Chapter 5 Consistency, Zero Stability, and the Dahlquist Equivalence Theorem

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2 Chapter 5 Conitency, Zero Stability, and the Dahlquit Equivalence Theorem In Chapter 2 we dicued convergence of numerical method and gave an experimental method for finding the rate of convergence (aka, global order of accuracy). When we devie a numerical method, however, we prefer to make a claim about the convergence propertie before implementing it. In thi chapter we will decribe the condition under which a numerical cheme can be claimed to be convergent. 18 Self-Aement Before reading thi chapter, you may wih to review... tability [Unified S/S II Lecture 12 Note] tability of difference equation [Wikipedia: Control Theory, Stability] baic of difference equation [6.003 Lecture 2 Slide] After reading thi chapter you hould be able to... evaluate the conitency of a numerical method evalute the zero tability of a numerical method decribe the relationhip between conitency, tability, and convergence 19 Zero tability A numerical method i zero table if the olution remain bounded a 0 for finite final time T. Recall the general form for an -tep multi-tep method a given in Definition 1: In the limit 0 we have α i v n+1 i = j=0 β j f n+1 j. (34) α i v n+1 i = 0. (35) Thi recurrence relationhip determine the characteritic, or unforced, behavior of the multi-tep method. The method i zero table if all olution to (35) remain bounded. Definition 1 (Zero tability). A multi-tep method i zero table if all olution to α i v n+1 i = 0 19

3 20 remain bounded a n. To determine if a method i zero table, we aume that the olution to the recurrence ha the following form, v n = v 0 z n, (36) where the upercript in the z n term i in fact a power. Note: z can be a complex number. If the recurrence relationhip ha olution with z > 1, then the multi-tep method would be untable. For our purpoe, a multi-tep method with a root of z =1 i zero-table provided the root i not repeated. (Note that in general, we need to be careful with the cae of z =1.) Example 1. In Example 2, the mot accurate two-tep, explicit method wa found to be, 4v n 5v n 1 = ( 4 f n + 2 f n 1). We will determine if thi algorithm i table. The recurrence relationhip i, 4v n 5v n 1 = 0. Then, ubtitution of v n = v 0 z n give, Factoring thi relationhip give, z n+1 + 4z n 5z n 1 = 0. z n 1( z 2 + 4z 5 ) = z n 1 (z 1)(z+5)=0. Thu, the recurrence relationhip ha root at z=1, z= 5, and z=0 (n 1 of thee root). The root at z= 5 will grow unbounded a n increae o thi method i not table. To demontrate the lack of convergence for thi method (due to it lack of tability), we again conider the olution of u t = u 2 with u(0)=1. Thee reult are hown in Figure 1. Thee reult clearly how the intability. Note that the olution ocillate a i expected ince the large paraitic root i negative (z = 5). Furthermore, decreaing from 0.1 to 0.05 only caue the intability to manifet itelf in horter time (though the ame number of tep). Clearly the method will not converge becaue of thi intability. 20 Conitency A numerical method i conitent if the multi-tep dicretization atifie the governing equation in the limit 0. Definition 2 (Conitency). A numerical method i conitent if it local truncation error ha the property lim 0 = 0. (37) Let v n+1 = g(v n+1,v n,...,t n+1,t n,...,) repreent the numerical method a we have een before. Then, if we write the ratio /, we find = g(un+1,u n,...,t n+1,t n,...,) u n+1, and ubtitute the Taylor erie of u n+1, = g(un+1,u n,...,t n+1,t n,...,) (u n + ut n +O( 2 )). We can rewrite thi in term of the lope of the numerical method = g(un+1,u n,...,t n+1,t n,...,) u n ut n +O()).

4 t = 0.1 v t t = 0.05 v t Fig. 3 Mot-accurate explicit, two-tep multi-tep method applied to u= u 2 with u(0)=1 with = 0.1 (upper plot) and 0.05 (lower plot). If we now take the limit 0, the lat term vanihe and we find lim 0 = g(un+1,u n,...,t n+1,t n,...,) u n u n t. It follow then that if the numerical method i conitent, we have lim 0 = 0 and therefore g(u n+1,u n,...,t n+1,t n,...,) u n = u n t. In other word, the multi-tep dicretization atifie the governing equation in the limit a 0. Thought Experiment For conitency we require lim 0 / = 0. Why i it inufficient to require the local truncation error to vanih in the limit, lim 0 = 0? We can extend the definition of conitency directly to a criterion on the order of accuracy of the numerical method. Requiring lim 0 / = 0 implie that = O( p+1 ) where p 1 ince then / = O( p ), which only vanihe in the limit if p 1. Thi mean that a conitent method mut be at leat firt-order accurate. Exercie 1. Which of the following numerical method i conitent?

5 22 (a) v n+1 = 1 2 vn ( f n + f n 1 ) (b) v n+1 = v n + f n+1 (c) v n+1 = v n f n (d) v n+1 = v n 1 21 Dahlquit Equivalence Theorem The Dahlquit Equivalence Theorem i the primary tool for aeing whether or not a numerical cheme i convergent. Uing the concept of conitency and zero tability alone, we can draw a concluion about the convergence. To ummarize from above, we have the following concept: Conitency: In the limit 0, the method give a conitent dicretization of the ordinary differential equation. Zero tability: In the limit 0, the method ha no olution that grow unbounded a N = T/. The Dahlquit Equivalence Theorem guarantee that a method that i conitent and table i convergent, and alo that a convergent method i conitent and table: Theorem 1 (Dahlquit Equivalence Theorem). A multi-tep method i convergent if and only if it i conitent and table. Exercie 2. The numerical cheme defined by the equation 4v n 5v n 1 = 4 f n + 2 f n 1 i (a) zero table, not conitent, and convergent (b) not zero table, conitent, and not convergent (c) zero table, conitent, and convergent (d) none of the above