April 1980 TR/96. Extrapolation techniques for first order hyperbolic partial differential equations. E.H. Twizell

Transcription

1 TR/96 Apri 980 Extrapoatio techiques for first order hyperboic partia differetia equatios. E.H. Twize

2 W96086

3 (0) 0. Abstract A uifor grid of step size h is superiposed o the space variabe x i the first order hyperboic partia differetia equatio u/ t a u/ x = 0 (a > 0, x > 0, t > 0). The space derivative is approxiated by its backward differece ad cetra differece repaceets ad the resutig iear systes of first order ordiary differetia equatios are soved epoyig Padé approxiats to the expoetia fuctio. A uber of differece schees for sovig the hyperboic equatio are thus deveoped ad each is extrapoated to give higher order accuracy. The schees, ad their extrapoated fors, are appied to two probes, oe of which has a discotiuity i the soutio across a characteristic.

4

5 (). The extrapoatios Give the firs t order hyperboic partia u u (0) t a x = 0 ; a>0, x>0, t>0 differetia equatio with iitia coditios u(x,0) = g(x) ad boudary coditios u(0,t)= v(t), repacig the space derivative with the backward differece forua u x = {u(x,t) u(x-h,t)}/h 0(h) eads to the syste of first order ordiary differetia equatios d () dt = - aa av t where (t) = [ (t), (t),..., N (t)] T, T deotig traspose, is the vector of coputed soutios of () at tie t > 0 at N poits whose x coordiates are x i = ih (i =,,..,N). Ceary that part of the x - axis for which the soutio is sought is divided ito N equa parts of width h. I equatio () A is a square atrix of order N give by ad v t is a vector with N copoets give by hv t = [v t,0,0,...,0] T where v t is the uerica (froze) vaue of the boudary coditio at tie t= (=0,,...) ad is a costat tie step. The soutio of () is () (t) = A - v t exp(-ata){g - A - v t }, where g is the vector of iitia vaues, ad it is easy to show

6 () that (t) () satisfies (t) = A v t exp(-aa){(t) A v t }. sig the (0,) Padé approxiat to the expoetia atrix fuctio i () eads to () (t) = A v t (I-aA) {(t) A v t } O ( ).. where I is the idetity atrix of order N. Appyig equatio () to the poit (h,) i the (x,t) pae, ad usig to deote the coputed vaue of u(h,), gives the three poit expicit schee (5) = (-ap) ap where p= /h. This schee appears i Mitche[;p.6] ad is kow to be first order accurate i tie ad to be stabe for 0 ap. as foows: (6) The schee ay be extrapoated to give secod order accuracy Writig () over a doube iterva gives () (t) = A v t (I aa){(t) A v t } O( ) ad writig () over two sige tie itervas gives (7) () (t) = A v t (I aa) (I aa){(t) A v t } O( ) Expadig the atrix products i (7) ad coparig (6) ad (7) with the Macauri expasio of the atrix expoetia ter i () writte over a doube tie step, that is () (8) (t ) = A v {I - a A a A - a A t a A - a 5 5 A }{(t) - A - v t }, () it is see that, Whist () ad are each oy first order (E) accurate, the extrapoated soutio defied by (E) () () (9) = - O ( ) is see to be secod order accurate whe copared with (M) (t).

7 () sig the (,0) Padè approxiat to the expoetia atrix fuctio i () gives (0) (t ) = A - v (I a A ) { (t) - A - v } O( t t ) Writig (0) i ipicit for ad repacig v t with the N-copoet vector v t = [v t, 0,.,0] T, where v t is the uerica vaue of the boudary coditio at tie t, gives () ( I a A)(t ) - av t (t) Appyig () to the poit (h,) gives the three poit, ipicit schee () ( ap) - ap = (see Mitche [ ;p.65]) which is aso first order accurate. The schee is ucoditioay stabe ad because of the for of the iitia ad boudary coditios ay be used expicity i the first quadrat of the (x,t) pae. The schee ca be extrapoated to give secod order accuracy: equatios (6) ad (7) becoe () () (t ) = A - v (I a A) - {(t) - A - v O( t t } ) ad () () (t ) = A - v (I a A) - (I a A) - {(t)-a - v O( t t } ) ad it ay be show that the extrapoated soutio (E) defied by (9) is secod order accurate. sig the (, ) Padè approxiat to the expoetia atrix fuctio i () gives - v t (5) (t ) = A (I aa) (I- aa)-{(t)-a v t } O( ) Writte ipicity, this eads to (6) (I a A)(t ) - aav t (I - aa) (t) av t, = - - which, whe appied to the esh poit ipicit schee (h,), gives the four poit

8 () (7) ( ap) - ap ( - ap) ap =. A stabiity aaysis shows that this ew schee is ucoditioay stabe. It, aso, ay be used expicity i the first quadrat because of the for of the iitia coditios ad boudary coditios Extrapoatio of the schee to give two ore powers of accuracy ay be achieved by writig (8) () (t - - (I - a A) { (t) - A - ) = A v t (I a A) v t } O( ) ad (9) () (t ) = A - v (I a A ) (I - a A ) (I a A ) t (I - a A ). { (t) - A - v t } O( ) ad defiig (0) (E) = () () O( 5). which is see to be fourth order accurate whe copared with (M) Suppose ow that istead of usig the backward differece repaceet of the space derivative, the cetra differece repaceet u = {u(x h, t) u(x h, t)}/h O(h ) x is used. Equatio () is the repaced by the syste d () = ab a w dt t where B is the atrix give by ad w t = [v,0,...0, ] T h t N is a N-copoet vector whose

9 (5) f i r s t eeet is the uerica (froze) vaue of the boudary coditio at tie t = (=0,,...) ad whose ast eeet is ius the vaue of the soutio at the poit ( ( N ) h, t ). This eas that kowedge of the soutio is required o soe "boudary" beyod that part of the x-axis uder cosideratio. A brief accout of how to dea with this difficuty is give by Mitche [;pp.67-68, together with further refereces therei. The soutio of () is () (t) = B w exp( atb){ g B t w t } which satisfies the reatio () (t ) = B w exp( a B){(t) B t W t }, ad repacig the expoetia atrix fuctio with its approxiat gives (0,) Pade () (t ) = B w (I a B){(t) B wt} O( t ). Appyig () to the poit (h,) gives the schee (5) = ap( ). A stabiity aaysis shows that schee (5) is ucoditioay ustabe ad woud ot be used. sig the (,0) Fade approxiat to the expoetia atrix fuctio i () gives (6) (t ) = B w (I a B) {(t) B w O( t t } ). W r i t i g (6) i ipicit for gives I a B) (t ) a w t = (t), (7) Where w = [v,0,...,0, ] T N, t h t ad appyig (7) to the poit (h.) gives the ew four poit, two eve, ipicit schee (8) ap( ) = which ay be show to be ucoditioay stabe.

10 (6) Cosiderig (6), (6) over a doube tie step to give (9) () (t ad over two sige tie steps to give (0) () the schee ay be extrapoated by first writig ) = B w (I a B) { (t) B t w t } O( ). (t ) = B w (I a B) (I a B) { (t) B w O( t t } ) The Macauri expasio (M) (t) of the expoetia atrix fuctio i () writte over a doube tie step is give by () (M) (t ) = B w t {I a B a B 6 a B a a 5 5 B 5...}{ (t) B w t }. 0 Expadig the atrix iverses i (9) ad (0) ad defiig (E) (t ) as i equatio (9), that is by (E) = () () O( ) it is cear that the first order schee give by (6) has bee extrapoated to give secod order accuracy. sig the (, ) Padé approxiat to the expoetia atrix fuctio i () gives () (t ) = B w (I a B) (I a B){(t) B w O( t t } ) which, whe writte ipicity, gives () (I a B) (t ) aw (I a B) (t) t a w t. = Appyig () to the poit (h,) gives the Crak-Nicoso type Ipicit schee () ap ap = ap ap which is kow to be ucoditioay stabe (Mitche [;p.67] ) but which aso requires kowedge of the soutio at the "boudary" x = ( N ) h (Mitche [;pp ] ). To extrapoate the schee, () is writte over a doube tie step ad over two sige tie steps givig, respectivey, (5) (6) () (t a B){ (t) B ) = B w t (I a B) (I w t } () (t ) = B w (I a B) (I a B)(I a B) t (I O( ). a B). { (t) B w O( t } ). The extrapoated soutio defied by (0) icreases the accuracy to fourth order. B

11 (7). Nuerica resuts To exaie the behaviour of the schees deveoped ad extrapoated i sectio, each was appied to equatio (0) with a =, that is u t u t = 0; with two differet sets of iitia coditios ad boudary coditios: x > 0, t > 0 Probe : u(x,0) = x, u(0,t) = t. The theoretica soutio of Probe i the first quadrat of the (x,t) pae was take to be u(x,t) = x - t, x t u(x,t) = t - x, x < t so that there exists a discotiuity i the soutio across the ie t = x i the (x, t) pae. Probe : u(x,0) = e x, u(0,t) - e t. The theoretica soutio of Probe i the first quadrat of the (x,t) pae was take to be u(x, t) = e x-t, x t u(x,t) = e t-x, x < t Each probe was ru with h = = 0. ad for foruas (8) ad () the theoretica vaue of the soutio aog the ie x=. was assued. For a = ad p - forua (5) reduces to = ; this schee propagates the iitia ad boudary coditios aog the characteristics ad wi ot be discussed further. Equatios () ad () have bee discussed previousy i the iterature (see, for exape, Mitche [;pp.65,67]) ad the behaviour of schees (7) ad (8), together with their respective extrapoated fors, are depicted i Fig; for Probe I ad i Fig:

12 (8) for Probe for 0 x.0 at tie t=0.. The two schees are see to behave i uch the sae aer whe appied to both probes ad the proxiity of each to the theoretica soutio is i agreeet with the theory of sectio I. The error odui for t = 0.(0.).0 at the poit x=0.5 usig a four schees give by foruas (), (7),(8),(), together with the error odui of their respective extrapoated foruatios, are give i Tabe for Probe ad i Tabe for Probe. I the case of Probe it was foud that a ethods icurred their worst errors at poits i the viciity of the discotiuity of the soutio across the ie t = x ; this pheoeo aso occurred with Probe which does ot have a discotiuity i the soutio but does have discotiuities i the first derivatives across this ie. It was geeray foud that a ethods behaved as predicted i sectio. I particuar, the ove forua (7) which is of the sae order of accuracy as the Crak-Nicoso type ethod (), was foud to icur saer errors with icreasig tie; this forua is aso superior i that it is easier to use as it ay be appied expicity, ad i that it does ot require kowedge of the soutio at a poit outside the iterva of x which is uder discussio. For just the sae reasos the we kow forua () ust be regarded as superior to forua (8). It is easiy see fro the figures ad tabes that extrapoatio of each schee to iprove accuracy is we justified.

13 (9) Theoretica soutio Forua (7) Extrapoated (7)... Forua (8) Extrapoated (8) Figure

14 (0) Theoretica soutio Forua (7) Extrapoated (7)... Forua (8) Extrapoated (8) Figure

15 () Tabe : Error odui for Probe at x=0.5 for t - 0,(0.).0 usig the schees give by foruas (),(7),(8),() ad their extrapoated foruatios. Method Tie Forua () Extrapoatio 8. (-).0(-).(-).0(-) 5.5(-).(-).7(-).(-).(-).0(-) Forua (7) Extrapoatio.0(-).7(-).7(-).(-).(-).(-).(-I).(-).(-).(-) Forua (8) Extrapoatio.8(-) 6.7(-).(-).7(-).8(-).8(-) 6.7(-).9(-) 9.(-) 7.6(-) Forua () Extrapoatio.(-).7(-) 5.6(-) 5.(-).7(-).8(-) 9.(-).(-).(-).5(-) Theoretica soutio

16 () Tabe : Error odui for Probe at x = 0. 5 for t = 0.(0.).0 usig the schees give by foruas (), (7), (-8), () ad their extrapoated foruatios. Method Forua () Extrapoatio Tie (-).0(-).8(-).0(-).(-).0(-).(-) 5.0(-).(-).6(-) Forua (7) Extrapoatio.0(-).0(-) 8.(-).6(-).(-).(-) 6.0(-).0(-).6(-).(-) Forua (8) Extrapoatio.(-).8(-).0(-).0(-).(-).0(-).(-).8(-).(-).7(-) Forua () Extrapoatio.(-).0(-) 5.(-).7<-).(-).0(-).9(-).0(-).8(-) 9.0(-) Theoretica soutio Referece. A.R. Mitche, Coputatioa ethods i partia differetia equatios, Wiey, Lodo, 969. XB 566