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September  2020, 13(9): 2603-2617. doi: 10.3934/dcdss.2020164

Functionally-fitted block $ \theta $-methods for ordinary differential equations

1. 

College of Information, Mechanical and Electrical Engineering, Shanghai Normal University, 100 Guilin Road, Shanghai 200234, China

2. 

Department of Mathematics, Shanghai Normal University, Scientific Computing Key Laboratory of Shanghai Universities, 100 Guilin Road, Shanghai 200234, China

* Corresponding author: Hongjiong Tian

Dedicated to the memory of Professor Christopher T. H. Baker

Received  December 2018 Revised  May 2019 Published  September 2020 Early access  December 2019

Fund Project: The work of the authors is supported in part by E-Institutes of Shanghai Municipal Education Commission under Grant No. E03004, the National Natural Science Foundation of China under Grant Nos. 11671266 and 11871343, and the Natural Science Foundation of Shanghai under Grant No. 16ZR1424900

We propose a new family of functionally-fitted block $ \theta $-methods for numerically solving ordinary differential equations which integrates a chosen set of linearly independent functions exactly. The advantage of such variable coefficient methods is that the basis functions can be chosen to exploit specific properties of the problem that may be known in advance. The basic theory for the proposed methods is established. First, we derive a sufficient condition to ensure the existence of the functionally-fitted block $ \theta $-methods, and discuss the independence on integration time for a set of separable basis functions. We then obtain some basic characteristics of the methods by Taylor series expansions, and show that the order of accuracy of $ r $-dimensional functionally-fitted block $ \theta $-method is at least $ r $ for ordinary differential equations. Numerical experiments are conducted to illustrate the efficiency of the functionally-fitted block $ \theta $-methods.

Citation: Jingwen Wu, Jintao Hu, Hongjiong Tian. Functionally-fitted block $ \theta $-methods for ordinary differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2603-2617. doi: 10.3934/dcdss.2020164
References:
[1]

G. D. AndriaG. D. Byrne and D. R. Hill, Natrual spline block implicit methods, BIT, 13 (1973), 131-144.  doi: 10.1007/bf01933485.

[2]

J. E. Bond and J. R. Cash, A block method for the numerical integration of stiff systems of ordinary differential equations, BIT, 19 (1979), 429-447.  doi: 10.1007/BF01931259.

[3]

L. Brugnano and D. Trigiante, Block implicit methods for ODEs, Recent Trends in Numerical Analysis, Nova Science Publishers, New York, 3 (2001), 81-105. 

[4]

L. Brugnano and D. Trigiante, Solving ODEs By Multistep Initial and Boundary Value Methods, Gordon & Breach: Amsterdam, 1998.

[5]

L. BrugnanoF. Iavernaro and D. Trigiante, Hamiltonian boundary value methods (energy preserving discrete line methods), JNAIAM, 5 (2010), 17-37. 

[6]

L. BrugnanoF. Iavernaro and D. Trigiante, A simple framework for the derivation and analysis of effective one-step methods for ODEs, Appl. Math. Comput., 218 (2012), 8475-8485.  doi: 10.1016/j.amc.2012.01.074.

[7]

J. C. Butcher, Numerical Methods for Ordinary Differential Equations, Second edition. John Wiley & Sons, Ltd., Chichester, 2008. doi: 10.1002/9780470753767.

[8]

J. R. Cash, A note on the exponential fitting of blended, extended linear multistep methods, BIT, 21 (1981), 450-453.  doi: 10.1007/BF01932841.

[9]

P. Chartier, $L$-stable parallel one-block methods for ordinary differential equations, SIAM J. Numer. Anal., 31 (1994), 552-571.  doi: 10.1137/0731030.

[10]

M. T. Chu and H. Hamilton, Parallel solution of ODEs by multi-block methods, SIAM J. Sci. Stat. Comput., 8 (1987), 342-353.  doi: 10.1137/0908039.

[11]

J. P. Coleman, P-stability and exponential-fitting methods for $y'' = f(x, y)$, IMA J. Numer. Anal., 16 (1996), 179-199.  doi: 10.1093/imanum/16.2.179.

[12]

R. D'AmbrosioE. Esposito and B. Paternoster, Parameter estimation in exponentially fitted hybrid methods for second order differential problems, J. Math. Chem., 50 (2012), 155-168.  doi: 10.1007/s10910-011-9903-7.

[13]

W. H. EnrightT. E. Hull and B. Lindberg, Comparing numerical methods for stiff systems of O.D.E.'s, BIT, 15 (1975), 10-48. 

[14]

W. Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials, Numer. Math., 3 (1961), 381-397.  doi: 10.1007/BF01386037.

[15]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Second edition, Springer Series in Computational Mathematics, 8. Springer-Verlag, Berlin, 1993.

[16]

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Second edition, Springer Series in Computational Mathematics, 14. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-05221-7.

[17]

N. S. Hoang and R. B. Sidje, On the stability of functionally fitted Runge-Kutta methods, BIT, 48 (2008), 61-77.  doi: 10.1007/s10543-007-0158-4.

[18]

N. S. HoangR. B. Sidje and N. H. Cong, On functionally-fitted Runge-Kutta methods, BIT, 46 (2006), 861-874.  doi: 10.1007/s10543-006-0092-x.

[19]

H. S. HoangR. B. Sidje and N. H. Cong, Analysis of trigonometric implicit Runge-Kutta methods, J. Comput. Appl. Math., 198 (2007), 187-207.  doi: 10.1016/j.cam.2005.12.006.

[20]

F. Iavernaro and F. Mazzia, Convergence and stability of multistep methods solving nonlinear initial value problems, SIAM J. Sci. Comput., 18 (1997), 270-285.  doi: 10.1137/S1064827595287122.

[21]

L. Gr. IxaruM. RizeaH. De Meyer and G. Vanden Berghe, Weights of the exponential fitting multistep algorithms for ODEs. Advanced numerical methods for mathematical modelling, J. Comput. Appl. Math., 132 (2001), 83-93.  doi: 10.1016/S0377-0427(00)00599-9.

[22]

L. Gr. IxaruG. Vanden Berghe and H. De Meyer, Frequency evaluation in exponential fitting multistep algorithms for ODEs, J. Comput. Appl. Math., 140 (2002), 423-434.  doi: 10.1016/S0377-0427(01)00474-5.

[23]

L. Gr. IxaruG. Vanden Berghe and H. De Meyer, Exponentially fitted variable two-step BDF algorithms for first order ODEs, Comput. Phys. Commun., 150 (2003), 116-128.  doi: 10.1016/S0010-4655(02)00676-8.

[24]

S. N. JatorS. Swindell and R. French, Trigonometrically fitted block Numerov type method for $y'' = f(x, y, y')$, Numer. Algor., 62 (2013), 13-26.  doi: 10.1007/s11075-012-9562-1.

[25]

J. D. Lambert, Numerical Methods for Ordinary Differential Systems. The Initial Value Problem, John Wiley & Sons, Inc., Chichester, 1991.

[26]

L. H. Lu, The stability of the block $\theta$-methods, IMA J. Numer. Anal., 13 (1993), 101-114.  doi: 10.1093/imanum/13.1.101.

[27]

S. MehrkanoonZ. A. Majid and M. Suleiman, A variable step implicit block multistep method for solving first-order ODEs, J. Comput. Appl. Math., 233 (2010), 2387-2394.  doi: 10.1016/j.cam.2009.10.023.

[28]

H. S. NguyenR. B. Sidje and N. H. Cong, Analysis of trigonometric implicit Runge-Kutta methods, J. Comput. Appl. Math., 198 (2007), 187-207.  doi: 10.1016/j.cam.2005.12.006.

[29]

F. F. Ngwane and S. N. Jator, Block hybrid method using trigonometric basis for initial value problems with oscillating solutions, Numer. Algor., 63 (2013), 713-725.  doi: 10.1007/s11075-012-9649-8.

[30]

K. Ozawa, A four-stage implicit Runge-Kutta-Nyström method with variable coefficients for solving periodic initial value problems, Japan J. Indust. Appl. Math., 16 (1999), 25-46.  doi: 10.1007/BF03167523.

[31]

K. Ozawa, A functional fitting Runge-Kutta Method with variable coefficients, Japan J. Indust. Appl. Math., 18 (2001), 107-130.  doi: 10.1007/BF03167357.

[32]

K. Ozawa, A functional fitting Runge-Kutta-Nyström method with variable coefficients, Japan J. Indust. Appl. Math., 19 (2002), 55-85.  doi: 10.1007/BF03167448.

[33]

B. Paternoster, Runge-Kutta(-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials, Appl. Numer. Math., 28 (1998), 401-412.  doi: 10.1016/S0168-9274(98)00056-7.

[34]

A. D. Raptis and T. E. Simos, A four-step phase-fitted method for the numerical integration of second order initial-value problems, BIT, 31 (1991), 160-168.  doi: 10.1007/BF01952791.

[35]

L. F. Shampine and H. A. Watts, Block implicit one-step methods, Math. Comput., 23 (1969), 731-740.  doi: 10.1090/S0025-5718-1969-0264854-5.

[36]

L. F. Shampine, Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, 1994.

[37]

T. E. Simos, Some new four-step exponential-fitting methods for the numerical solution of the radial Schrödinger equation, IMA J. Numer. Anal., 11 (1991), 347-356.  doi: 10.1093/imanum/11.3.347.

[38]

B. P. SommeijerW. Couzy and P. J. van der Houwen, $A$-stable parallel block methods for ordinary and integro-differential equations, Appl. Numer. Math., 9 (1992), 267-281.  doi: 10.1016/0168-9274(92)90021-5.

[39]

R. M. ThomasT. E. Simos and G. V. Mitsou, A family of Numerov type exponential fitted predictor-corrector methods for the numerical integration of the radial Schrodinger equation, J. Comput. Appl. Math., 67 (1996), 255-270.  doi: 10.1016/0377-0427(94)00126-X.

[40]

H. J. TianK. T. Shan and J. X. Kuang, Continuous block $\theta$-methods for ordinary and delay differential equations, SIAM J. Sci. Comput., 31 (2009/10), 4266-4280.  doi: 10.1137/080730779.

[41]

H. J. TianQ. H. Yu and C. L. Jin, Continuous block implicit hybrid one-step methods for ordinary and delay differential equations, Appl. Numer. Math., 61 (2011), 1289-1300.  doi: 10.1016/j.apnum.2011.09.001.

[42]

J. VanthournoutG. Vanden Berghe and H. De Meyer, Families of backward differentiation methods based on a new type of mixed interpolation, Comput. Math. Appl., 20 (1990), 19-30.  doi: 10.1016/0898-1221(90)90215-6.

[43]

H. Van de Vyver, Frequency evaluation for exponentially fitted Runge-Kutta methods, J. Comput. Appl. Math., 184 (2005), 442-463.  doi: 10.1016/j.cam.2005.01.020.

[44]

D. S. Watanabe, Block implicit one-step methods, Math. Comput., 32 (1978), 405-414.  doi: 10.1090/S0025-5718-1978-0494959-0.

[45]

H. A. Watts and L. F. Shampine, $A$-stable block implicit one-step methods, BIT, 12 (1972), 252-266.  doi: 10.1007/bf01932819.

[46]

J. W. Wu and H. J. Tian, Functionally-fitted block methods for ordinary differential equations, J. Comput. Appl. Math., 271 (2014), 356-368.  doi: 10.1016/j.cam.2014.04.013.

[47]

J. W. Wu and H. J. Tian, Functionally-fitted block methods for second order ordinary differential equations, Comput. Phys. Communc., 197 (2015), 96-108.  doi: 10.1016/j.cpc.2015.08.010.

[48]

L. Xie and H. J. Tian, Continuous parallel block methods and their applications, Appl. Math. Comp., 241 (2014), 356-370.  doi: 10.1016/j.amc.2014.05.026.

[49]

Y. XuJ. J. Zhao and Z. N. Sui, Exponential Runge-Kutta methods for delay differential equations, Math. Comput. Simulation, 80 (2010), 2350-2361.  doi: 10.1016/j.matcom.2010.05.016.

[50]

B. Zhou, $A$-stable and $L$-stable block implicit one-step methods, J. Comput. Math., 3 (1985), 328-341. 

show all references

References:
[1]

G. D. AndriaG. D. Byrne and D. R. Hill, Natrual spline block implicit methods, BIT, 13 (1973), 131-144.  doi: 10.1007/bf01933485.

[2]

J. E. Bond and J. R. Cash, A block method for the numerical integration of stiff systems of ordinary differential equations, BIT, 19 (1979), 429-447.  doi: 10.1007/BF01931259.

[3]

L. Brugnano and D. Trigiante, Block implicit methods for ODEs, Recent Trends in Numerical Analysis, Nova Science Publishers, New York, 3 (2001), 81-105. 

[4]

L. Brugnano and D. Trigiante, Solving ODEs By Multistep Initial and Boundary Value Methods, Gordon & Breach: Amsterdam, 1998.

[5]

L. BrugnanoF. Iavernaro and D. Trigiante, Hamiltonian boundary value methods (energy preserving discrete line methods), JNAIAM, 5 (2010), 17-37. 

[6]

L. BrugnanoF. Iavernaro and D. Trigiante, A simple framework for the derivation and analysis of effective one-step methods for ODEs, Appl. Math. Comput., 218 (2012), 8475-8485.  doi: 10.1016/j.amc.2012.01.074.

[7]

J. C. Butcher, Numerical Methods for Ordinary Differential Equations, Second edition. John Wiley & Sons, Ltd., Chichester, 2008. doi: 10.1002/9780470753767.

[8]

J. R. Cash, A note on the exponential fitting of blended, extended linear multistep methods, BIT, 21 (1981), 450-453.  doi: 10.1007/BF01932841.

[9]

P. Chartier, $L$-stable parallel one-block methods for ordinary differential equations, SIAM J. Numer. Anal., 31 (1994), 552-571.  doi: 10.1137/0731030.

[10]

M. T. Chu and H. Hamilton, Parallel solution of ODEs by multi-block methods, SIAM J. Sci. Stat. Comput., 8 (1987), 342-353.  doi: 10.1137/0908039.

[11]

J. P. Coleman, P-stability and exponential-fitting methods for $y'' = f(x, y)$, IMA J. Numer. Anal., 16 (1996), 179-199.  doi: 10.1093/imanum/16.2.179.

[12]

R. D'AmbrosioE. Esposito and B. Paternoster, Parameter estimation in exponentially fitted hybrid methods for second order differential problems, J. Math. Chem., 50 (2012), 155-168.  doi: 10.1007/s10910-011-9903-7.

[13]

W. H. EnrightT. E. Hull and B. Lindberg, Comparing numerical methods for stiff systems of O.D.E.'s, BIT, 15 (1975), 10-48. 

[14]

W. Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials, Numer. Math., 3 (1961), 381-397.  doi: 10.1007/BF01386037.

[15]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Second edition, Springer Series in Computational Mathematics, 8. Springer-Verlag, Berlin, 1993.

[16]

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Second edition, Springer Series in Computational Mathematics, 14. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-05221-7.

[17]

N. S. Hoang and R. B. Sidje, On the stability of functionally fitted Runge-Kutta methods, BIT, 48 (2008), 61-77.  doi: 10.1007/s10543-007-0158-4.

[18]

N. S. HoangR. B. Sidje and N. H. Cong, On functionally-fitted Runge-Kutta methods, BIT, 46 (2006), 861-874.  doi: 10.1007/s10543-006-0092-x.

[19]

H. S. HoangR. B. Sidje and N. H. Cong, Analysis of trigonometric implicit Runge-Kutta methods, J. Comput. Appl. Math., 198 (2007), 187-207.  doi: 10.1016/j.cam.2005.12.006.

[20]

F. Iavernaro and F. Mazzia, Convergence and stability of multistep methods solving nonlinear initial value problems, SIAM J. Sci. Comput., 18 (1997), 270-285.  doi: 10.1137/S1064827595287122.

[21]

L. Gr. IxaruM. RizeaH. De Meyer and G. Vanden Berghe, Weights of the exponential fitting multistep algorithms for ODEs. Advanced numerical methods for mathematical modelling, J. Comput. Appl. Math., 132 (2001), 83-93.  doi: 10.1016/S0377-0427(00)00599-9.

[22]

L. Gr. IxaruG. Vanden Berghe and H. De Meyer, Frequency evaluation in exponential fitting multistep algorithms for ODEs, J. Comput. Appl. Math., 140 (2002), 423-434.  doi: 10.1016/S0377-0427(01)00474-5.

[23]

L. Gr. IxaruG. Vanden Berghe and H. De Meyer, Exponentially fitted variable two-step BDF algorithms for first order ODEs, Comput. Phys. Commun., 150 (2003), 116-128.  doi: 10.1016/S0010-4655(02)00676-8.

[24]

S. N. JatorS. Swindell and R. French, Trigonometrically fitted block Numerov type method for $y'' = f(x, y, y')$, Numer. Algor., 62 (2013), 13-26.  doi: 10.1007/s11075-012-9562-1.

[25]

J. D. Lambert, Numerical Methods for Ordinary Differential Systems. The Initial Value Problem, John Wiley & Sons, Inc., Chichester, 1991.

[26]

L. H. Lu, The stability of the block $\theta$-methods, IMA J. Numer. Anal., 13 (1993), 101-114.  doi: 10.1093/imanum/13.1.101.

[27]

S. MehrkanoonZ. A. Majid and M. Suleiman, A variable step implicit block multistep method for solving first-order ODEs, J. Comput. Appl. Math., 233 (2010), 2387-2394.  doi: 10.1016/j.cam.2009.10.023.

[28]

H. S. NguyenR. B. Sidje and N. H. Cong, Analysis of trigonometric implicit Runge-Kutta methods, J. Comput. Appl. Math., 198 (2007), 187-207.  doi: 10.1016/j.cam.2005.12.006.

[29]

F. F. Ngwane and S. N. Jator, Block hybrid method using trigonometric basis for initial value problems with oscillating solutions, Numer. Algor., 63 (2013), 713-725.  doi: 10.1007/s11075-012-9649-8.

[30]

K. Ozawa, A four-stage implicit Runge-Kutta-Nyström method with variable coefficients for solving periodic initial value problems, Japan J. Indust. Appl. Math., 16 (1999), 25-46.  doi: 10.1007/BF03167523.

[31]

K. Ozawa, A functional fitting Runge-Kutta Method with variable coefficients, Japan J. Indust. Appl. Math., 18 (2001), 107-130.  doi: 10.1007/BF03167357.

[32]

K. Ozawa, A functional fitting Runge-Kutta-Nyström method with variable coefficients, Japan J. Indust. Appl. Math., 19 (2002), 55-85.  doi: 10.1007/BF03167448.

[33]

B. Paternoster, Runge-Kutta(-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials, Appl. Numer. Math., 28 (1998), 401-412.  doi: 10.1016/S0168-9274(98)00056-7.

[34]

A. D. Raptis and T. E. Simos, A four-step phase-fitted method for the numerical integration of second order initial-value problems, BIT, 31 (1991), 160-168.  doi: 10.1007/BF01952791.

[35]

L. F. Shampine and H. A. Watts, Block implicit one-step methods, Math. Comput., 23 (1969), 731-740.  doi: 10.1090/S0025-5718-1969-0264854-5.

[36]

L. F. Shampine, Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, 1994.

[37]

T. E. Simos, Some new four-step exponential-fitting methods for the numerical solution of the radial Schrödinger equation, IMA J. Numer. Anal., 11 (1991), 347-356.  doi: 10.1093/imanum/11.3.347.

[38]

B. P. SommeijerW. Couzy and P. J. van der Houwen, $A$-stable parallel block methods for ordinary and integro-differential equations, Appl. Numer. Math., 9 (1992), 267-281.  doi: 10.1016/0168-9274(92)90021-5.

[39]

R. M. ThomasT. E. Simos and G. V. Mitsou, A family of Numerov type exponential fitted predictor-corrector methods for the numerical integration of the radial Schrodinger equation, J. Comput. Appl. Math., 67 (1996), 255-270.  doi: 10.1016/0377-0427(94)00126-X.

[40]

H. J. TianK. T. Shan and J. X. Kuang, Continuous block $\theta$-methods for ordinary and delay differential equations, SIAM J. Sci. Comput., 31 (2009/10), 4266-4280.  doi: 10.1137/080730779.

[41]

H. J. TianQ. H. Yu and C. L. Jin, Continuous block implicit hybrid one-step methods for ordinary and delay differential equations, Appl. Numer. Math., 61 (2011), 1289-1300.  doi: 10.1016/j.apnum.2011.09.001.

[42]

J. VanthournoutG. Vanden Berghe and H. De Meyer, Families of backward differentiation methods based on a new type of mixed interpolation, Comput. Math. Appl., 20 (1990), 19-30.  doi: 10.1016/0898-1221(90)90215-6.

[43]

H. Van de Vyver, Frequency evaluation for exponentially fitted Runge-Kutta methods, J. Comput. Appl. Math., 184 (2005), 442-463.  doi: 10.1016/j.cam.2005.01.020.

[44]

D. S. Watanabe, Block implicit one-step methods, Math. Comput., 32 (1978), 405-414.  doi: 10.1090/S0025-5718-1978-0494959-0.

[45]

H. A. Watts and L. F. Shampine, $A$-stable block implicit one-step methods, BIT, 12 (1972), 252-266.  doi: 10.1007/bf01932819.

[46]

J. W. Wu and H. J. Tian, Functionally-fitted block methods for ordinary differential equations, J. Comput. Appl. Math., 271 (2014), 356-368.  doi: 10.1016/j.cam.2014.04.013.

[47]

J. W. Wu and H. J. Tian, Functionally-fitted block methods for second order ordinary differential equations, Comput. Phys. Communc., 197 (2015), 96-108.  doi: 10.1016/j.cpc.2015.08.010.

[48]

L. Xie and H. J. Tian, Continuous parallel block methods and their applications, Appl. Math. Comp., 241 (2014), 356-370.  doi: 10.1016/j.amc.2014.05.026.

[49]

Y. XuJ. J. Zhao and Z. N. Sui, Exponential Runge-Kutta methods for delay differential equations, Math. Comput. Simulation, 80 (2010), 2350-2361.  doi: 10.1016/j.matcom.2010.05.016.

[50]

B. Zhou, $A$-stable and $L$-stable block implicit one-step methods, J. Comput. Math., 3 (1985), 328-341. 

Figure 1.  Global errors $ \log_{2}(E) $ of FFBT1, FFBT2, FFBT3 and BT with various $ \varepsilon $
Figure 2.  Global errors $ \log_{10}E $ (left) and Execution times (right) of FFBT and FFB
Table 1.  Maximum absolute errors of FFBT(Ⅰ)-FFBT(Ⅲ), BT(Ⅰ)-BT(Ⅲ)
$ h $ FFBT(Ⅰ) FFBT(Ⅱ) FFBT(Ⅲ) BT(Ⅰ) BT(Ⅱ) BT(Ⅲ)
0.4 3.66e-19 4.04e-13 2.07e-01 1.00e-04 2.51e-05 3.16e-01
0.2 6.30e-19 2.87e-18 3.20e-03 2.51e-05 5.01e-06 1.58e-06
0.1 2.74e-18 1.69e-18 2.41e-11 5.01e-06 1.26e-06 3.98e-07
0.05 1.11e-17 9.41e-18 2.91e-18 1.26e-06 3.16e-07 1.00e-07
$ h $ FFBT(Ⅰ) FFBT(Ⅱ) FFBT(Ⅲ) BT(Ⅰ) BT(Ⅱ) BT(Ⅲ)
0.4 3.66e-19 4.04e-13 2.07e-01 1.00e-04 2.51e-05 3.16e-01
0.2 6.30e-19 2.87e-18 3.20e-03 2.51e-05 5.01e-06 1.58e-06
0.1 2.74e-18 1.69e-18 2.41e-11 5.01e-06 1.26e-06 3.98e-07
0.05 1.11e-17 9.41e-18 2.91e-18 1.26e-06 3.16e-07 1.00e-07
Table 2.  Maximum absolute errors of FFBT(A)-FFBT(B), BT(A)-BT(B).
$ h $ FFBT(A) FFBT(B) BT(A) BT(B)
0.4 2.93e-04 1.39e-04 5.92e-02 1.41e-04
0.2 1.12e-04 6.58e-05 6.35e-02 6.60e-05
0.1 5.72e-05 4.46e-05 6.57e-02 4.47e-05
0.05 2.86e-05 1.11e-05 6.68e-02 1.11e-05
$ h $ FFBT(A) FFBT(B) BT(A) BT(B)
0.4 2.93e-04 1.39e-04 5.92e-02 1.41e-04
0.2 1.12e-04 6.58e-05 6.35e-02 6.60e-05
0.1 5.72e-05 4.46e-05 6.57e-02 4.47e-05
0.05 2.86e-05 1.11e-05 6.68e-02 1.11e-05
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