Salmon: Lectures on partial differential equations. 3. First-order linear equations as the limiting case of second-order equations

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1 3. First-order liear equatios as the limitig case of secod-order equatios We cosider the advectio-diffusio equatio (1) v = 2 o a bouded domai, with boudary coditios of prescribed. The coefficiets ( ) (2) v = u 1 ( x),u 2 ( x),k,u ( x) are give fuctios which we regard as the compoets of a velocity v. ( / x 1,K, / x ). We assume, o physical grouds, that this problem is well-posed; the solutio exists ad is uique. We are iterested i the case, i which the solutios of (1) must, i some sese, approach the solutios of the first-order equatio (3) v = studied i the previous sectio. The characteristics of (3) the streamlies of v are called subcharacteristics of (1). Cosider the case i which these subcharacteristics itersect the boudary: I that case, the limit must be subtle, because the theory of (3) tells us that we caot prescribe at 2 poits o the same charcteristic. What really happes as? Cosider the oe-dimesioal example, (4) u x = xx, 1 < x <+1 ( 1)= a, ( +1) = b I the case u=costat, the solutio is ux / (5) = A + Be where the costats A ad B are determied by the boudary coditios. We fid that 3-1

2 (6) A = aeu/ u / be, B = e u/ e u / b a e u / u/ e The ature of the solutio depeds very much o the sig of u. If u>, e u/ ad e u /, so A a, B (b a)e u/ ad (7) a + ( b a)e u ( x 1) The solutio looks like this: If, o the other had, u<, the A b, B (a b)e u/, ad (8) b + ( a b)e u ( x +1) Thus, i both cases, as the solutio approaches the solutio of u x = (amely =costat) almost everywhere, with the costat chose to satisfy the boudary coditio at the iflow boudary. A boudary layer of thickess occurs at the outflow boudary. I the boudary layer, chages rapidly to satisfy the outflow boudary coditio. Next we cosider the slightly more iterestig cases of u =±x. I the case u=x of divergig flow, x x = xx x = l x x x2 = l x + cost (9) = A e y2 / dy + B. x 3-2

3 Oce agai we choose A ad B to satisfy the boudary coditios, fidig that (1) A = 1 b a 2 e y 2 / dy, B = a + b 2. Thus (11) = b a 2 where Rx, ( )+ a + b 2 (12) Rx, ( )= x 1 e y2 / dy e y2 / dy (u =+x) As, the ratio R is tiy except ear x =±1. Thus, throughout the iterior of the domai, is uiform at the average of the 2 boudary values. Boudary layers occur at both ( outflow ) boudaries. The solutio looks like this: I the case u = x of covergig flow, we obtai the solutio simply by replacig by i (12). Thus the solutio of (4) with u = x is (11) with (13) Rx, ( )= x 1 e y 2 / dy e y 2 / dy (u = x) As R chages rapidly from 1 to +1 as x passes through zero. The solutio looks like this: 3-3

4 There are o boudary layers at x =±1, but there is a iteral boudary layer at x=. As (11,13) may, by a chage of variables, be writte as (14) = ( b a) 2 x / 1/ 2 e y 2 /2 dy e y 2 /2 dy + ( a + b) 2 (u = x) Thus the iteral boudary layer has a thickess 1/ 2. I the divergig (u=x) case, the boudary layer thickess is, the same as for the case of costat u. This follows from (12). These thickesses could also be obtaied by a crude balace of terms. The emergig picture is oe i which the limit correspods to a solutio of v = almost everywhere, with the value of o each streamlie (i.e. subcharacteristic) equal to the value at the boudary poit of iflow. The diffusio xx is egligible outside of the boudary layers, but it determies which boudary iflueces the iterior. That is, the diffusio determies the directio i which the boudary iformatio travels alog subcharacteristics. Whe = either directio is preferred, ad the boudary value problem is simply ill-posed. The iteral boudary layer of example (11,13) correspods to the eardiscotiuity that develops whe iformatio flows to the same poit from widely differet locatios o the same boudary. This also happes i 2 or more dimesios. However, i 2 or more dimesios ew pheomea occur: 1.) The boudary ca coicide with a subcharacteristic. 2.) Closed subcharacteristics ca occur withi the domai. We illustrate the first of these with a example dear to the hearts of oceaographers: Stommel s theory of ocea circulatio. I this example, it is especially importat to distiguish betwee the velocity v ad the real, physical velocity to which the problem refers. Example. The equatios of motio for a rotatig, two-dimesioal fluid are 3-4

5 (15) u t v t p x fv = u + F x p + fu = v + Fy y u x + v y = where (u,v) is the fluid velocity i the (east, orth) directio, p is the pressure, is a drag coefficiet, ad ( F x, F y ) is the prescribed wid force. The Coriolis parameter (16) f = 2 si latitude day ( ) equals twice the local vertical compoet of the Earth s rotatio vector. As a rough approximatio, f = f + y. To satisfy the last of (15) we take (u,v) = ( y, x ). The, elimiatig p betwee the first ad secod of (15) we obtai the vorticity equatio (17) t 2 + x = 2 + W x, y ( ) where W = F y / x F x / y is a give fuctio. We seek steady solutios of (18) x = 2 W i a square ocea <x,y<l. The boudary coditio is =. Now forget everythig you kow about the physics of (18). The importat mathematical fact is that (18) fits the geeral form of a advectio-diffusio equatio, (19) v = 2 + Q with velocity v = (,) ad source Q = W. Thus the subcharacteristics are lies of costat y, ad boudary iformatio flows from east to west. A boudary layer of thickess must occur at the wester ( outflow ) boudary. 3-5

6 (2) x Outside the boudary layer, is egligible, ad = W ( x, y ), correspodig to the characteristic equatios, (21) dx ds =, dy ds =, d ds = W ( x, y ) which imply x (22) = 1 W( x',y)dx' I x, y L ( ) which is called the iterior solutio. Note that the costat of itegratio i (22) has bee chose to satisfy the boudary coditio at x=l. I the wester boudary layer we let = I + W, where W is the wester boudary layer correctio. W approximately obeys (23) w x = 2 w x 2, the boudary coditio W = I at x=, ad the matchig coditio W as x /. The solutio is (24) W = I (, y)e x /. If W( x,), there must also be a souther boudary layer, because i that case the iterior solutio (22) does ot satisfy the boudary coditio at y=. From a more mathematical viewpoit, the souther boudary coicides with a subcharacteristic, alog which it is forbidde to prescribe whe =. Let = I + S i the souther boudary layer. The (25) S x = 2 S y 2 with boudary coditio (26) S = I ( x,) o y = ad iitial coditio (27) S ( L,y)= Except for the boudary at y=l, which is too distat to affect the solutio for S, the problem (25-27) is equivalet to the problem of heat diffusio i a semi-ifiite bar with a 3-6

7 prescribed temperature at the ed of the bar. The solutio was give by eqs (52-53) i Sectio 1. The aalogy is betwee: (28) S, L x t, Thus, y x, I g (29) S ( x, y)= dx I x, L x [ ( )] y 4 x x exp ( ) 3/2 y 2 ( ) 4 x x From this we see that the boudary layer thickess is proportioal to ( L x). The solutio is summarized by the diagram o the left. I the subtropical ocea, W< correspodig to Q>. The -lies look like as o the right, with the arrows poitig i the directio of the actual, physical velocity. The wester boudary layer correspods to the Gulf Stream. It is a amazig fact that the fully 3-dimesioal set of equatios correspodig to liear ocea circulatio theory ca also be studied as a advectio-diffusio problem. I the geeral 3d case, the advected quatity is the fluid pressure. Example. Cosider the advectio-diffusio equatio (3) ur, ( ) o the aulus ( ) (31) a < r < b, < <. r + vr, r = 1 r r r r r

8 with boudary coditios (32) ( a,) = a ( ) ad ( b,) = b ( ) where a ( ) ad b ( ) are give fuctios. Solve this problem i the four distict cases: (i) u 1, v (ii) u 1, v (iii) u, v 1 (iv) u,v such that the equatio takes the form x = ( xx + yy ) i Cartesia coordiates, but i the same domai, ad with the same boudary coditios, as above. Solutios. (i) I the case (33) r = 2 the iterior equatio is (34) I r =. Thus I = I ( ). Sice there is o boudary layer at r=a, the iterior solutio must satisfy the boudary coditio there. Thus I = a ( ). This of course does ot satisfy the boudary coditio at r=b. Near r=b, the balace must be (35) r = 2 r 2 which implies (36) = A( )+ B( )e ( r b)/. To match the iterior we choose A( ) = a ( ), ad to satisfy the boudary coditio at r=b, we choose B( )= b ( ) a ( ). The uiformly valid approximatio is (37) = a ( )+ ( b ( ) a ( ) )e r b The boudary layer thickess is. ( )/. 3-8

9 (ii) The case (38) r = 2 is similar except that the boudary layer is at r=a, ad the boudary layer equatio is (39) r = 2 r 2. We fid that (4) = b ( )+ ( a ( ) b ( ) )e (iii) I the case ( r a )/. (41) 1 r = 2, the iterior solutio I = I ( r) satisfies either boudary coditio. Thus we aticipate boudary layers at both boudaries. The most straight-forward approach is to expad the -depedece i a Fourier series, (42) r, where ( ) = r = (43) ()= r 1 ( )e i ( r,)e i d ad similarly for the boudary coditios, e.g. (44) a ( )= ( a ) e i. = Note that the = compoet correspods to the azimuthal average, (45) ()= r 1 ( r,)d. As we have already see, the = compoet is the oly compoet preset outside the boudary layers. For each, the exact equatio takes the form 3-9

10 (46) 1 r i ()= r 1 r r r r 2 r 2 () r with boudary coditios (47) ( a) = ( a ) 1 ad (48) ( b) = b ( ) 1 a The case = is clearly special. If =, (49) = 1 r r r r ( )e i d b ( )e i d. ad the diffusio term is importat throughout the domai. Two itegratios ad use of the boudary coditios give us the solutio (5) ()= r l( r / b) a b l( r / a). l( a / b) For, ()= r i the iterior (as we have already see), but this satisfies either boudary coditio. I the boudary layer ear r=a, (51) 1 a i ()= r 2 r 2 which has solutios of the form e r where (52) 2 = i a. Thus if >, (53) =± ad if <, (54) =± a a 12 +i i 1 2. We cover both cases by writig (55) =± ( 2a 1 + isg( ) ). 3-1

11 Oly the mius-alterative produces a acceptable match, ad oce agai we fid that the amplitude is determied by the boudary coditio. Thus (56) ()= r ( a ) exp ( 2a 1 + i sg( ) )( r a). ( ) implies that ( a ) is cojugate symmetric; it the follows We ote that reality of a from the equatio above that is cojugate symmetric; hece is real. The other boudary layer proceeds similarly ad we obtai the uiformly valid approximatio (57) ( ) = l r / b a l( a / b) r, ( ) exp + a + b ( ) exp + ( ) b l( r / a) ( 2a 1 + i sg( ) ) ( r a )+ i ( 2b 1 + i sg( ) ) ( r b )+ i where the first lie represets the iterior solutio ad the secod ad third lies are the boudary layer correctios. We see that the boudary layers have thickess. This solutio ca be writte i various ways. If we take a ca be writte as ( ) = A + ib, the the secod term (58) ( A + ib )exp 2a ( r a) cos + i si ( ) where (59) ( r,) = 2a sg( ) ( r a )+ Sice the sum is clearly real, it ca be writte as (6) 2exp > 2a ( r a) A cos B si ( ) where (61) = ( 2a r a )+. Physically, diffusio i the boudary layers wipes out the -depedece of there, allowig oly the radially averaged value of to peetrate to the iterior. The behavior i the boudary layers is a oscillatory decay oscillatory because the -depedece of the 3-11

12 boudary coditio is swirled may times aroud the boudary layer before the small diffusio wipes it out. (iv) This case is very similar to the ocea circulatio problem doe i lecture. There are -thickess boudary layers o the west coast of the basi ad o the east coast of the cetral islad. These wester boudary layers thicke as oe goes orth or south from the ceter of the domai. The ew features are the boudary layers that emaate from the top ad bottom of the islad ad exted to the west coast as show: These layers arise to smooth the discotiuity betwee the boudary values A ad B show o the sketch: I the (s,) coordiates depicted, the boudary layer equatio is the heat equatio (62) s = 2 2 with iitial coditio (63) (,) = A < B > Usig the basic Gree s fuctio for the heat equatio, the solutio is (64) ( s, )= { } 1 4s A ( e ) 2 /4s d + B e ( ) 2 /4s d (This is strictly valid oly for s>> 1/ 2 ; the small regio ear s= demads special treatmet.) Refereces. Beder ad Orszag. Cole. 3-12