UCLA Math 135, Winter 2015 Ordinary Differential Equations Review of First-Order Equations from Math 33b

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1 UCLA Mah 35, Winer 05 Ordinary Differenial Equaions Review of Firs-Order Equaions from Mah 33b 7. Numerical Mehods C. David Levermore Deparmen of Mahemaics Universiy of Maryland January 9, Numerical Mehods Conens 7.. Numerical Approximaion 7.. Explici and Implici Euler Mehods 7.3. Explici One-Sep Mehods Based on Taylor Approximaion Explici Euler Mehod Revisied Local and Global Errors Higher-Order Taylor-Based Mehods (no covered) 7.4. Explici One-Sep Mehods Based on Quadraure Explici Euler Mehod Revisied Again Runge Trapeziodal Mehod Runge Midpoin Mehod Runge-Kua Mehod (no covered) General Runge-Kua Mehods (no covered) c 05 C. David Levermore Reurn o Main Webpage

2 7. Numerical Mehods 7.. Numerical Approximaion. Analyic mehods are eiher difficul or impossible o apply o many firs-order differenial equaions. In such cases direcion fields are he only graphical mehod which we have covered ha can be applied. However, i can be hard o undersand how any paricular soluion behaves from he direcion field of is governing equaion. If we are ineresed in undersanding how a paricular soluion behaves hen a numerical mehod can be used o consruc an accurae approximaion o he soluion. This approximaion hen can be graphed much like an explici soluion. Suppose we are ineresed in he soluion Y () of he iniial-value problem dy (7.) d = f(, y), y( I) = y I, over he ime inerval [ I, F ] i.e. for I F. Here I is called he iniial ime while F is called he final ime. A numerical mehod selecs imes { n } N n=0 such ha and compues values {y n } N n=0 such ha I = 0 < < < < N < N = F, y 0 = Y ( 0 ) = y I, y n approximaes Y ( n ) for n =,,, N. For good numerical mehods, hese approximaions will improve as N increases. So for sufficienly large N we can plo he poins {( n, y n )} N n=0 in he (, y)-plane and connec he dos o ge an accurae picure of how Y () behaves over he ime inerval [ I, F ]. Here we will inroduce a few basic numerical mehods in simple seings. The numerical mehods used in sofware packages such as MATLAB are generally far more sophisicaed han hose we will sudy here. They are however buil upon he same fundamenal ideas as he simpler mehods we will sudy. Throughou his chaper we will make he following wo basic simplificaions. We will employ uniform ime seps. This means ha given N we se (7.) h = F I N, and n = I + nh for n = 0,,, N, where h is called he sep size. We will employ one-sep mehods. This means ha given f(, y) and h he value of y n+ for n = 0,,, N will depend only on y n. Sophisicaed sofware packages use mehods in which he sep size is chosen adapively. In oher words, he choice of n+ will depend on he behavior of recen approximaions for example, on ( n, y n ) and ( n, y n ). Employing uniform ime seps grealy simplifies he algorihms, and hereby simplifies he programming we have o do. If we do no like he way a run looks, we will simply ry again wih a larger N. Similarly, sophisicaed sofware packages someimes use so-called muli-sep mehods for which he value of y n+ for n = m, m +,, N will depend on y n, y n,, and y n m for some posiive ineger m. Employing one-sep mehods again simplifies he algorihms, and hereby simplifies he programming we have o do.

3 7.. Explici and Implici Euler Mehods. The simples (and leas accurae) numerical mehods are he Euler mehods. These can be derived many ways. Here we give a simple approach based on he definiion of he derivaive hrough difference quoiens. If we sar wih he fac ha Y ( + h) Y () lim = Y () = f(, Y ()), h 0 h hen for small posiive h we have Y ( + h) Y () f(, Y ()). h Upon solving his for Y ( + h) we find ha Y ( + h) Y () + hf(, Y ()). If we le = n above (so ha + h = n+ ) his is equivalen o Y ( n+ ) Y ( n ) + hf( n, Y ( n )). Because y n and y n+ approximae Y ( n ) and Y ( n+ ) respecively, his suggess seing (7.3) y n+ = y n + hf( n, y n ) for n = 0,,, N. This so-called Euler mehod was inroduced by Leonhard Euler in 768. Alernaively, we could have sared wih he fac ha Y () Y ( h) lim h 0 h Then for small posiive h we have = Y () = f(, Y ()). Y () Y ( h) f(, Y ()). h Upon solving his for Y ( h) we find ha Y ( h) Y () hf(, Y ()). If we le = n+ above (so ha h = n ) his is equivalen o Y ( n+ ) hf( n+, Y ( n+ )) Y ( n ). Because y n and y n+ approximae Y ( n ) and Y ( n+ ) respecively, his suggess seing (7.4) y n+ hf( n+, y n+ ) = y n for n = 0,,, N. This mehod is called he implici Euler or backward Euler mehod. I is called he implici Euler mehod because equaion (7.4) implicily relaes y n+ o y n. I is called he backward Euler mehod because he difference quoien upon which i is based seps backward in ime (from o h). In conras, he Euler mehod (7.3) someimes called he explici Euler or forward Euler mehod because i gives y n+ explicily and because he difference quoien upon which i is based seps forward in ime (from o + h). Remark. One sep of he implici Euler mehod will be much slower han one sep of he explici Euler mehod unless equaion (7.4) can be explicily solved for y n+. This can be done when f(, y) is a fairly simple funcion if y. For example, his can be done when f(, y) is linear or quadraic in eiher y or y. In general equaion (7.4) mus be 3

4 4 solved for y n+ numerically (say by he Newon mehod), which akes ime. However, here are equaions for which he implici Euler mehod ouperforms he explici Euler mehod because he explici Euler mehod has o ake so many more ime seps ha is speed advanage per ime sep canno compensae Explici One-Sep Mehods Based on Taylor Approximaion. The explici (or forward) Euler mehod can be undersood as he firs in a sequence of explici mehods ha can be derived from he Taylor approximaion formula Explici Euler Mehod Revisied. The explici Euler mehod can be derived from he firs-order Taylor approximaion, which is also known as he angen line approximaion. This approximaion saes ha if Y () is wice coninuously differeniable hen (7.5) Y ( + h) = Y () + hy () + O(h ). Here he O(h ) means ha he remainder vanishes a leas as fas as h as h ends o zero. I is clear from (7.5) ha for small posiive h we have Y ( + h) Y () + hy (). Because Y () saisfies (7.), his is he same as Y ( + h) Y () + hf(, Y ()). If we le = n above (so ha + h = n+ ) his is equivalen o Y ( n+ ) Y ( n ) + hf( n, Y ( n )). Because y n and y n+ approximae Y ( n ) and Y ( n+ ) respecively, his suggess seing (7.6) y n+ = y n + hf( n, y n ) for n = 0,,, N, which is exacly he Euler mehod (7.3). This view of he Euler mehod is illusraed by he following diagram Local and Global Errors. One advanage of viewing he Euler mehod hrough he angen line approximaion (7.5) is ha we gain some undersanding of how is error behaves as we increase N, he number of ime seps or wha is equivalen by (7.), as we decrease h, he sep size. The O(h ) erm in (7.5) represens he local error, which is error he approximaion makes a each sep. Roughly speaking, if we halve he sep size h hen by (7.5) he local error will reduce by a facor of one quarer, while by (7.) he number of seps N we mus ake o ge o a prescribed ime (say F ) will double. If we assume ha errors add (which is ofen he case) hen he error a F will reduce by a facor of one half. In oher words, doubling he number of ime seps will reduce he error by abou a facor of one half. Similarly, ripling he number of ime seps will reduce he error by abou a facor of one hird. Indeed, i can be shown (bu we will no do so) ha he error of he explici Euler mehod is O(h) over he inerval [ I, F ]. The bes way o hink abou his is ha if we ake N seps and he error made a each sep is O(h ) hen we can expec ha he acummulaion of he local errors will lead o a global error of O(h )N. This is

5 5 y y n+ y n+ Y () y n n n+ n+ Figure 7.. Illusraion of he explici Euler mehod. y y n y n+ O(h ) y N h Y () O(h) n n+... N Figure 7.. Illusraion of global error arising hrough he accumulaion of local errors for he explici Euler mehod. illusraed in he following figure. Because (7.) saes ha hn = F I, which is a number ha is independen of h and N, we see ha global error of he explici Euler mehod is O(h). This was shown by Cauchy in 84. Moreover, i can be shown ha he error of he implici Euler mehod behaves he same way. Global error is a more meaningful concep han local error because i ells us how fas a mehod converges over he enire inerval [ I, F ]. Therefore we idenify he order of

6 6 a mehod by he order of is global error. In paricular, mehods like he Euler mehods wih global errors of O(h) are firs-order mehods. By reasoning similar o ha given in he previous paragraph, mehods whose local error is O(h m+ ) will have a global error of O(h m+ )N = O(h m ) and hereby are m h -order mehods. Higher-order mehods are more complicaed han he explici Euler mehod. The hope is ha his cos is overcome by he fac ha is error improves faser as you increase N or wha is equivalen by (7.), as you decrease h. For example, if we halve he sep size h of a fourh-order mehod hen he global error will reduce by a facor of /6. Similarly, ripling he number of ime seps will reduce he error by abou a facor of / Higher-Order Taylor-Based Mehods. The second-order Taylor approximaion saes ha if Y () is hrice coninuously differeniable hen (7.7) Y ( + h) = Y () + hy () + h Y () + O(h 3 ). Here he O(h 3 ) means ha he remainder vanishes a leas as fas as h 3 as h ends o zero. I is clear from (7.7) ha for small posiive h one has (7.8) Y ( + h) Y () + hy () + h Y (). Because Y () saisfies (7.), we see by he chain rule from mulivariable calculus ha Y () = d d( Y () ) = d d f(, Y ()) = f(, Y ()) + Y () y f(, Y ()) = f(, Y ()) + f(, Y ()) y f(, Y ()). Hence, equaion (7.8) is he same as ) Y ( + h) Y () + hf(, Y ()) + ( h f(, Y ()) + f(, Y ()) y f(, Y ()). If we le = n above (so ha + h = n+ ) his is equivalen o ) Y ( n+ ) Y ( n ) + hf( n, Y ( n )) + ( h f( n, Y ( n )) + f( n, Y ( n )) y f( n, Y ( n )). Because y n and y n+ approximae Y ( n ) and Y ( n+ ) respecively, his suggess seing ) y n+ = y n + hf( n, y n ) + ( (7.9) h f( n, y n ) + f( n, y n ) y f( n, y n ) for n = 0,,, N. We call his he second-order Taylor-based mehod. Remark. We can generalize our derivaion of he second-order Taylor-based mehod by using he m h -order Taylor approximaion o derive an explici numerical mehod whose error is O(h m ) over he inerval [ I, F ] a so-called m h -order mehod. However, he formulas for hese mehods grow in complexiy. For example, he hird-order mehod

7 7 is (7.0) ) y n+ = y n + hf( n, y n ) + ( h f( n, y n ) + f( n, y n ) y f( n, y n ) + [ 6 h3 f( n, y n ) + f( n, y n ) y f( n, y n ) + f( n, y n ) yy f( n, y n ) ( ) ] + f( n, y n ) + f( n, y n ) y f( n, y n ) y f( n, y n ) for n = 0,,, N. This complexiy of hese mehods makes hem far less pracical for general algorihms han he nex class of mehods we will sudy Explici One-Sep Mehods Based on Quadraure. The saring poin for our nex class of mehods will be he Fundamenal Theorem of Calculus specifically, he fac Y ( + h) Y () = Because Y () saisfies (7.), his becomes (7.) Y ( + h) = Y () + +h +h Y (s) ds. f(s, Y (s)) ds. In 895 Carl Runge proposed using quadraure rules (numerical inegraion) o consruc approximaions o he definie inegral above in he form (7.) +h f(s, Y (s)) ds = K(h,, Y ()) + O(h m+ ), where m is some posiive ineger. The key poin here is ha K(h,, Y ()) depends on Y (), bu does no depend on Y (s) for any s. When approximaion (7.) is placed ino (7.) we obain Y ( + h) = Y () + K(h,, Y ()) + O(h m+ ). If we le = n above (so ha + h = n+ ) his is equivalen o Y ( n+ ) = Y ( n ) + K(h, n, Y ( n )) + O(h m+ ). Because y n and y n+ approximae Y ( n ) and Y ( n+ ) respecively, his suggess seing (7.3) y n+ = y n + K(h, n, y n ) for n = 0,,, N, Hence, every approximaion of he form (7.) yields he m h -order explici one-sep mehod (7.3) for approximaing soluions of (7.). Here we will presen mehods associaed wih four basic quadraure rules ha are covered in mos calculus courses: he lef-hand rule, he rapezoidal rule, he midpoin rule, and he Simpson rule.

8 Explici Euler Mehod Revisied Again. The lef-hand rule approximaes he definie inegral on he lef-hand side of (7.) as +h f(s, Y (s)) ds = hf(, Y ()) + O(h ). This approximaion is already in he form (7.) wih K(h,, y) = hf(, y). Mehod (7.3) hereby becomes y n+ = y n + hf( n, y n ) for n = 0,,, N, which is exacly he explici Euler mehod (7.3). In pracice, he explici Euler mehod is implemened by iniializing y 0 = y I and hen for n = 0,, N cycling hrough he insrucions where n = I + nh. f n = f( n, y n ), y n+ = y n + hf n, Example. Le Y () be he soluion of he iniial-value problem dy d = + y, y(0) =. Use he explici Euler mehod wih h =. o approximae Y (.). Soluion. We iniialize 0 = 0 and y 0 =. The explici Euler mehod hen gives Therefore Y (.) y =.. f 0 = f( 0, y 0 ) = 0 + = y = y 0 + hf 0 = +. =. f = f(, y ) = (.) + (.) =.0 +. =. y = y + hf =. +.. =. +. =. The explici Euler mehod is implemened by he following MATLAB funcion M-file. funcion [,y] = EulerExplici(f, I, yi, F, N) = zeros(n +, ); y = zeros(n +, ); () = I; y() = yi; h = (F - I)/N; for j = :N (j + ) = (j) + h; y(j + ) = y(j) + h*f((j), y(j)); end Remark. There are some hings you should noice. Firs, (j) is j and y(j) is y j, he approximaion of y( j ). In paricular, y(j) is no he same as Y (j), which denoes he soluion Y () evaluaed a = j. (You mus pay aenion o he fon in which a leer is wrien!) The shif of he indices by one is needed because indexed variables in MATLAB begin wih he index. In paricular, () and y() denoe he iniial ime

9 0 and value y 0. Consequenly, all subsequen indices are shifed oo, so ha () and y() denoe and y, (3) and y(3) denoe and y, ec Runge-Trapezoidal Mehod. The rapezoidal rule approximaes he definie inegral on he lef-hand side of (7.) as +h f(s, Y (s)) ds = h [ f(, Y ()) + f( + h, Y ( + h)) ] + O(h 3 ). This approximaion is no in he form (7.) because of he Y ( + h) on he righ-hand side. If we approximae his Y ( + h) by he explici Euler mehod hen we obain +h f(s, Y (s)) ds = h [ f(, Y ()) + f ( + h, Y () + hf(, Y ()) )] + O(h 3 ). This approximaion is in he form (7.) wih Mehod (7.3) hereby becomes K(h,, y) = h [ f(, y) + f ( + h, y + hf(, y) )]. y n+ = y n + h [ f(n, y n ) + f ( n+, y n + hf( n, y n ) )] for n = 0,,, N. This is someimes called he improved Euler mehod. However, ha name is also used for oher mehods and is no very descripive. Raher, we will call his he Rungerapezoidal mehod because i was proposed by Runge based on he rapeziodal rule. This name makes he origins of he mehod clear. In pracice, he Runge-rapezoidal mehod is implemened by iniializing y 0 = y I and hen for n = 0,, N cycling hrough he insrucions f n = f( n, y n ), f n+ = f( n+, ỹ n+ ), ỹ n+ = y n + hf n, y n+ = y n + h[f n + f n+ ], where n = I + nh. Example. Le y() be he soluion of he iniial-value problem dy d = + y, y(0) =. Use he Runge-rapezoidal mehod wih h =. o approximae y(.). Soluion. We iniialize 0 = 0 and y 0 =. The Runge-rapezoidal mehod hen gives f 0 = f( 0, y 0 ) = 0 + = ỹ = y 0 + hf 0 = +. =. f = f(, ỹ ) = (.) + (.) = =.48 y = y 0 + h[ f 0 + f ] = +. ( +.48) = =.48 We hen have y(.) y =.48. Remark. Noice ha wo seps of he explici Euler mehod wih h =. gave y(.)., while one sep of he Runge-rapezoidal mehod wih h =. gave y(.).48, 9

10 0 y ỹ n+ y n+ Y () y n y n + h f n+ n n+ Figure 7.3. Illusraion of he Runge-rapazoidal mehod. The mehod is described as follows: Firs evaluae y n+ using he explici Euler mehod, hen find f n+ by evaluaing f(y, ) a ỹ n+ and n+. Finally y n+ is he midpoin of ỹ n+ and he correcion y n + h f n+. Noice how he line leaving ỹ n+ is parallel o he segmen beween y n and y n +h f n+. which is much closer o he exac value. As hese wo calculaions required roughly he same compuaional effor, his shows he advanange of using he second-order mehod. The Runge-rapezoidal mehod is implemened by he following MATLAB funcion M-file. funcion [,y] = RungeTrap(f, I, yi, F, N) = zeros(n +, ); y = zeros(n +, ); () = I; y() = yi; h = (F - I)/N; hhalf = h/; for j = :N (j + ) = (j) + h; fnow = f((j), y(j)); yplus = y(j) + h*fnow; fplus = f((j + ), yplus); y(j + ) = y(j) + hhalf*(fnow + fplus); end Remark. Here (j) and y(j) have he same meaning as hey did in he M-file for he explici Euler mehod. In paricular, we have he same shif of he indices by one. Here we have inroduced he so-called working variables fnow, yplus, and fplus o emporarily hold he values of f j, ỹ j, and f j during each cycle of he loop. These values do no

11 have o be saved, and so are overwrien wih each new cycle. Here we have isolaed he funcion evaluaions for fnow and fplus ino wo separae insrucions. This is good coding pracice ha makes adapaions easier. For example, you can replace he funcion calls o f(,y) by explici formulas in hose wo lines wihou changing he res of he coding Runge-Midpoin Mehod. The midpoin rule approximaes he definie inegral on he lef-hand side of (7.) as +h f(s, Y (s)) ds = hf ( + h, Y ( + h)) + O(h 3 ). This approximaion is no in he form (7.) because of he Y (+ h) on he righ-hand side. If we approximae his Y ( + h) by he explici Euler mehod hen we obain +h f(s, Y (s)) ds = hf ( + h, Y () + hf(, Y ())) + O(h 3 ). This approximaion is in he form (7.) wih Mehod (7.3) hereby becomes K(h,, y) = hf ( + h, y + hf(, y)). y n+ = y n + hf ( n+, y n + hf( n, y n ) ) for n = 0,,, N. This is someimes called he modified Euler mehod. However, ha name is also used for oher mehods and is no very descripive. Raher, we will call his he Rungemidpoin mehod because i was proposed by Runge based on he midpoin rule. This name makes he origins of he mehod clear. In pracice, he Runge-midpoin mehod is implemened by iniializing y 0 = y I and hen for n = 0,, N cycling hrough he insrucions f n+ where n = I + nh and n+ f n = f( n, y n ), = f( n+, y n+ ), = I + (n + )h. Remark. The half-ineger subscrips on n+ variables are associaed wih he ime = n + y n+ = y n + hf n, y n+ = y n + hf n+,, y n+, and f n+ indicae ha hose h, which is halfway beween he imes n and n+. While i may seem srange a firs, his noaional device is a handy way o help keep rack of he meanings of differen variables. Example. Le y() be he soluion of he iniial-value problem dy d = + y, y(0) =. Use he Runge-midpoin mehod wih h =. o approximae y(.).

12 y y n+ y n+ y n Y () n n+ n+ Figure 7.4. Illusraion of he Runge-midpoin mehod. The mehod is described as follows: Firs evalue y n+ by aking he midpoin of he segmen beween y n and he value y n +hf(y n, n ) prediced by he explici Euler mehod. Nex, find f n+ by evaluaing f(y, ) a y n+ and n+. Fnally, find y n+ by sepping from y n in he direcion of f n+, ha is y n+ = y n + hf n+. Noice how he line leaving y n+ is parallel o he segmen beween y n and y n+. Soluion. We iniialize 0 = 0 and y 0 =. Then he Runge-midpoin mehod gives f 0 = f( 0, y 0 ) = 0 + = y f = y 0 + hf 0 = +. =. = f(, y ) = (.) + (.) =.0 +. =., y = y 0 + hf = +. (.) = +.44 =.44. We hen have y(.) y =.44. Remark. Noice ha he Runge-rapezoidal mehod gave y(.).48 while he Runge-midpoin mehod gave y(.).44. The resuls are abou he same because boh mehods are second-order. Here he Runge-rapezoidal mehod gave a beer approximaion. For oher problems he Runge-midpoin mehod migh give a beer approximaion. The Runge-midpoin mehod is implemened by he following MATLAB funcion M-file. funcion [,y] = RungeMid(f, I, yi, F, N) = zeros(n +, ); y = zeros(n +, ); () = I; y() = yi; h = (F - I)/N; hhalf = h/;

13 3 for j = :N half = (j) + hhalf; (j + ) = (j) + h; fnow = f((j), y(j)); yhalf = y(j) + hhalf*fnow; fhalf = f(half, yhalf); y(j + ) = y(j) + h*fhalf; end Remark. Here (j) and y(j) have he same meaning as hey did in he M-file for he explici Euler mehod. In paricular, we have he same shif of he indices by one. Here we have inroduced he working variables fnow, half, yhalf, and fhalf o emporarily hold he values of f j, j, y j, and f j during each cycle of he loop. These values do no have o be saved, and so are overwrien wih each new cycle Runge-Kua Mehod. The Simpson rule approximaes he definie inegral on he lef-hand side of (7.) as +h f(s, Y (s)) ds = h 6 [ f(, Y ())+4f ( + h, Y (+ h)) +f ( +h, Y (+h) )] +O(h 5 ). This approximaion is no in he form (7.) because of he Y ( + h) and Y ( + h) on he righ-hand side. If we approximae hese wih he explici Euler mehod as we did before hen we will degrade he local error o O(h 3 ). We would like o find an approximaion ha is consisen wih he O(h 5 ) local error of he Simpson rule. In 90 Wilhelm Kua found such an approximaion, which led o he so-called Runge-Kua mehod. We will no give a derivaion of his mehod here. Such derivaions can be found in numerical analysis books. In pracice he Runge-Kua mehod is implemened by iniializing y 0 = y I and hen for n = 0,, N cycling hrough he insrucions f n+ f n+ f n = f( n, y n ), = f( n+, ỹ n+ ), ), = f( n+, y n+ f n+ = f( n+, ỹ n+ ), ỹ n+ y n+ = y n + hf n, = y n + h f n+,, ỹ n+ = y n + hf n+ y n+ = y n + h[ f 6 n + f n+ + f n+ + f ] n+, where n = I + nh and n+ = I + (n + )h. Remark. The Runge-Kua mehod requires four evaluaions of f(, y) o advance each ime sep, whereas he second-order mehods each required only wo. Therefore i requires roughly wice as much compuaional work per ime sep as hose mehods.

14 4 Remark. Noice ha because we see ha ỹ n+ y n+ y n Y ( n ), Y ( n + h), Y ( n + h), ỹ n+ Y ( n + h), f n+ f n+ f n f ( n, Y ( n ) ) f ( n + h, Y ( n + h)), f ( n + h, Y ( n + h)), f n+ f ( n + h, Y ( n + h) ), y n+ Y ( n ) + h 6 [ f(n, Y ( n )) + 4f ( n + h, Y ( n + h)) + f ( n + h, Y ( n + h) )]. The Runge-Kua mehod hereby looks consisan wih he Simpson rule approximaion. This argumen does no show ha he Runge-Kua mehod is fourh order, bu i is. Example. Le y() be he soluion of he iniial-value problem dy d = + y, y(0) =. Use he Runge-Kua mehod wih h =. o approximae y(.). Soluion. We iniialize 0 = 0 and y 0 =. The Runge-Kua mehod hen gives f 0 = f( 0, y 0 ) = 0 + = ỹ f y f = y 0 + hf 0 = +. =. = f( = y 0 + h f = f(, ỹ ) = (.) + (.) =.0 +. =., y = +.. =. ) = (.) + (.) = = ỹ = y 0 + hf = = = f = f(, ỹ ) = (.) + (.57768) = y = y 0 + h[ f f + f + f ] [ ]. We hen have y(.) y Of course, you would no be expeced o carry ou such arihmeic calculaions o nine decimal places on an exam. Remark. One sep of he Runge-Kua mehod wih h =. yielded he approximaion y(.) This is more accurae han he approximaions we had obained wih eiher second-order mehod. However, ha is no a fair comparison because he Runge-Kua mehod required roughly wice he compuaional work. A beer comparison would be wih he approximaion produced by wo seps of eiher second-order mehod wih h =.. Remark. You will no be required o memorize he Runge-Kua mehod. You also will no be required o carry ou one sep of i on an exam or quiz because, as he above example illusraes, he arihmeic ges messy even for fairly simple differenial equaions.

15 However, you should undersand he implicaions of i being a fourh-order mehod namely, he relaionship beween is error and he sep size h. You also should be able o recognize he Runge-Kua mehod if i is presened o you in MATLAB code. The Runge-Kua mehod is implemened by he following MATLAB funcion M-file. funcion [,y] = RungeKua(f, I, yi, F, N) = zeros(n +, ); y = zeros(n +, ); () = I; y() = yi; h = (F - I)/N; hhalf = h/; hsixh = h/6; for j = :N half = (j) + hhalf; (j + ) = (j) + h; fnow = f((j), y(j)); yhalfone = y(j) + hhalf*fnow; fhalfone = f(half, yhalfone); yhalfwo = y(j) + hhalf*fhalfone; fhalfwo = f(half, yhalfwo); yplus = y(j) + h*fhalfwo; fplus = f((j + ), yplus); y(j + ) = y(j) + hsixh*(fnow + *fhalfone + *fhalfwo + fplus); end Remark. Here (j) and y(j) have he same meaning as hey did in he M-file for he explici Euler mehod. In paricular, we have he same shif of he indices by one. Here we have inroduced he working variables fnow, half, yhalfone, fhalfone, yhalfwo, fhalfwo, yplus, and fplus o emporarily hold he values of f j, j, ỹ j, f j, y j, f j, ỹ j, and f j General Runge-Kua Mehods. All he mehods presened in his secion are members of he family of general Runge-Kua mehods. The MATLAB command ode45 uses he Dormand-Prince mehod, which is anoher member of his Runge- Kua family ha was discovered in 980! The Runge-Kua family coninues o be enlarged by new mehods, some of which migh replace he Dormand-Prince mehod in fuure versions of MATLAB. An inroducion o hese modern mehods requires a graduae course in numerical analysis. Here we have he more modes goal of inroducing hose family members presened by Wilhelm Kua in his 90 paper. Carl Runge had described jus a few mehods in his 895 paper, including he Runge rapezoid and midpoin mehods. In 900 Karl Heun presened a family of mehods ha included all hose sudied by Runge as special cases. Heun characerized he compuaional effor of hese mehods by how many evaluaions of f(, y) are needed o compue K(h,, y). We say a mehod ha requires s evaluaions of f(, y) is an s-sage mehod. The explici Euler mehod, for which K(h,, y) = hf(, y), is he only one-sage mehod.

16 6 Heun considered he family of wo-sage mehods in he form (7.4a) K(h,, y) = α k + α k, wih α + α =, where k and k are given by wo evaluaions of f(, y) as (7.4b) k = hf(, y), k = hf( + βh, y + βk ), for some β > 0. Heun showed he wo-sage mehod (7.4) is second-order for every f(, y) if and only if α = β, α = β. These include he Runge rapeziodal mehod, which is given by α = α = and β =, and he Runge midpoin mehod, which is given by α = 0, α =, and β =. Heun also showed ha no wo-sage mehod (7.4) is hird-order for every f(, y). Remark. Second-order, wo-sage mehods are ofen called Heun mehods in recogniion of his work. Of hese Heun favored he mehod given by α =, α 4 = 3, and 4 β =, which is hird order in he special case when 3 yf = 0. Heun also considered families of hree- and four-sage mehods in his 900 paper. However in 90 Kua inroduced families of s-sage mehods ha where more general when s 3. For example, Kua considered he family of hree-sage mehods in he form (7.5a) K(h,, y) = α k + α k + α 3 k 3, wih α + α + α 3 =, where k, k, and k 3 are given by hree evaluaions of f(, y) as (7.5b) k = hf(, y), k = hf( + β h, y + γ k ), wih β = γ, k 3 = hf( + β 3 h, y + γ 3 k + γ 3 k ), wih β 3 = γ 3 + γ 3. Kua showed he hree-sage mehod (7.5) is second-order for every f(, y) if and only if α β + α 3 β 3 = ; and is hird-order for every f(, y) if and only if in addiion α β + α 3 β 3 = 3, α 3γ 3 β = 6. Kua also showed ha no hree-sage mehod (7.5) is fourh-order for every f(, y). Heun had shown he analogus resuls resriced o he case γ 3 = 0. He favored he hird-order mehod given by α = 4, α = 0, α 3 = 3 4, β = γ = 3, β 3 = γ 3 = 3, γ 3 = 0, which is he hird-order mehod requiring he fewes arihmeic operaions. favored he hird-order mehod given by α = 6, α = 3, α 3 = 6, β = γ =, β 3 =, γ 3 =, γ 3 =, which agrees wih he Simpson rule in he special case when y f = 0. Similarly, Kua considered he family of four-sage mehods in he form (7.6a) K(h,, y) = α k + α k + α 3 k 3 + α 4 k 4, wih α + α + α 3 + α 4 =, Kua

17 where k, k, k 3, and k 4 are given by four evaluaions of f(, y) as (7.6b) k = hf(, y), k = hf( + β h, y + γ k ), wih β = γ, k 3 = hf( + β 3 h, y + γ 3 k + γ 3 k ), wih β 3 = γ 3 + γ 3, k 4 = hf( + β 4 h, y + γ 4 k + γ 4 k + γ 43 k 3 ), wih β 4 = γ 4 + γ 4 + γ 43. Kua showed he four-sage mehod (7.6) is second-order for every f(, y) if and only if α β + α 3 β 3 + α 4 β 4 = ; is hird-order for every f(, y) if and only if in addiion α β + α 3 β 3 + α 4 β 4 = 3, α 3γ 3 β + α 4 ( γ4 β + γ 43 β 3 ) = 6 ; and is fourh-order for every f(, y) if and only if in addiion α β 3 + α 3 β3 3 + α 4 β4 3 =, α ( ) 4 3γ 3 β + α 4 γ4 β + γ 43 β3 =, ( ) α 3 β 3 γ 3 β + α 4 β 4 γ4 β + γ 43 β 3 =, α 8 4γ 43 γ 3 β =. 4 Kua also showed ha no four-sage mehod (7.6) is fifh-order for every f(, y). Heun had shown he analogus resuls resriced o he case γ 3 = γ 4 = γ 4 = 0. Kua favored he classical Runge-Kua mehod presened in he previous subsecion, which is given by α = 6, α = 3, α 3 = 3, α 4 = 6, β = γ =, β 3 = γ 3 =, γ 3 = 0, β 4 = γ 43 =, γ 4 = γ 4 = 0. This is he fourh-order mehod ha boh requires he fewes arihmeic operaions and is consisan wih he Simpson rule. More generally, Kua considered he family of s-sage mehods in he form s s (7.7a) K(h,, y) = α j k j, wih α j =, j= where k j for j =,, s are given by s evaluaions of f(, y) as (7.7b) k = hf(, y), ( ) j j k j = hf + β j h, y + γ ji k i, wih β j = γ ji, for j =,, s. i= Kua showed ha no five-sage mehod (7.7) is fifh-order for every f(, y). This resul was surprising because for s =,, 3, and 4 here were s-sage mehods ha were s h - order. Kua hen characerized hose six-sage mehods (7.7) ha are fifh-order for every f(, y). We will no give he condiions he found here. Remark. Programmable elecrionic compuers were invened over fify years afer Runge, Heun, and Kua carried ou heir work. Early numerical compuaions had less precision han hey do oday. Higher-order mehods suffer from round-off error j= i= 7

18 8 more han lower-order mehods. Because round-off error is larger on machines wih less precision, here was lile advanage o using higher-order mehods on early machines. As machines became more precise, he classical Runge-Kua mehod became widely used o solve differenial equaions because i offers a nice balance beween order and round-off error. Remark. One of he mos imporan developmens in Runge-Kua mehods since heir invenion is embedded mehods, which emerged in he 950s. These mehods mainain a prescribed error olerance by selecing a differen h for each ime sep based upon an error esimae made by compairing relaed Runge-Kua mehods of orders m and m +. By relaed we mean ha he mehods are buil from he same evaluaions of f(, y), so ha hey can be compued simulaniously. The MATLAB command ode45 uses a fourh-order/fifh-order Runge-Kua embedded mehod. Originally i used a fourhorder mehod invened by Fehlberg in 969, someimes denoed RKF4(5). Currenly i uses a fifh-order mehod invened by J.R. Dormand and P.J. Prince in 980, someimes denoed RKDP5(4). This mehod migh be replaced by a higher-order embedded mehod as faser machines wih smaller round-off error become more common. One candidae o fill his role is an eighh-order mehod invened by Dormand and Prince in 98, a sevenh-order/eighh-order Runge-Kua embedded mehod someimes denoed RKDP8(7). There are oher candidaes.