\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Adaptive order of block backward differentiation formulas for stiff ODEs

Abstract Full Text(HTML) Figure(5) / Table(4) Related Papers Cited by
  • In this paper, Adapative Order of Block Backward Differentiation Formulas (ABBDFs) are formulated using uniform step size for the numerical solution of stiff ordinary differential equations (ODEs). These ABBDF methods are of order four, five and six. The benefit of the ABBDF methods is the computation time in the computation of solutions. Numerical results are presented to demonstrate the advantage of implementing adaptive order selection in a single code.

    Mathematics Subject Classification: Primary: 65L04, 65L05; Secondary: 65L20.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Stability region of BBDF order 4, 5 and 6

    Figure 2.  k-order ABBDF method

    Figure 3.  Problem 3.1

    Figure 4.  Problem 3.2

    Table 1.  Coefficients for ABBDF

    $k$$y_{n-4}$$y_{n-3}$$y_{n-2}$$y_{n-1}$$y_{n}$$y_{n+1}$$y_{n+2}$$f_{n+1}$$f_{n+2}$
    400$\frac{1}{10}$$-\frac{3}{5}$ $\frac{9}{5}$1 $-\frac{3}{10}$ $\frac{12}{10}$0
    00$-\frac{3}{25}$$\frac{16}{25}$ $-\frac{36}{25}$ $\frac{48}{25}$10 $\frac{12}{25}$
    50$-\frac{3}{65}$$\frac{4}{13}$$-\frac{12}{13}$ $\frac{24}{13}$1 $\frac{12}{65}$ $\frac{12}{13}$0
    0$\frac{12}{137}$$-\frac{75}{137}$$\frac{200}{137}$ $-\frac{300}{137}$ $\frac{300}{137}$10 $\frac{16}{137}$
    6$\frac{2}{77}$-$\frac{15}{77}$$\frac{50}{77}$$-\frac{100}{77}$ $\frac{150}{77}$1 $-\frac{10}{77}$ $\frac{60}{77}$0
    -$\frac{10}{147}$$\frac{24}{49}$$-\frac{75}{49}$$\frac{400}{147}$ $-\frac{150}{49}$ $\frac{120}{49}$10 $\frac{60}{147}$
     | Show Table
    DownLoad: CSV

    Table 2.  Comparison between the ABBDF and BBDF for solving Problem 3.1

    $H$MethodMAXETIME
    $10^{-2}$BBDF(4)5.34238e-60.008280686
    BBDF(5)8.60443e-60.008490485
    BBDF(6)1.09196e-50.0089108066
    ABBDF6.29854e-60.0082606264
    $10^{-4}$BBDF(4)7.07097e-120.0145041159
    BBDF(5)3.99341e-110.01477376
    BBDF(6)5.75778e-110.01497369
    ABBDF2.51638e-110.014267131
    $10^{-6}$BBDF(4)1.19693e-120.331891367
    BBDF(5)1.27653e-120.35464837
    BBDF(6)1.54599e-120.366591181
    ABBDF1.32749e-120.3299601339
    $10^{-8}$BBDF(4)5.70600e-1132.31383741
    BBDF(5)6.27814e-1133.986499
    BBDF(6)8.38467e-1135.61432122
    ABBDF6.87739e-1132.222682490
     | Show Table
    DownLoad: CSV

    Table 3.  Comparison between the ABBDF and BBDF for solving Problem 3.2

    $H$MethodMAXETIME
    $10^{-2}$BBDF(4)6.13882e+10.008227781
    BBDF(5)2.14167e-10.0082600851
    BBDF(6)2.96326e+00.00828741
    ABBDF1.44159e-20.008064681
    $10^{-4}$BBDF(4)7.76160e-80.01543074
    BBDF(5)9.48795e-80.016383168
    BBDF(6)1.11123e-70.0171090074
    ABBDF7.76160e-80.0142214673
    $10^{-6}$BBDF(4)1.93315e-112.97284740
    BBDF(5)1.07853e-113.12728618
    BBDF(6)2.66430e-113.8798845185
    ABBDF1.93315e-112.31853289
    $10^{-8}$BBDF(4)2.35567e-984.36481221
    BBDF(5)4.41047e-1085.76521578
    BBDF(6)2.59949e-987.69974205
    ABBDF2.35567e-984.36227574
     | Show Table
    DownLoad: CSV

    Table 4.  Comparison between the ABBDF and mathod in [13] for solving Problem 3.3

    $H$MethodMAXE
    $10^{-3}$BBDF(5)7.43088e-3
    BDF4.97642e+43
    ode15s6.30000e-3
    ABBDF2.02095e-3
    $10^{-5}$BBDF(5)7.68218e-4
    BDF7.88641e-4
    ode15s6.40000e-3
    ABBDF6.22269e-4
    $10^{-7}$BBDF(5)8.09607e-8
    BDF8.39232e-8
    ode15s4.30000e-5
    ABBDF6.49214e-8
     | Show Table
    DownLoad: CSV
  • [1] L. G. Birta and O. Abou-Rabia, Parallel block predictor corrector methods for ODEs, IEEE Transactions on Computer, 36 (1987), 299-311. 
    [2] L. Chu and H. Hamilton, Parallel solution of ODEs by multi-block methods, SIAM J. Sci. Statist. Computation, 8 (1987), 342-353.  doi: 10.1137/0908039.
    [3] S. O. Fatunla, Numerical Methods for Initial Value Problems for Ordinary Differential equations 1st edition, U. S. A Academy Press, Boston, 1988.
    [4] P. J. Houwen and B. P. Sommeijer, Block Runge-Kutta Methods on Parallel Computers Report NM-R8906, Centre for Mathematics and Computer Science, Amsterdam, 1989.
    [5] P. J. Houwen and B. P. Sommeijer, Block Runge-Kutta methods on parallel computers Report NM-R8906, Centre for Mathematics and Computer Science, Amsterdam.
    [6] Z. B. IbrahimK. I. Othman and M. B. Suleiman, Implicit r-point block backward differentiation formula for solving first-order stiff ODEs, Applied Mathematics and Computation, 186 (2007), 558-565.  doi: 10.1016/j.amc.2006.07.116.
    [7] Z. B. IbrahimK. I. Othman and M. B. Suleiman, Variable step size block backward differentiation formula for solving stiff ODEs, Proc. of World Congress on Engineering, LONDON, U.K, 2 (2007), 785-789. 
    [8] Z. B. IbrahimM. B. Suleiman and K. I. Othman, Fixed coefficients block backward differentiation formulas for the numerical solution of stiff ordinary differential equations, European Journal of Scientific Research, 21 (2008), 508-520. 
    [9] J. D. Lambert, Numerical Methods for Ordinary Differential Systems 2nd edition, John Willey and Sons, New York, 1991.
    [10] N. E. Mastorakis, An extended Crank-Nicholson method and its Applications in the Solution of Partial Differential Equations: 1-D and 3-D Conduction Equations, 10th WSEAS International Conference on Applied Mathematics, (2006), 134-143. 
    [11] Z. A. Majid, Parallel Block Methods for Solving Ordinary Differential Equations Ph. D thesis, Faculty of Science, Universiti Putra Malaysia, 2004.
    [12] N. E. Mastorakis and O. Martin, About the numerical solution of a stationary transport equation, 5th WSEAS Int. Conf. on Simulation, Modeling and Optimization, (2005), 419-426. 
    [13] N. A. A. M. NasirZ. B. IbrahimK. I. Othman and M. B. Suleiman, Fifth order two-point block backward differentiation formulas for solving ordinary differential equations, Applied Mathematical Sciences, 5 (2011), 3505-3518. 
    [14] L. F. Shampine, Order selection in ODE codes based on implicit formulas, Applied Mathematics Letters, 4 (1991), 53-55.  doi: 10.1016/0893-9659(91)90168-U.
    [15] L. F. Shampine and H. A. Watts, Block implicit one-step methods, Math. Comp., 23 (1969), 731-740. 
  • 加载中

Figures(5)

Tables(4)

SHARE

Article Metrics

HTML views(627) PDF downloads(204) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return