x() = xo + (-)dx; = zeros(n+,); A = zeros(n+,n+); B = zeros(n+,);

13 A(,) =.; B() = o; for = ::n+ B() =.5dx(funkeval_b(x()) + funkeval_b(x(-))); A(,) =. -.5dxfunkeval_a(x()); A(,-) = dxfunkeval_a(x(-)); nva = nv(a); = nvab; close all; plot = ; f (plot == ) plot (x,,'k-o'), xlabel( 'x' ), label ( '' ); fd = fopen('lnode_o_sm out.txt','w'); fprntf(fd,'x \n'); fprntf(fd,'%3.5e %3.5e \n', [x,]'); fclose(fd); % ddx = a(x)(x) + b(x); functon a = funkeval_a(x); a = -x; functon b = funkeval_b(x); b = sn(x); An example of usng lnode_o_sm short s gven below. >> [x,]=lnode_o_sm short(,,,); Code A.4. Heun s ethod n ODEs (heunn_short) functon [x,]=heunn_short(n,xo,xf,o); dx = (xf-xo)/n; x = zeros(n+,); for = ::n+ x() = xo + (-)dx; m=max(sze(o)); = zeros(n+,m); (,:m) = o(:m); ddx = zeros(,m); temp = zeros(,m); k = zeros(,m); k = zeros(,m); for = ::n x = x(); temp(:m) = (,:m); k(:m) = funkeval(x,temp); x = x() + dx; temp(:m) = (,:m) + dxk(:m); k(:m) = funkeval(x,temp); ddx(:m) =./.(k(:m) + k(:m)); (+,:m) = (,:m) + dxddx(:m); 3

14 close all; plot = ; f (plot == ) for = ::m color_ndex = get_plot_color(); plot (x(:),(:,),color_ndex); hold on; hold off; xlabel( 'x' ); label ( '' ); leg (ntstr([:m]')); fd = fopen('heunn_out.txt','w'); fprntf(fd,'x ()... (m) \n'); for = ::n+ fprntf(fd,'%3.5e ', x()); for j = ::m fprntf(fd,'%3.5e ', (,j)); fprntf(fd,' \n'); fclose(fd); functon ddx = funkeval(x,); ddx() = -.() -.() -.5(3)^; ddx() = -.() - 4.() -.5(3); ddx(3) = -.5() -.4() -.(3); An example of usng heunn_short s gven below. >> [x,]=heunn_short(,,,[,,]); Code A.5. Implct ethod, Sequental Soluton n ODEs (lnode_o_seq_n_short) functon [x,]=lnode_o_seq_n(n,xo,xf,o); dx = (xf-xo)/n; x = zeros(n+,); for = ::n+ x() = xo + (-)dx; m=max(sze(o)); = zeros(n+,m); (,:m) = o(:m); for k = ::n+ AA = zeros(m,m); BB = zeros(m,); for = ::m BB() =.5dx(funkeval_b(,x(k)) + funkeval_b(,x(k-))); for j = ::m AA(,j) = funkeval_c(,j,x(k)) -.5dxfunkeval_a(,j,x(k)); term = funkeval_c(,j,x(k-)) +.5dxfunkeval_a(,j,x(k-)); BB() = BB() + term(k-,j); nvaa = nv(aa); 4

15 (k,:m) = nvaabb; close all; plot = ; f (plot == ) for = ::m color_ndex = get_plot_color(); plot (x(:),(:,),color_ndex); hold on; hold off; xlabel( 'x' ); label ( '' ); leg (ntstr([:m]')); fd = fopen('lnode_o_seq_n_out.txt','w'); fprntf(fd,'x ()... (m) \n'); for = ::n+ fprntf(fd,'%3.5e ', x()); for j = ::m fprntf(fd,'%3.5e ', (,j)); fprntf(fd,' \n'); fclose(fd); % c(x)d_/dx = A(x)_ + b(x); functon aout = funkeval_a(,j,x); amat = [ ; - -]; aout = amat(,j); functon bout = funkeval_b(,x); bvec = [/(+x); ]; bout = bvec(); functon cout = funkeval_c(,j,x); cmat = [ ; ]; cout = cmat(,j); An example of usng lnode_o_seq_n_short s gven below. >> [x,]=lnode_o_seq_n_short(,,,[,]); 5