American Journal of Computational and Applied Mathematics 2014, 4(2): 51-59 DOI: 10.5923/j.ajcam.20140402.03
A Family of One-Block Implicit Multistep Backward Euler Type Methods
Ajie I. J.1,*, Ikhile M. N. O.2, Onumanyi P.1
1National Mathematical Centre, Abuja, Nigeria 2Department of Mathematics, University of Benin, Benin City, Nigeria
Abstract A family of k-step Backward Differentiation Formulas (BDFs)and the additional methods required to form blocks that possess L(α)-stability properties are derived by imposing order k on the general formula of continuous BDF. This leads to faster derivation of the coefficients compare to collocation and integrand approximation methods. Linear stability analysis of the resultant blocks show that they are L(α)-stable. Numerical examples are presented to show the efficacy of the methods. Keywords Backward Differentiation Formulas (BDFs), One-Block Implicit Backward Euler Type, L(α)-stable, A(α)-stable, Region of Absolute Stability (RAS)
We require that the methods belonging to (1.4) converge 1. Introduction efficiently like (1.1) but with better accuracy of order k, k 1 . The popular one-step Backward Euler Method (BEM) is given by the formula
yn11 y n h n f n n = 1, 2, 3, … (1.1) 2. Derivation of Formulae (1.4) (1.1) is known to possess a correct behaviour when the Consider the general linear multistep method given by stiffness ratio is severe in a stiff system of initial value kk problem of ordinary differential equations given in the form ry n r h n r f t n r, y n r (2.1) y f( t , y ); t [ a , b ] (1.2) rr00 where the step number k0, h t t is a variable y(), t00 y (1.3) n n1 n step length, αk and βk are both not zero. Following Lambert Our main concern in this paper is to propose a new family (1973) by making use of Taylor expansion of(2.1) and the of One-Block Implicit Backward Euler Type method of the associated linear operator L defined as form k A Y A Y h B F (1.4) (1)n 1 (0) n m (1) n 1 L[y(t);hn ] r y ( t rh ) r0 where (2.2) k T Y ( y , y , y , ..., y ) n1 n 1 n 2 n 3 n k hnr y() t rh T r0 Yn ( y n k 1 , ..., y n 2 , y n 1 , y n ) where y(t) is an arbitrary function which is continuously T Fn1 ( f n 1 , f n 2 , f n 3 , ..., f n k ) differentiable on the interval [a, b] and expanding the test and function y(t+rh) and its derivative y'(t+rh) about t gives (2.3) if we collect like terms, A(1), A (0) and B (1) are of order k by k. Lyxh ( );nn cyx01 ( ) chyx ( ) (2.3) * Corresponding author: qq() [email protected] (Ajie I. J.) cqn h y Published online at http://journal.sapub.org/ajcam Copyright © 2014 Scientific & Academic Publishing. All Rights Reserved where
52 Ajie I. J. et al.: A Family of One-Block Implicit Multistep Backward Euler Type Methods
c , 0 0 1 2 k 0 1 1 1 1. . .1 0 ck10 0 1 1 2 2 kk 0 1 0 1 2 3. . .k 1 1 2jk 0 1 22 3 2 . . . 2 and 2 1 ... (2.7) qq ckqk00 1 1 2 2 ... q!
1 ... qq11 12 2 k k k1 k k k (q 1)! kj0 1 2 3 . .. k k Equation (2.7) is used to derive k-step BDFs at point qk 2,3, , are constants. (2.4) For backward differentiation formulas (BDFs), we put in tt nk , additional methods are obtained by evaluating the derivative function at the points (2.1) i 0,ik 0,1, 2, ..., 1; and k 1 This gives t tnj ; j 1, 2, ..., k 1. This is done by computing k the vandermonde matrix at jk1, 2, ..., 1. The k-step y h f t, y (2.5) r n r n k n r n r Backward Differentiation Formula and the additional r0 methods are then combined to obtain a self-starting block If continuous coefficients of (2.5) are assumed, we have that can simultaneously generate the solutions k {yn1 , y n 2 , y n 3 , ..., y n k } of (1.2)-(1.3) at points ()(),t y h t f t y (2.6) r n r n j n r n r {tn1 , t n 2 , t n 3 , ..., t n k }. r0 We then express the block methods for a fixed value of k like the one-step Backward Euler Method (BEM) of the form where the coefficients i (t ), i 0,1, 2, ..., k , in formula (1.4) where A(1), A (0) and B (1) for the first six members (2.6), are evaluated at t tj j1, 2 ,..., k by are as given in this work. The computer finds it easier to imposing order k on it, that is c0 c 1 c 2 ... ck 0. compute the inverse of the vandermonde matrix with real This leads to the following k by k vandermonde matrix (see 1 constants in (2.7) than the matrix inverse D associated Brugnano and Trigiante (1998)). with the direct construction of interpolation and collocation matrix 1 1x x2 x 3 . . . xkk 1 x n n n n n 2 3kk 1 1xn1 x n 1 x n 1 . . . x n 1 x n 1 2 3kk 1 1xn2 x n 2 x n 2 . . . x n 2 x n 2 1 ...... D C (2.8) ...... ...... 1x x2 x 3 . . . xkk 1 x n k 1 n k 1 n k 1 n k 1 n k 1 3kk 2 1 0 1 2xn k 3 x n k . . . ( k 1) x n k kx n k such that CD = Ikxk identity matrix. The columns of C contains the constant coefficients of the continuous multistep method. (see Onumanyi et al. (2001), Akinfenwa et al (2011)). The first six members of the family expressed in form (1.4) have the following as A(1), A (0) and B (1) Case k = 1
Case k = 2
American Journal of Computational and Applied Mathematics 2014, 4(2): 51-59 53
Case k = 3
Case k = 4
Case k = 5
Case k = 6
54 Ajie I. J. et al.: A Family of One-Block Implicit Multistep Backward Euler Type Methods
3. Linear Stability Analysis of 1.4 (kz 1) k2 Applying (1.4) to the test equation 00 zzik 1 0 2 i k yy ; < 0 (3.1) R(z) i2 (3.6) k1 0 (kz 1) ik gives 00 i zz 2 i2 Y1 D(); z Y z h (3.2) where Case k = 1 1 D()() z A(1) zB (1) A (0) (3.3) From (3.3), we obtain the stability function R(z) which is a rational function of real coefficients given by Case k = 2 R( z ) Det [ IR D ( z )] (3.4) where I is a kk identity matrix. The stability domain S of a one-step block method is Sz{ C:R(z) 1} (3.5) Case k = 3 From the analysis given we easily obtain as follows the rational functions R(z) for k 1,2,3,...,11 which satisfies (3.5).
American Journal of Computational and Applied Mathematics 2014, 4(2): 51-59 55
Case k = 4
Case k = 5
Case k = 6
Case k = 7
Case k =8
Case k = 9
Case k = 10
Case k = 11
56 Ajie I. J. et al.: A Family of One-Block Implicit Multistep Backward Euler Type Methods
The characteristics polynomial of the method is (r, z) Det(A(1) r A (0) -zB (1) r) 0
The methods are absolutely stable in the region where rj (z ) 1, j 1,2,..., k . Using the boundary locus method, the regions of instabilities are as given below
Figure 1. Boundary loci of the proposed blocked methods of order k, k = 1, 2, 3, …, 12
L() stable family
Let k denote the step number, p = k the order and (0, ) 2
K 1 2 3 4 5 6 7 8 9 10 11 12 90o 90o 89o 85o 78o 75o 90 89o 89o 88o 85o 81.5o
The methods are A() stable since its region of absolute stability (RAS) contains an infinite wedge w z: arg( z ) and L() stable since (D ( ) ) 0. (see Chartier (1994))
4. Numerical Examples Example 4.1
American Journal of Computational and Applied Mathematics 2014, 4(2): 51-59 57
x The test equation y y; y (0) 1, with solution ye is solved with h = 0.01 and 5 using different order of the proposed methods. The results of the absolute errors are displayed in table 1 below
Table 1. Absolute errors in example 4.1 when solved with different order of the proposed methods
X Order 1 Order 3 Order 5 Order 7 Order 8 Order 9 0.02 4.10e-002 1.72e-006 3.82e-009 8.57e-012 4.22e-013 1.90e-014 0.04 3.52e-002 3.53e-006 3.45e-009 7.79e-012 3.84e-013 1.72e-014 0.06 3.01e-002 3.21e-006 6.69e-009 7.00e-012 3.47e-013 1.53e-014 0.08 2.57e-002 4.18e-006 5.95e-009 1.33e-011 3.02e-013 1.44e-014 0.10 2.19e-002 5.25e-006 5.50e-009 1.18e-011 5.56e-013 2.49e-014 0.12 1.85e-002 4.75e-006 7.30e-009 1.07e-011 5.05e-013 2.45e-014
Example 4.2 Consider the following linear constant coefficient initial value problem taken from Lambert (1973), 21 19 20 1 y19 21 20 y ; y (0) 0 40 40 40 1 Its theoretical solution is e2tt e 40 (cos(40 t ) sin(40 t )) 1 y( t ) e2tt e 40 (cos(40 t ) sin(40 t )) 2 2e40t (sin(40 t ) cos(40 t )) The above problem was solved by Brugnano and Trigiante (1998) using Generalized Backward Differentiation Formula (GBDF) of different order. GBDF is a boundary value method (BVM). We report here in table 2 the maximum absolute error obtained using our proposed methods and GBDF of the same order (7 and 8).
Table 2. Absolute errors in example 4.2 using the two methods proposed methods GBDF proposed methods GBDF H order 7 order 7 order 8 order 8 1e-002 5.683e-004 1.187e-003 1.386e-004 4.494e-004 5e-003 3.373e-006 1.389e-005 8.828e-007 2.683e-006 2.5e-003 2.789e-008 1.079e-007 4.845e-009 1.538e-008 1.25e-003 2.067e-010 1.079e-009 1.442e-011 8.543e-011 6.25e-0004 1.577e-012 9.409e-012 5.274e-014 4.431e-013
Example 4.3 yy12
2 y2 y 1 y 2(1 y 1 ); y 1 (0) 2, y 2 (0) 0 The Vander Pol problem is one of the problems used to demonstrate the ability of a method to solve stiff nonlinear problems. The above Vander Pol is solved for 30, using order 8 of our method and step size h = 0.005. The phase diagram of the solution computed using ode 15s is displayed in figure 1 while the one by our proposed method is in figure 2 below:
58 Ajie I. J. et al.: A Family of One-Block Implicit Multistep Backward Euler Type Methods
Figure 2. Solution of problem 4.3 in phase plane, computed with order 8 of the proposed method ( 30 )
Figure 3. Solution of problem 4.3 in phase plane, computed with order 8 of the proposed method ( 30 and h = 0.005)
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5. Conclusions Computational and Applied Mathematics, Bangkok, Thailand, p. 425-428. A family of efficient high order L(α)-Stable block [3] Akinfenwa, O., Yao, N. and Jator, S., (2011) “A Self-Starting methods for solving stiff ordinary differential equations has Block Adams Mathods for Solving Stiff Ordinary Differential been proposed and implemented for k = 1, 2, …, 12 as Equation,” in Computational Science and Engineering (CSE), self-starting methods. The methods which are derived from 2011 IEEE 14th International Conference, p. 127-136. Backward Differentiation Formula is based on order [4] Brugnano, L. and Trigiante, D. (1998), Solving differential definition. It circumvented the collocation and integrand problems by multistep initial and boundary value methods. approximation approaches which have been widely used in Published by Gordon and Beach Science Publisherspp the past by many authors. The numerical experiments show 136-139. that our methods have high order and can compete [5] Charter, P. (1994) „L-stable parallel one-block methods for favourably with existing ones. Ordinary differential equations‟ SIAM J. numer. anal. Vol. 31, No. 2, pp. 552-571.
[6] Fatunla, S. O., (1991) “Block Method for Second Order IVPs” J. Compt. Maths, vol. 41, p. 55-63. REFERENCES [7] Lambert, J. D., (1973) Computational Methods in Ordinary [1] Ajie, I. J., Onumanyi P., and Ikhile, M. N. O., (2011) Journal Differential Equations. New York: John Wiley, Sons, Inc. pp of The Nigerian Mathematical Society, vol. 13, p. 6-9 228-230. [2] Akinfenwa, O., Yao N. and Jator, S., (2011) „Implicit two step [8] Onumanyi, P., Sirisena, W.U. and Chollom, J. P., (2001) continuous hybrid block methods with four Off-Steps point Continuous hybrid methods through multistep collocation. for solving stiff ordinary differential equation.‟ In Abacus, Journal Mathematical Association of Nigeriapp Proceedings of the International Conference on 58-64.