A Class of Linearly Implicit Numerical Methods for Solving Stiff Ordinary Differential Equations

Transcription

1 The Open Numerical Method Journal, 2010, 2, Open Acce A Cla o Linearl Implicit Numerical Method or Solving Sti Ordinar Dierential Equation S.S. Filippov * and A.V. Tglian Keldh Intitute o Applied Mathematic, Ruian Academ o Science, Mocow , Ruia Abtract: We introduce ABC-cheme, a new cla o linearl implicit one-tep method or numerical integration o ti ordinar dierential equation tem. Formula o ABC-cheme invoke the Jacobian o dierential tem imilar to the method o Roenbrock tpe, but unlike the latter the include alo the quare o the Jacobian matrix. Keword: Solving ti ordinar dierential equation, Linearl implicit method, ABC-cheme, A-tabilit, L-tabilit. 1. INTRODUCTION We propoe a new cla o one-tep numerical method or olving ti ordinar dierential equation. Thee method emplo the Jacobian o a dierential tem and, in ditinction rom Roenbrock method [1], the quare o Jacobian i alo involved in their ormula. The irt two onetage method o thi kind were reported b S.S. Filippov and M.V. Bulatov (Conerence on Scientiic Computation, Geneva, Switzerland, June 26-29, 2002, p. 26); ee alo [2]. The term ABC-cheme or uch method wa uggeted later in [3]. In Section 2 one-tage ABC-cheme are deined and ome reult obtained or them are preented. Section 3 contain everal example o one-tage ABC-cheme. Multitage ABC-cheme are introduced in Section 4. Two example o two-tage ABC-cheme are given in Section 5. Some reult o a numerical experiment with ABC-cheme compared with thoe obtained b the ue o implicit Runge- Kutta method are preented in Section ONE-STAGE ABC-SCHEMES Deinition 1 A one-tage ABC-cheme or numerical integration o a Cauch problem or an autonomou tem o n ordinar dierential equation (x) = ((x)), (x 0 ) = 0 (1) i deined a ollow: (I + Ah + Bh 2 2 )[ 1 (h) 0 ] = (I + Ch )h (2) Here, A, B, and C are the coeicient that determine a particular method, 1 (h) i the deired numerical olution ater one tep o integration with the tep ize h, (x) and *Addre correpondence to thi author at the Keldh Intitute o Applied Mathematic o the Ruian Academ o Science, Miukaa q. 4, Mocow , Ruia; Tel: +7(495) ; Fax: +7(499) ; ilippov@ keldh.ru () are n-dimenional vector unction, i the Jacobian matrix, and I i the identit matrix. We conider the irt tep o integration a a repreentative one or the ubequent tep and write,,... without argument or ( 0 ), ( 0 ),.... The ollowing tatement or one-tage ABC-cheme can be eail proved in tandard wa (ee e.g. [4] and [5]). Theorem 1 The convergence order o method (2) i not le then one at an choice o real coeicient A, B, and C. Theorem 2 The order o method (2) equal two, i C = A In thi cae, the principal error term i equal to (x 0 + h) 1 (h) = h3 3! ( + ), where = 1+ 3A + 6B (3) and n 2 = i j k, j, k=1 j k n 2 = i j j, k=1 j k or i = 1, K,n. k Theorem 3 The tabilit unction o ABC-cheme (2) i given b 1+ (1 + A)z + (B + C)z2 1+ Az+ Bz / Bentham Open

2 2 The Open Numerical Method Journal, 2010, Volume 2 Filippov and Tglian Theorem 4 The ABC-cheme (2) o order two are A-table, i A 1 2, B A Theorem 5 The ABC-cheme (2) o order two are L-table, i B = A 1 2. Furthermore, ome important reult or linear autonomou tem ollow immediatel rom the above theorem. Corollar 1 ABC-cheme (2) approximate olution to linear tem (1) having contant coeicient with order three, i B = A 2 1 6, A 1 2. In thi cae, Eq. (3) ield = 0, and we have a amil o method depending on the ingle parameter A: I + Ah A h 2 2 ( (h) ) = h + A h 2 (4) with the principal error term (x 0 + h) 1 (h) = h4 4! (1 + 2A) 3 and tabilit unction 1+ (1 + A)z + A z 2 1+ Az A z 2 Thee method are A-table or A 12, and at A = 23 the method i alo L-table. Corollar 2 With A = 1 2 method (4) take the orm I 1 2 h h2 2 ( 1 (h) 0 ) = h. It give ourth order approximation or the olution o linear autonomou tem o dierential equation. It principal term o local error i then equal to (x 0 + h) 1 (h) = h5 1 5! 6 4. Thi method i A-table with the tabilit unction z z z z2 Remark 1 Solving linear tem o algebraic equation (2) in it original orm eem to be rather expenive. Indeed, in addition to ~ n 3 /3 multiplication and diviion that are needed or the LU- decompoition o the matrix in the let-hand ide 3 o Eq. (2), extra n multiplication are required or quaring a ull matrix o nth order, i.e. totall ~4n 3 /3 multiplicative operation. However, it i poible to avoid matrix multiplication b decompoing the matrix in the let-hand ide o Eq. (2) a ollow: I + Ah + Bh 2 2 = (I + Fh )(I + Gh ) where the new coeicient F and G are real or complex number depending on the value o A and B. In thi cae, onl two LU- decompoition o the matrice (I + Fh ) and (I + Gh ) are needed, i.e. totall ~2n 3 /3 multiplicative operation. Moreover, the amount o arithmetical operation can be once more halved, i we conine ourelve with the choice B = A 2 /4, becaue in thi cae F = G = A 2, and onl one LU- decompoition i needed ( cheap ABCcheme). Fig. (1) illutrate ome eential reult obtained or the one-tage ABC-cheme o order 2. Region A contain all pair o coeicient (A, B) correponding to A-table method (Theorem 4). The thick line L indicate L-table method (Theorem 5). The dahed line i the locu o all the method (4) with = 0 (Corollar 1). The dotted parabola B = A 2 /4 repreent the cheap ABC-cheme (Remark 1). The example rom the next ection are indicated b mall circle with correponding number. 3 EXAMPLES OF ONE-STAGE ABC-SCHEMES Each o the example given below i indicated in Fig. (1) b a mall circle with the number o the correponding example. Example 1 The choice A = 1 2 and B = C = 0 give an A-table method o the orm I h 2 [ 1 (h) 0 ] = h with = 1/ 2 in the principal term o the local error and the tabilit unction z z Actuall, thi i a Roenbrock tpe method, though it i not mentioned in [1]. Example 2 Now let u et A = 1, B = C = 12. In thi cae, we get an L-table method

3 A Cla o Linearl Implicit Numerical Method or Solving The Open Numerical Method Journal, 2010, Volume 2 3 Fig. (1). A-table and L-table one-tage econd order ABC-cheme. I h + h2 2 2 [ 1(h) 0 ] = h h2 2 Eq. (3) give = 1 or thi method, and the tabilit unction o it i given b 1 1 z z2 The method wa derived in other wa and dicued in [2]. Example 3 The choice A = 23, B = C = 16 give the L-table method I 2h 3 + h2 6 2 [ (h) h2 1 0 ] = h 6 with = 0 in the principal term o local error. Thereore, thi method i a member o the amil deined b Eq. (4). It tabilit unction ha the orm z z z2 It wa alo mentioned in [3]. Example 4 The choice A = 1 2, B = 112, and C = 0 give an A- table method decribed in Corollar 2 (ee above). The correponding value o rom Eq. (3) i equal to zero. Thi method i alo a member o the amil decribed b Eq. (4). It give ourth order approximation or the olution o linear autonomou tem o dierential equation. Example 5 With the choice B = A 2 /4, we get cheap ABCcheme that minimize the cot o olving the tem o linear algebraic equation (2) (ee Remark 1 above). Then Eq. (2) take the ollowing orm or the econd order method: 2 I Ah [ (h) ] = h + A h 2 Setting A = give an L-table method. The tabilit unction o thi method i 1+ ( 2 1)z z 2 Eq. (3) ield the value o equal to Example 6 Another example o a cheap ABC-cheme give the choice A = 1 3 1/ Thi time the value o i equal to zero. Thi mean that the method i alo a member o the amil decribed b Eq. (4), and it give third order approximation or the olution o linear autonomou tem o dierential equation. 4. MULTISTAGE ABC-SCHEMES Deinition 2 A multitage ABC-cheme or numerical integration o a Cauch problem or an autonomou tem o n ordinar dierential equation (1) i deined a ollow:

4 4 The Open Numerical Method Journal, 2010, Volume 2 Filippov and Tglian (I + A i h + B i h 2 2 )[u i (h) 0 ] = ( i I + C i h )h (u i1 (h))(i = 1,K.) 1 (h) = i = 1 i u i (h) (5) Here, A i, B i, C i, i, and i are the coeicient that determine a particular method, i the number o tage ( 1), 1 (h) i the deired numerical olution ater one tep o integration with the tep ize h ( 1 (h) i the weighted um o partial olution u i (h) obtained on the ith tage, u 0 (h) 0 ); (x) and () are n-dimenional vector unction, i the Jacobian matrix, and I i the identit matrix. We conider the irt tep o integration a a repreentative one or the ubequent tep and write,,... without argument or ( 0 ), ( 0 ),.... Note that the number o coeicient that deine a particular multitage ABC-cheme in the cae 2 i ubtantiall more then or one-tage ABC-cheme. Thi act enable one to contruct method o order higher then 2, but it i alo the caue o diicultie that encounter in the anali o order condition and tabilit unction. Theorem 6 Stabilit unction o multitage ABC-cheme can be written in the ollowing orm: i R i (z) The tabilit unction R i (z) o equential tage are evaluated recurivel: R i (z) = 1+ P i (z) Q i (z) R i1 (z) (i = 1, K, ) where R 0 (z) = 1, P i (z) = i z + C i z 2, Q i (z) = 1+ A i z + B i z 2 The proo o thi theorem i traightorward. One ha to appl the ormula rom Deinition 2 to Dahlquit tet equation ' =, where i a complex number (ee e.g. [5]), and then put h = z. 5. EXAMPLES OF TWO-STAGE ABC-SCHEMES The gain o uing cheap ABC-cheme (ee Remark 1 above) become till more in the cae o multitage ABCcheme. I we put A i = A, B i = B or all tage and aume B = A 2 4, then onl a ingle LU-decompoition will be needed on each tep o integration, ince the matrix in the let-hand ide o Eq. (5) i the ame or all tage. For thi reaon, we conine ourelve with two example o cheap two-tage ABC-cheme. Example 1 The choice 1 = 2 = 1, 1 = 2/3, 2 = 1/3, A A = A B 1 = B 2 = A 2 /4 give a amil o two-tage third order method depending on a ingle parameter A. In thi cae, C 1 = 3 4 A A C 2 = 3 2 A2 + 2A The tabilit unction at z take the orm 1 = 2, R() = A A 3 Thee method are A-table at the value o A between approximatel 0.75 and 0.4. The value A 0.59 correpond to an L-table method. Example 2 The choice 1 = 1/ 3, 2 = 1, 1 = 0, 2 = 1, A 1 = A 2 = A, B 1 = B 2 = A 2 /4 give again a amil o twotage third order method depending on a ingle parameter A, but in thi cae C 1 = 1 4 A A and the tabilit unction at C 2 = A z now take the orm R() = A A A A 4 Further reult or thee method will be preented elewhere. 6. NUMERICAL EXPERIMENT For our numerical experiment, we have choen a particular cae o the ingularl perturbed tet problem uggeted b Kap [6], namel, the initial value problem 1 (x) = (2 + 1 ) 1 (x) (x), 2 (x) = 1 (x) 2 (x) 2 2 (x), 1 (0) = 2 (0) = 1, 0 x 1. The exact olution o thi problem 1 (x) = e 2 x, 2 (x) = e x doe not depend on. However, the problem become ver ti, a 0. We compare the reult o numerical integration perormed with the ue o our method: - method 1 i the one-tage ABC-cheme rom Example 3 o Section 3; - method 2 i the implicit midpoint rule (one-tage Gau method [4, 5]); - method 3 i the two-tage cheap ABC-cheme rom Example 1 o Section 5 with A = 0.59 ; - method 4 i the two-tage Gau method [5].

5 A Cla o Linearl Implicit Numerical Method or Solving The Open Numerical Method Journal, 2010, Volume 2 5 Table 1. The Comparion o Reult Obtained with the Ue o ABC-Scheme and Gau Method Method 1 Method 2 Method 3 Method Method 1, 2, 3, and 4 have claical order 2, 2, 3, and 4, repectivel. Thee method were ued or the numerical integration o the above problem with everal diminihing value o and with two contant value o tep ize, h = 1/40 and h = 1/80. Table 1 contain the ollowing value: i the Euclidean norm o the abolute value o error or h = 1/80at the endpoint o the integration interval; = log 2 ( e 40 / e 80 ) i the actual order o accurac etimated uing the reult o integration with h = 1/ 40 and h = 1/80. In the cae o method 1 and 3 the computation wa perormed uing double preciion. In the cae o method 2 and 4 the data (evaluated with comparable preciion) are taken rom Table in [7]. Oberve that, or mall value o, ABC-cheme give better reult than implicit Runge-Kutta method. Note that the implicit Runge-Kutta method emplo Newton iteration, i.e. the are more expenive then the ABCcheme. One can clearl ee the phenomenon o lowering o the actual order o accurac at mall value o or the method 3 and 4, which i in accordance with the theor o B-convergence [5, 7]. ACKNOWLEGEMENTS The author are grateul to Proeor J.G. Verwer or hi encouraging criticim and to Proeor E. Hairer or drawing our attention to a poible wa to eliminate matrix multiplication. We are alo thankul to Dr M.V. Bulatov or a ueul dicuion o one-tage ABC-cheme and to Dr G.Yu. Kulikov or hi valuable obervation. REFERENCES [1] Roenbrock HH. Some general implicit procee or the numerical olution o dierential equation. Comput J 1962/63; 5: [2] Bulatov MV. Contruction o a one-tage L-table econd-order method. Dierential Equation 2003; 39: [3] Filippov SS. ABC-cheme or ti tem o ordinar dierential equation. Doklad Mathematic 2004; 70: [4] Hairer E, Nørett SP, Wanner G. Solving ordinar dierential equation, I: Nonti problem. Springer-Verlag: Berlin [5] Hairer E, Wanner G. Solving ordinar dierential equation, II: Sti and dierential-algebraic problem. Springer-Verlag: Berlin [6] Kap P. Roenbrock-tpe method. In: Dahlquit G, Jeltch R. Ed., Numerical method or olving ti initial value problem. Int. ür Geometrie und praktiche Math. der RWTH Aachen 1981; Bericht No. 9. [7] Dekker K, Verwer JG. Stabilit o Runge-Kutta method or ti nonlinear dierential equation. North-Holland: Amterdam Received: October 30, 2009 Revied: Januar 28, 2010 Accepted: Februar 02, 2010 Filippov and Tglian; Licenee Bentham Open. Thi i an open acce article licened under the term o the Creative Common Attribution Non-Commercial Licene ( which permit unretricted, non-commercial ue, ditribution and reproduction in an medium, provided the work i properl cited.