LECTURE Review. In this lecture we shall study the errors and stability properties for numerical solutions of initial value.

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1 LECTURE 24 Error Analysis for Multi-step Methods 1. Review In this lecture we shall study the errors and stability properties for numerical solutions of initial value problems of the form è24.1è dx = fèt; xè è24.2è x èt0è =x0 Recall that the starting point for multi-step methods such as the Adams-Bashforth and Adams-Moulton methods was actually the integral of equation è24.1è Z tn+1 Z dx tn+1 xèt n+1 è, x èt n è= tn = fèt; xè tn By replacing fèt; xè by its polynomial intepolation at the points t n ;t n,2 ;::: ;t n,4 one obtains the æfth order Adams-Bashforth formula where x n+1 = x n + Z tn+1 tn è nx i=n,4 f i`iètè! = x n + h 720 è1901f n, 2774f n, f n,2, 1274f n, f n,4 è t n = t 0 + nh x n = x èt n è f n = f èt n ;x n è and the numerical coeæcients in the second line are just the integrals of the cardinal functions `iètè. Similarly, by replacing the function fèx; tè by its polynomial interpolation at the points t n+1 ;t n; ::: ;t n,3. we obtained the Adams-Moulton formula x n+1 = x n + Z tn+1 tn è n+1 X i=n,3 f i`iètè! = x n + h 720 è251f n f n, 264f n, f n,2, 19f n,3 è 2. Linear Multi-step Methods Of course, there's nothing to prevent us from calculating even higher order analogs of the Adams-Bashforth and Adams-Moulton formulae. But instead of doing so explicitly, we'll now assume that we have in our hands a k-step linear multi-step method given by a formula of the form è24.3è a k x n + a k=1 x n,1 + æææ+ a 1 x n,k,1 + a 0 x n,k = h ëb k f n + b k,1 f n,1 + æææ+ b 1 f n,k+1 + b 0 f n,k ë 1

2 2. LINEAR MULTI-STEP METHODS 2 relating k successive values of the xètè tok successive values of the function fèt; xè èthe function that deænes the diæerential equationè. The utility ofsuch an equation is, of course, to compute x n from the k preceding values, x n,k ;x n,k+1 ;::: ;x n,1, and so we shall henceforth assume that the constant a k 6= 0. The coeæcient b k on the right hand side may or not equal zero. If b k = 0 then we say the multi-step method corresponding to equation è24.3è is explicit; because, in this case we can solve the equation explicitly for x n x n =, 1 a k èa k=1 x n,1 + æææ+ a 1 x n,k,1 + a 0 x n,k è+ h a k ëb k f n + b k,1 f n,1 + æææ+ b 1 f n,k+1 + b 0 f n,k ë If b k 6=0wesay that the corresponding method is implicit; in this case, the term f n = f èt n ;x n è on the right hand side also depends on x n and so we have an implicit algebraic equation for x n Convergence of Multi-Step Methods. Definition A multi-step method deæned by the a formula of the form è24.3è is said to be convergent inaregion ët 0 ;t 1 ë if è24.4è provided only that lim x hètè =xètè ; t 2 ët 0 ;t 1 ë h!0 è24.5è lim h!0 x hèt + jhè =x 0 ; 0 jék Here x h ètè is the numerical solution computed using a step size of h and xètè is the exact solution. This deænition is natural enough. The following deænitions are not so natural, but nevertheless extremely useful. Definition Consider a multi-step method corresponding to a relation of the form è24.3è. Set P èzè =a k z k + a k,1 z k,1 + æææ+ a 1 z + a 0 Qèzè =b k z k + b k,1 z k,1 + æææ+ b 1 z + b 0 We shall say that the method è24.3è is stable if the roots of the polynomial P èzè lie in disk jzj 1, and if each root such that jzj =1is simple. The method è24.3è is said to be consistent if P è1è = 0 and P 0 è1è = Qè1è: Theorem Consider a multi-step method corresponding to a relation of the form è24.3è. Then this method is convergent if and only if it is both stable and consistent. A partial proof èthat of the necessity of stability and consistency for convergenceè is given in the text The Order of Multi-step Method. The order of a multi-step method is an interger that corresponds to the number of terms in the Taylor expansion of the solution that a multi-step method simulates. Let us represent the multi-step method è24.3è as a linear functional Lëxë = = j=0 j=0 ëa j x èjhè, hb j fèjhèë ëa j x èjhè, hb j x 0 èjhèë

3 2. LINEAR MULTI-STEP METHODS 3 Here we let k = n to simplify our notation and assume that the ærst value of in equation è24.3è begins at t = 0, rather than t =èn, kèh. Now and so we can write Lëxë = " j=0 xèjhè = 1X 1X èjhè i x èiè è0è i! x 0 èjhè èjhè = i x èi+1è è0è i! a j 1 X èjhè i X 1 x èiè è0è, hb j i! è èjhè i x èi+1è è0è i! Collecting terms proportional to xè0è;x 0 è0è;::: èor equivalently by their degree in hè wehave where è24.6è è24.7è è24.8è è24.9è Lëxë=d 0 xè0è + d 1 hx 0 è0è + d 2 h 2 x 00 è0è + æææ d 0 = d 1 = d 2 =. d j = a i èia i, b i è 1 2 i2 a i, ib i i j j,1 j! a i, i èj, 1è! b j Theorem The following three properties of the multi-step method è24.3è are equivalent: 1. d 0 = d 1 = æææ= d m =0. 2. LëP ë=0for each polynomial P of degree m. 3. Lëxë is O, h m+1æ for all x 2 C m+1. Proof. æ è1è è è2è: If è1è is true then Lëxë=0+0+æææ+0+d m+1 h m+1 x èm+1è è0è + d m+2 h m+2 x èm+2è è0è + æææ But if P is a polynomial of degree m then P èm+1è èxè = 0. Therefore, for such a polynomial LëP ë=0+0+æææ+0+d m+1 h m+1 P èm+1è è0è + d m+2 h m+2 P èm+2è è0è + æææ =0 æ è2è è è3è: If x 2 C m+1, then by Taylor's Theorem we can write where P ètè is a polynomial of degree m and xètè =P ètè+rètè Rètè = xèm+1è èè èm + 1è! tm+1

4 2. LINEAR MULTI-STEP METHODS 4 Notice that Hence, Lëxë =LëP ë+lërë d j R j æ ææææt=0 =0 =0+d m+1 h m+1 x èm+1è è0è + d m+2 h m+2 x èm+2è è0è + æææ = O, h m+1æ æ è3è è è1è If è3è is true, then we must have d 0 = d 1 = æææ= d m = 0. Hence, è3è implies è1è. Definition The order of a multi-step method is the unique natural number m such that 0=d 0 = d 1 = æææ= d m 6= d m+1 Example What is the order of the following multistep method x n, x n,2 = h 3 èf n +4f n,1 + f n,2 è æ This is a three step method èk = 2è, with a 2 =1 ; a 1 =0 ; a 0 =,1 b 2 = 1 3 ; b 1 = 4 3 ; b 0 = 1 3 We have d 0 = d 1 = d 2 = d 3 = d 4 = d 5 = 2X k=0 a i =1+0, 1=0 èia i, b i è= 1 2 i2 a i, ib i 1 6 i3 a i, 1 2 i2 b i 1 24 i4 a i, 1 6 i3 b i 0, 1 + 0, 4 + 2, 1 = =è0, 0è + 0, , 2 =0 3 =è0, 0è + 0, =è0, 0è + 0, i5 a i, 1 24 i4 b i =è0, 0è + And so the order is m =4. 8 6, 4 6 0, è16èè4è 24 è16èè1è è24è + =0, è8èè1è =0 è6èè3è è32èè1è 120, è16èè1è è24èè3è =, Local Truncation Error. By a local trunction error we mean the error induced at a particular stage of an iterative numerical procedure. In the case at hand, this means the error induced by using a diæerence relation of the form è24.3è instead to obtain an estimate x n of x èt n è instead of evaluating the exact solution at t n.

5 3. GLOBAL TRUNCATION ERROR 5 Theorem Consider a multi-step method corresponding to a relation of the form è24.3è. xètè 2 C k+2 èrè is continuous, we have x èt n è, x n = d, k+1 h m+1 x èm+1è èt n,k è+o m+2æ h a k where the coeæcients are deæned byequations è??è - è24.9è Then if Proof. It suæces to prove the statement for n = k, since x n can be expressed as a solution with initial conditions imposted at t n,k. Using the linear function L of the preceding section we have for the exact solution xètè è24.10è Lëxë = ëa i x èt i è, hb i x 0 èt i èë = On the other hand, the numerical solution fx i g should satisfy è24.11è Subtracting è24.11è from è24.10è yields Lëxë = 0= ëa i x i, hb i f èt i ;x i èë ëa i x èt i è, hb i f èt i ;xèt i èèë ëa i èx èt i è, x i è, hb i èf èt i ;xèt i èè, fèt i ;x i èë Since we are interested here only in the error induced by the current iteration of the multi-step method we shall assume that the previously determined x i are all exactly correct: x i = x èt i è. In this case, all but the ænal terms of the summand vanish and we have Writing we have We thus have x èt k è, x k = Lëxë=a k èx èt k è, x k è, hb k èf èt k ;xèt k èè, fèt k ;x k è f èt k ;x k è=f èt k ;xèt k èèèx èt kè, x k è ; for some 2 ëxèt k è;x k ë Lëxë =a k èx èt k è, x k è, èèèèx èt kè, x k èè =ëa k, hb k F ëèx èt k è, x k è Lëxë a k, hb k F = d m+1h m+1 x èm+1è èt n,k è+æææ d m+1h m+1 x èm+1è èt n,k è, + O m+2æ h a k, hb k F a k 3. Global Truncation Error We shall now establish a bound on total error induced by the local truncation errors that occur at each stage of a multi-step numerical method. As before we let x n represent the value of the numerical solution after n iterations of the multi-step method and let xèt n è denote the corresponding value of the exact solution èfor the same time t n è. Now the ærst thing one should realize is that the global truncation error is not simply the sum of the local truncation errors that occur at each stage of the iterative method. For the accuracy of a successive stage depends crucially on the accuracy of the stage that preceded it. To get a handle on how the error of a preceding stage eæects the error of a latter stage we consider a family of initial value problems: è24.12è dx = fèt; xè è24.13è xè0è = s ; s 2 R

6 3. GLOBAL TRUNCATION ERROR 6 We shall assume that f is continuous and f x èt; xè ; for all t 2 ë0;të and all x 2 R The exact solution of è24.12è and è24.13è is, of course, a function of t, but in order to make its dependence on the initial conion è24.13è explicit, we shall denote it by xèt; sè. Thus, we x èt; sè =f èt; xèt; xè0;sè=s If we diæerentiate these equations with respect to s @x Writing and noting that the second equation tells us that uè0;sè does not depend on s, wehave è24.14è è24.15è u 0 = f x u uè0è = 1 Theorem If f x éthen the solution of è24.14è and è24.15è satisæes the inequality juètèj e t ; t 0 Proof. Write æètè =, f x ètè Since f x ètè é, æètè is a positive function. And so we have u 0 u = f x =, æètè or d èln jujè, æètè Integrating both sides between 0 and t yields Z t Z t Z t d ln juètèj,ln juè0èj = èln jujè 0 = è, æètèè = t, æètè 0 0 or è24.16è ln juètèj ét where on the left hand side we have used the initial condition uè0è = 1 and on the right hand side we have used the fact that Z t t, æètè é t 0 since æètè is a positive function. Noting that the exponential function is monotonically increasing, we can maintain the inequality if we exponentiate both sides of è24.16è. We thus have uètè ée t Theorem If the inital value problem corresponding to è24.12è and è24.13è is solved with initial values s and s + æ, then the solution curves at time t diæer by at most jæje t.

7 Proof. By the Mean Value Theorem we have jxèt; sè, xèt; s + æèj = 3. GLOBAL TRUNCATION ERROR 7 æ æ xèt + æè jæj ; for some 2 ë0; 1ë uèt + æèjæj é jæj e t Theorem If the local truncation errors at t 1 ;t 2 ;::: ;t n = t 0 + nh do not exceed æ in magnitude, then the global truncation error " does not exceed Proof. e tn, 1 e h, 1 æ At the ærst iteration the total error is just the local truncation error so " 1 = æ At the second iteration the total error is the sum of the local truncation error and the error arising from the fact that initial condition might beoæby " 1.Thus Similarly at level 3 we would have and so on and so on until one has " 2 = æ + e h æ " 3 = æ + e h " 2 = æ +, æ + æe hæ e h = æ + æe h + æe 2h " n = æ + æe h + æe 2h + æææ+ æe èn,1èh X n,1 = æ e ih = æ enh, 1 e h, 1 = æ etn, 1 e h, 1 where in passing from the second to the third equation we have used the fact that x n, 1=èx, 1èèx n,1 + x n,2 + æææ+ x +1è Theorem If the local truncation error in a numerical solution of an initial value problem is of order Oèh m è then the global truncation error is of order Oèh m,1 è. Proof. According to the preceding theorem, " n = æ etn, 1 e h, 1 Now for, assuming we keep the ænal time t n æxed, we have for h éé 1 e tn, 1 e h, 1 = e tn, 1 1+h èhè2 + æææ, 1 e = tn, 1 h èhè2 + æææ = 1 h Oèh,1 è e tn, èhè+æææ

8 3. GLOBAL TRUNCATION ERROR 8 and so " n Oèh m è æo, h,1æ = O, h m,1æ