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Preface: "Structural Dynamical Systems: Computational aspects"
A comparison of boundary correction methods for Strang splitting
Department of Mathematics, University of Innsbruck, A-6020 Innsbruck, Austria |
In this paper we investigate splitting methods in the presence of non-homogeneous boundary conditions. In particular, we consider the corrections that have been described and analyzed in Einkemmer, Ostermann 2015 and Alonso-Mallo, Cano, Reguera 2016. The latter method is extended to the non-linear case, and a rigorous convergence analysis is provided. We perform numerical simulations for diffusion-reaction, advection-reaction, and dispersion-reaction equations in order to evaluate the relative performance of these two corrections. Furthermore, we introduce an extension of both methods to obtain order three locally and evaluate under what circumstances this is beneficial.
References:
[1] |
I. Alonso-Mallo, B. Cano and N. Reguera, Avoiding order reduction when integrating linear initial boundary value problems with exponential splitting methods, IMA J. Numer. Anal., (2017), in press.
doi: 10.1093/imanum/drx047. |
[2] |
I. Alonso-Mallo, B. Cano and N. Reguera,
Avoiding order reduction when integrating linear initial boundary value problems with Lawson methods, IMA J. Numer. Anal., 37 (2017), 2091-2119.
|
[3] |
W. Bao, S. Jin and P. A. Markowich,
On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), 487-524.
doi: 10.1006/jcph.2001.6956. |
[4] |
B. Cano and N. Reguera,
Avoiding order reduction when integrating nonlinear Schrödinger equation with Strang method, J. Comput. Appl. Math., 316 (2017), 86-99.
doi: 10.1016/j.cam.2016.09.033. |
[5] |
M. H. Carpenter, D. Gottlieb, S. Abarbanel and W. S. Don,
The theoretical accuracy of Runge--Kutta time discretizations for the initial boundary value problem: a study of the boundary error, SIAM J. Sci. Comput., 16 (1995), 1241-1252.
doi: 10.1137/0916072. |
[6] |
F. Casas, N. Crouseilles, E. Faou and M. Mehrenberger,
High-order Hamiltonian splitting for Vlasov--Poisson equations, Numer. Math., 135 (2017), 769-801.
doi: 10.1007/s00211-016-0816-z. |
[7] |
C. Cheng and G. Knorr,
The integration of the Vlasov equation in configuration space, J. Comput. Phys., 22 (1976), 330-351.
doi: 10.2172/4200114. |
[8] |
J. M. Connors, J. W. Banks, J. A. Hittinger and C. S. Woodward,
Quantification of errors for operator-split advection--diffusion calculations, Comput. Methods Appl. Mech. Engrg., 272 (2014), 181-197.
doi: 10.1016/j.cma.2014.01.005. |
[9] |
N. Crouseilles, L. Einkemmer and E. Faou,
A Hamiltonian splitting for the Vlasov--Maxwell system, J. Comput. Phys., 283 (2015), 224-240.
doi: 10.1016/j.jcp.2014.11.029. |
[10] |
N. Crouseilles, L. Einkemmer and E. Faou,
An asymptotic preserving scheme for the relativistic Vlasov--Maxwell equations in the classical limit, Comput. Phys. Commun., 209 (2016), 13-26.
doi: 10.1016/j.cpc.2016.08.001. |
[11] |
C. N. Dawson and M. F. Wheeler, Time-splitting methods for advection-diffusion-reaction equations arising in contaminant transport, In R. O'Malley, editor, Proceedings of ICIAM 91 (Washington, DC, 1991), pages 71-82. SIAM, Philadelphia, 1992. |
[12] |
S. Descombes,
Convergence of a splitting method of high order for reaction-diffusion systems, Math. Comp., 70 (2001), 1481-1501.
|
[13] |
L. Einkemmer and A. Ostermann,
Convergence analysis of Strang splitting for Vlasov-type equations, SIAM J. Numer. Anal., 52 (2014), 140-155.
doi: 10.1137/130918599. |
[14] |
L. Einkemmer and A. Ostermann,
A splitting approach for the Kadomtsev--Petviashvili equation, J. Comput. Phys., 299 (2015), 716-730.
doi: 10.1016/j.jcp.2015.07.024. |
[15] |
L. Einkemmer and A. Ostermann,
Overcoming order reduction in diffusion-reaction splitting. Part 1: Dirichlet boundary conditions, SIAM J. Sci. Comput., 37 (2015), A1577-A1592.
doi: 10.1137/140994204. |
[16] |
L. Einkemmer and A. Ostermann,
Overcoming order reduction in diffusion-reaction splitting. Part 2: Oblique boundary conditions, SIAM J. Sci. Comput., 38 (2016), A3741-A3757.
doi: 10.1137/16M1056250. |
[17] |
E. Faou, Geometric Numerical Integration and Schrödinger Equations, European Mathematical Society, Zürich, 2012. |
[18] |
A. Gerisch and J. G. Verwer,
Operator splitting and approximate factorization for taxis-diffusion-reaction models, Appl. Numer. Math., 42 (2002), 159-176.
doi: 10.1016/S0168-9274(01)00148-9. |
[19] |
V. Grandgirard, M. Brunetti, P. Bertrand, N. Besse, X. Garbet, P. Ghendrih, G. Manfredi, Y. Sarazin, O. Sauter, E. Sonnendrücker, J. Vaclavik and L. Villard,
A drift-kinetic Semi-Lagrangian 4D code for ion turbulence simulation, J. Comput. Phys., 217 (2006), 395-423.
doi: 10.1016/j.jcp.2006.01.023. |
[20] |
V. Grimm and M. Hochbruck,
Error analysis of exponential integrators for oscillatory second-order differential equations, J. Phys. A: Math. Gen., 39 (2006), 5495-5507.
doi: 10.1088/0305-4470/39/19/S10. |
[21] |
M. Hochbruck and C. Lubich,
Exponential integrators for quantum-classical molecular dynamics, BIT, 39 (1999), 620-645.
doi: 10.1023/A:1022335122807. |
[22] |
H. Holden, C. Lubich and N. Risebro,
Operator splitting for partial differential equations with Burgers nonlinearity, Math. Comp., 82 (2013), 173-185.
|
[23] |
W. Hundsdorfer and J. G. Verwer,
A note on splitting errors for advection-reaction equations, Appl. Numer. Math., 18 (1995), 191-199.
doi: 10.1016/0168-9274(95)00069-7. |
[24] |
W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer, Berlin, 2003. |
[25] |
C. Klein and K. Roidot,
Fourth order time-stepping for Kadomtsev--Petviashvili and Davey--Stewartson equations, SIAM J. Sci. Comput., 33 (2011), 3333-3356.
doi: 10.1137/100816663. |
[26] |
R. J. LeVeque and J. Oliger,
Numerical methods based on additive splittings for hyperbolic partial differential equations, Math. Comp., 40 (1983), 469-497.
doi: 10.1090/S0025-5718-1983-0689466-8. |
[27] |
C. Lubich,
On splitting methods for Schrödinger--Poisson and cubic nonlinear Schrödinger equations, Math. Comp., 77 (2008), 2141-2153.
doi: 10.1090/S0025-5718-08-02101-7. |
[28] |
E. J. Spee, J. G. Verwer, P. M. deZeeuw, J. G. Blom and W. Hundsdorfer,
A numerical study for global atmospheric transport-chemistry problems, Math. Comput. Simulat., 48 (1998), 177-204.
|
show all references
References:
[1] |
I. Alonso-Mallo, B. Cano and N. Reguera, Avoiding order reduction when integrating linear initial boundary value problems with exponential splitting methods, IMA J. Numer. Anal., (2017), in press.
doi: 10.1093/imanum/drx047. |
[2] |
I. Alonso-Mallo, B. Cano and N. Reguera,
Avoiding order reduction when integrating linear initial boundary value problems with Lawson methods, IMA J. Numer. Anal., 37 (2017), 2091-2119.
|
[3] |
W. Bao, S. Jin and P. A. Markowich,
On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), 487-524.
doi: 10.1006/jcph.2001.6956. |
[4] |
B. Cano and N. Reguera,
Avoiding order reduction when integrating nonlinear Schrödinger equation with Strang method, J. Comput. Appl. Math., 316 (2017), 86-99.
doi: 10.1016/j.cam.2016.09.033. |
[5] |
M. H. Carpenter, D. Gottlieb, S. Abarbanel and W. S. Don,
The theoretical accuracy of Runge--Kutta time discretizations for the initial boundary value problem: a study of the boundary error, SIAM J. Sci. Comput., 16 (1995), 1241-1252.
doi: 10.1137/0916072. |
[6] |
F. Casas, N. Crouseilles, E. Faou and M. Mehrenberger,
High-order Hamiltonian splitting for Vlasov--Poisson equations, Numer. Math., 135 (2017), 769-801.
doi: 10.1007/s00211-016-0816-z. |
[7] |
C. Cheng and G. Knorr,
The integration of the Vlasov equation in configuration space, J. Comput. Phys., 22 (1976), 330-351.
doi: 10.2172/4200114. |
[8] |
J. M. Connors, J. W. Banks, J. A. Hittinger and C. S. Woodward,
Quantification of errors for operator-split advection--diffusion calculations, Comput. Methods Appl. Mech. Engrg., 272 (2014), 181-197.
doi: 10.1016/j.cma.2014.01.005. |
[9] |
N. Crouseilles, L. Einkemmer and E. Faou,
A Hamiltonian splitting for the Vlasov--Maxwell system, J. Comput. Phys., 283 (2015), 224-240.
doi: 10.1016/j.jcp.2014.11.029. |
[10] |
N. Crouseilles, L. Einkemmer and E. Faou,
An asymptotic preserving scheme for the relativistic Vlasov--Maxwell equations in the classical limit, Comput. Phys. Commun., 209 (2016), 13-26.
doi: 10.1016/j.cpc.2016.08.001. |
[11] |
C. N. Dawson and M. F. Wheeler, Time-splitting methods for advection-diffusion-reaction equations arising in contaminant transport, In R. O'Malley, editor, Proceedings of ICIAM 91 (Washington, DC, 1991), pages 71-82. SIAM, Philadelphia, 1992. |
[12] |
S. Descombes,
Convergence of a splitting method of high order for reaction-diffusion systems, Math. Comp., 70 (2001), 1481-1501.
|
[13] |
L. Einkemmer and A. Ostermann,
Convergence analysis of Strang splitting for Vlasov-type equations, SIAM J. Numer. Anal., 52 (2014), 140-155.
doi: 10.1137/130918599. |
[14] |
L. Einkemmer and A. Ostermann,
A splitting approach for the Kadomtsev--Petviashvili equation, J. Comput. Phys., 299 (2015), 716-730.
doi: 10.1016/j.jcp.2015.07.024. |
[15] |
L. Einkemmer and A. Ostermann,
Overcoming order reduction in diffusion-reaction splitting. Part 1: Dirichlet boundary conditions, SIAM J. Sci. Comput., 37 (2015), A1577-A1592.
doi: 10.1137/140994204. |
[16] |
L. Einkemmer and A. Ostermann,
Overcoming order reduction in diffusion-reaction splitting. Part 2: Oblique boundary conditions, SIAM J. Sci. Comput., 38 (2016), A3741-A3757.
doi: 10.1137/16M1056250. |
[17] |
E. Faou, Geometric Numerical Integration and Schrödinger Equations, European Mathematical Society, Zürich, 2012. |
[18] |
A. Gerisch and J. G. Verwer,
Operator splitting and approximate factorization for taxis-diffusion-reaction models, Appl. Numer. Math., 42 (2002), 159-176.
doi: 10.1016/S0168-9274(01)00148-9. |
[19] |
V. Grandgirard, M. Brunetti, P. Bertrand, N. Besse, X. Garbet, P. Ghendrih, G. Manfredi, Y. Sarazin, O. Sauter, E. Sonnendrücker, J. Vaclavik and L. Villard,
A drift-kinetic Semi-Lagrangian 4D code for ion turbulence simulation, J. Comput. Phys., 217 (2006), 395-423.
doi: 10.1016/j.jcp.2006.01.023. |
[20] |
V. Grimm and M. Hochbruck,
Error analysis of exponential integrators for oscillatory second-order differential equations, J. Phys. A: Math. Gen., 39 (2006), 5495-5507.
doi: 10.1088/0305-4470/39/19/S10. |
[21] |
M. Hochbruck and C. Lubich,
Exponential integrators for quantum-classical molecular dynamics, BIT, 39 (1999), 620-645.
doi: 10.1023/A:1022335122807. |
[22] |
H. Holden, C. Lubich and N. Risebro,
Operator splitting for partial differential equations with Burgers nonlinearity, Math. Comp., 82 (2013), 173-185.
|
[23] |
W. Hundsdorfer and J. G. Verwer,
A note on splitting errors for advection-reaction equations, Appl. Numer. Math., 18 (1995), 191-199.
doi: 10.1016/0168-9274(95)00069-7. |
[24] |
W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer, Berlin, 2003. |
[25] |
C. Klein and K. Roidot,
Fourth order time-stepping for Kadomtsev--Petviashvili and Davey--Stewartson equations, SIAM J. Sci. Comput., 33 (2011), 3333-3356.
doi: 10.1137/100816663. |
[26] |
R. J. LeVeque and J. Oliger,
Numerical methods based on additive splittings for hyperbolic partial differential equations, Math. Comp., 40 (1983), 469-497.
doi: 10.1090/S0025-5718-1983-0689466-8. |
[27] |
C. Lubich,
On splitting methods for Schrödinger--Poisson and cubic nonlinear Schrödinger equations, Math. Comp., 77 (2008), 2141-2153.
doi: 10.1090/S0025-5718-08-02101-7. |
[28] |
E. J. Spee, J. G. Verwer, P. M. deZeeuw, J. G. Blom and W. Hundsdorfer,
A numerical study for global atmospheric transport-chemistry problems, Math. Comput. Simulat., 48 (1998), 177-204.
|


Local error | |||||||||
step size | order | order | order | ||||||
6.40e-02 | 3.14e-02 | - | 4.08e-04 | - | 4.54e-04 | - | |||
3.20e-02 | 1.54e-02 | 1.03 | 9.93e-05 | 2.04 | 6.13e-05 | 2.89 | |||
1.60e-02 | 7.51e-03 | 1.03 | 2.48e-05 | 2.00 | 7.72e-06 | 2.99 | |||
8.00e-03 | 3.64e-03 | 1.04 | 6.21e-06 | 2.00 | 9.69e-07 | 2.99 | |||
4.00e-03 | 1.75e-03 | 1.06 | 1.55e-06 | 2.00 | 1.22e-07 | 2.99 | |||
2.00e-03 | 8.24e-04 | 1.08 | 3.88e-07 | 2.00 | 1.54e-08 | 2.99 | |||
Global error | |||||||||
step size | order | order | order | ||||||
6.40e-02 | 3.15e-02 | - | 6.75e-04 | - | 9.66e-04 | - | |||
3.20e-02 | 1.54e-02 | 1.03 | 1.71e-04 | 1.98 | 2.41e-04 | 2.00 | |||
1.60e-02 | 7.52e-03 | 1.03 | 4.34e-05 | 1.98 | 6.01e-05 | 2.00 | |||
8.00e-03 | 3.65e-03 | 1.04 | 1.09e-05 | 1.99 | 1.50e-05 | 2.00 | |||
4.00e-03 | 1.75e-03 | 1.06 | 2.75e-06 | 1.99 | 3.76e-06 | 2.00 | |||
2.00e-03 | 8.29e-04 | 1.08 | 6.91e-07 | 1.99 | 9.40e-07 | 2.00 |
Local error | |||||||||
step size | order | order | order | ||||||
6.40e-02 | 3.14e-02 | - | 4.08e-04 | - | 4.54e-04 | - | |||
3.20e-02 | 1.54e-02 | 1.03 | 9.93e-05 | 2.04 | 6.13e-05 | 2.89 | |||
1.60e-02 | 7.51e-03 | 1.03 | 2.48e-05 | 2.00 | 7.72e-06 | 2.99 | |||
8.00e-03 | 3.64e-03 | 1.04 | 6.21e-06 | 2.00 | 9.69e-07 | 2.99 | |||
4.00e-03 | 1.75e-03 | 1.06 | 1.55e-06 | 2.00 | 1.22e-07 | 2.99 | |||
2.00e-03 | 8.24e-04 | 1.08 | 3.88e-07 | 2.00 | 1.54e-08 | 2.99 | |||
Global error | |||||||||
step size | order | order | order | ||||||
6.40e-02 | 3.15e-02 | - | 6.75e-04 | - | 9.66e-04 | - | |||
3.20e-02 | 1.54e-02 | 1.03 | 1.71e-04 | 1.98 | 2.41e-04 | 2.00 | |||
1.60e-02 | 7.52e-03 | 1.03 | 4.34e-05 | 1.98 | 6.01e-05 | 2.00 | |||
8.00e-03 | 3.65e-03 | 1.04 | 1.09e-05 | 1.99 | 1.50e-05 | 2.00 | |||
4.00e-03 | 1.75e-03 | 1.06 | 2.75e-06 | 1.99 | 3.76e-06 | 2.00 | |||
2.00e-03 | 8.29e-04 | 1.08 | 6.91e-07 | 1.99 | 9.40e-07 | 2.00 |
Local error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
1.60e-02 | 7.49e-03 | - | 1.25e-04 | - | 1.06e-04 | - | ||
8.00e-03 | 3.64e-03 | 1.04 | 3.25e-05 | 1.94 | 2.76e-05 | 1.94 | ||
4.00e-03 | 1.75e-03 | 1.06 | 8.17e-06 | 1.99 | 6.91e-06 | 2.00 | ||
2.00e-03 | 8.24e-04 | 1.08 | 2.04e-06 | 2.00 | 1.73e-06 | 2.00 | ||
1.00e-03 | 3.79e-04 | 1.12 | 5.13e-07 | 2.00 | 4.31e-07 | 2.00 | ||
5.00e-04 | 1.68e-04 | 1.18 | 1.27e-07 | 2.01 | 1.07e-07 | 2.00 | ||
Global error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
1.60e-02 | 7.52e-03 | - | 3.13e-05 | - | 4.15e-05 | - | ||
8.00e-03 | 3.65e-03 | 1.04 | 7.72e-06 | 2.02 | 1.04e-05 | 2.00 | ||
4.00e-03 | 1.75e-03 | 1.06 | 1.91e-06 | 2.02 | 2.60e-06 | 2.00 | ||
2.00e-03 | 8.29e-04 | 1.08 | 4.69e-07 | 2.02 | 6.49e-07 | 2.00 | ||
1.00e-03 | 3.82e-04 | 1.12 | 1.15e-07 | 2.03 | 1.62e-07 | 2.00 | ||
5.00e-04 | 1.70e-04 | 1.17 | 2.81e-08 | 2.03 | 4.06e-08 | 2.00 |
Local error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
1.60e-02 | 7.49e-03 | - | 1.25e-04 | - | 1.06e-04 | - | ||
8.00e-03 | 3.64e-03 | 1.04 | 3.25e-05 | 1.94 | 2.76e-05 | 1.94 | ||
4.00e-03 | 1.75e-03 | 1.06 | 8.17e-06 | 1.99 | 6.91e-06 | 2.00 | ||
2.00e-03 | 8.24e-04 | 1.08 | 2.04e-06 | 2.00 | 1.73e-06 | 2.00 | ||
1.00e-03 | 3.79e-04 | 1.12 | 5.13e-07 | 2.00 | 4.31e-07 | 2.00 | ||
5.00e-04 | 1.68e-04 | 1.18 | 1.27e-07 | 2.01 | 1.07e-07 | 2.00 | ||
Global error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
1.60e-02 | 7.52e-03 | - | 3.13e-05 | - | 4.15e-05 | - | ||
8.00e-03 | 3.65e-03 | 1.04 | 7.72e-06 | 2.02 | 1.04e-05 | 2.00 | ||
4.00e-03 | 1.75e-03 | 1.06 | 1.91e-06 | 2.02 | 2.60e-06 | 2.00 | ||
2.00e-03 | 8.29e-04 | 1.08 | 4.69e-07 | 2.02 | 6.49e-07 | 2.00 | ||
1.00e-03 | 3.82e-04 | 1.12 | 1.15e-07 | 2.03 | 1.62e-07 | 2.00 | ||
5.00e-04 | 1.70e-04 | 1.17 | 2.81e-08 | 2.03 | 4.06e-08 | 2.00 |
Local error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
1.60e-02 | 7.49e-03 | - | 1.71e-04 | - | 8.81e-05 | - | ||
8.00e-03 | 3.64e-03 | 1.04 | 1.86e-05 | 3.20 | 1.44e-05 | 2.61 | ||
4.00e-03 | 1.75e-03 | 1.06 | 2.29e-06 | 3.02 | 2.11e-06 | 2.77 | ||
2.00e-03 | 8.24e-04 | 1.08 | 3.11e-07 | 2.88 | 2.87e-07 | 2.88 | ||
1.00e-03 | 3.79e-04 | 1.12 | 4.06e-08 | 2.94 | 3.75e-08 | 2.94 | ||
5.00e-04 | 1.68e-04 | 1.18 | 5.18e-09 | 2.97 | 4.80e-09 | 2.97 | ||
Global error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
1.60e-02 | 7.52e-03 | - | 2.32e-04 | - | 6.85e-05 | - | ||
8.00e-03 | 3.65e-03 | 1.04 | 3.30e-05 | 2.81 | 1.67e-05 | 2.04 | ||
4.00e-03 | 1.75e-03 | 1.06 | 5.89e-06 | 2.49 | 4.11e-06 | 2.02 | ||
2.00e-03 | 8.29e-04 | 1.08 | 1.22e-06 | 2.27 | 1.02e-06 | 2.01 | ||
1.00e-03 | 3.82e-04 | 1.12 | 2.77e-07 | 2.14 | 2.54e-07 | 2.00 | ||
5.00e-04 | 1.70e-04 | 1.17 | 6.59e-08 | 2.07 | 6.34e-08 | 2.00 |
Local error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
1.60e-02 | 7.49e-03 | - | 1.71e-04 | - | 8.81e-05 | - | ||
8.00e-03 | 3.64e-03 | 1.04 | 1.86e-05 | 3.20 | 1.44e-05 | 2.61 | ||
4.00e-03 | 1.75e-03 | 1.06 | 2.29e-06 | 3.02 | 2.11e-06 | 2.77 | ||
2.00e-03 | 8.24e-04 | 1.08 | 3.11e-07 | 2.88 | 2.87e-07 | 2.88 | ||
1.00e-03 | 3.79e-04 | 1.12 | 4.06e-08 | 2.94 | 3.75e-08 | 2.94 | ||
5.00e-04 | 1.68e-04 | 1.18 | 5.18e-09 | 2.97 | 4.80e-09 | 2.97 | ||
Global error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
1.60e-02 | 7.52e-03 | - | 2.32e-04 | - | 6.85e-05 | - | ||
8.00e-03 | 3.65e-03 | 1.04 | 3.30e-05 | 2.81 | 1.67e-05 | 2.04 | ||
4.00e-03 | 1.75e-03 | 1.06 | 5.89e-06 | 2.49 | 4.11e-06 | 2.02 | ||
2.00e-03 | 8.29e-04 | 1.08 | 1.22e-06 | 2.27 | 1.02e-06 | 2.01 | ||
1.00e-03 | 3.82e-04 | 1.12 | 2.77e-07 | 2.14 | 2.54e-07 | 2.00 | ||
5.00e-04 | 1.70e-04 | 1.17 | 6.59e-08 | 2.07 | 6.34e-08 | 2.00 |
Local error | ||||||||
step size | order | order | order | |||||
2.40e-01 | 1.26e-01 | - | 7.70e-03 | - | 2.41e-03 | - | ||
1.20e-01 | 6.08e-02 | 1.05 | 1.84e-03 | 2.07 | 2.94e-04 | 3.04 | ||
6.00e-02 | 2.94e-02 | 1.05 | 4.44e-04 | 2.05 | 3.63e-05 | 3.02 | ||
3.00e-02 | 1.41e-02 | 1.06 | 1.06e-04 | 2.06 | 4.51e-06 | 3.01 | ||
1.50e-02 | 6.53e-03 | 1.11 | 2.47e-05 | 2.10 | 5.62e-07 | 3.00 | ||
7.50e-03 | 2.76e-03 | 1.24 | 5.45e-06 | 2.18 | 6.98e-08 | 3.01 | ||
Global error | ||||||||
step size | order | order | order | |||||
2.40e-01 | 1.25e-01 | - | 1.11e-02 | - | 1.14e-02 | - | ||
1.20e-01 | 5.98e-02 | 1.07 | 2.16e-03 | 2.36 | 2.92e-03 | 1.96 | ||
6.00e-02 | 2.85e-02 | 1.07 | 4.44e-04 | 2.28 | 7.32e-04 | 1.99 | ||
3.00e-02 | 1.31e-02 | 1.12 | 1.06e-04 | 2.06 | 1.83e-04 | 2.00 | ||
1.50e-02 | 5.54e-03 | 1.24 | 2.55e-05 | 2.06 | 4.56e-05 | 2.00 | ||
7.50e-03 | 1.94e-03 | 1.51 | 6.37e-06 | 2.00 | 1.14e-05 | 2.00 |
Local error | ||||||||
step size | order | order | order | |||||
2.40e-01 | 1.26e-01 | - | 7.70e-03 | - | 2.41e-03 | - | ||
1.20e-01 | 6.08e-02 | 1.05 | 1.84e-03 | 2.07 | 2.94e-04 | 3.04 | ||
6.00e-02 | 2.94e-02 | 1.05 | 4.44e-04 | 2.05 | 3.63e-05 | 3.02 | ||
3.00e-02 | 1.41e-02 | 1.06 | 1.06e-04 | 2.06 | 4.51e-06 | 3.01 | ||
1.50e-02 | 6.53e-03 | 1.11 | 2.47e-05 | 2.10 | 5.62e-07 | 3.00 | ||
7.50e-03 | 2.76e-03 | 1.24 | 5.45e-06 | 2.18 | 6.98e-08 | 3.01 | ||
Global error | ||||||||
step size | order | order | order | |||||
2.40e-01 | 1.25e-01 | - | 1.11e-02 | - | 1.14e-02 | - | ||
1.20e-01 | 5.98e-02 | 1.07 | 2.16e-03 | 2.36 | 2.92e-03 | 1.96 | ||
6.00e-02 | 2.85e-02 | 1.07 | 4.44e-04 | 2.28 | 7.32e-04 | 1.99 | ||
3.00e-02 | 1.31e-02 | 1.12 | 1.06e-04 | 2.06 | 1.83e-04 | 2.00 | ||
1.50e-02 | 5.54e-03 | 1.24 | 2.55e-05 | 2.06 | 4.56e-05 | 2.00 | ||
7.50e-03 | 1.94e-03 | 1.51 | 6.37e-06 | 2.00 | 1.14e-05 | 2.00 |
Local error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
2.40e-01 | 1.25e-01 | - | 1.51e-02 | - | 8.80e-03 | - | ||
1.20e-01 | 5.98e-02 | 1.07 | 2.14e-03 | 2.82 | 1.93e-03 | 2.19 | ||
6.00e-02 | 2.84e-02 | 1.07 | 4.73e-04 | 2.18 | 4.42e-04 | 2.13 | ||
3.00e-02 | 1.31e-02 | 1.12 | 1.15e-04 | 2.04 | 1.01e-04 | 2.13 | ||
1.50e-02 | 5.53e-03 | 1.24 | 2.84e-05 | 2.02 | 2.20e-05 | 2.19 | ||
7.50e-03 | 1.89e-03 | 1.55 | 6.91e-06 | 2.04 | 4.68e-06 | 2.24 | ||
Global error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
2.40e-01 | 3.73e-01 | - | 1.09e-01 | - | 2.59e-02 | - | ||
1.20e-01 | 9.07e-02 | 2.04 | 3.56e-02 | 1.62 | 4.72e-03 | 2.46 | ||
6.00e-02 | 2.84e-02 | 1.67 | 1.02e-02 | 1.80 | 1.30e-03 | 1.86 | ||
3.00e-02 | 1.31e-02 | 1.12 | 2.74e-03 | 1.90 | 4.15e-04 | 1.65 | ||
1.50e-02 | 5.54e-03 | 1.24 | 7.07e-04 | 1.95 | 1.16e-04 | 1.83 | ||
7.50e-03 | 1.94e-03 | 1.51 | 1.80e-04 | 1.98 | 3.08e-05 | 1.92 |
Local error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
2.40e-01 | 1.25e-01 | - | 1.51e-02 | - | 8.80e-03 | - | ||
1.20e-01 | 5.98e-02 | 1.07 | 2.14e-03 | 2.82 | 1.93e-03 | 2.19 | ||
6.00e-02 | 2.84e-02 | 1.07 | 4.73e-04 | 2.18 | 4.42e-04 | 2.13 | ||
3.00e-02 | 1.31e-02 | 1.12 | 1.15e-04 | 2.04 | 1.01e-04 | 2.13 | ||
1.50e-02 | 5.53e-03 | 1.24 | 2.84e-05 | 2.02 | 2.20e-05 | 2.19 | ||
7.50e-03 | 1.89e-03 | 1.55 | 6.91e-06 | 2.04 | 4.68e-06 | 2.24 | ||
Global error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
2.40e-01 | 3.73e-01 | - | 1.09e-01 | - | 2.59e-02 | - | ||
1.20e-01 | 9.07e-02 | 2.04 | 3.56e-02 | 1.62 | 4.72e-03 | 2.46 | ||
6.00e-02 | 2.84e-02 | 1.67 | 1.02e-02 | 1.80 | 1.30e-03 | 1.86 | ||
3.00e-02 | 1.31e-02 | 1.12 | 2.74e-03 | 1.90 | 4.15e-04 | 1.65 | ||
1.50e-02 | 5.54e-03 | 1.24 | 7.07e-04 | 1.95 | 1.16e-04 | 1.83 | ||
7.50e-03 | 1.94e-03 | 1.51 | 1.80e-04 | 1.98 | 3.08e-05 | 1.92 |
Local error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
2.40e-01 | 1.25e-01 | - | 1.51e-02 | - | 1.39e-02 | - | ||
1.20e-01 | 5.98e-02 | 1.07 | 2.14e-03 | 2.82 | 1.61e-03 | 3.11 | ||
6.00e-02 | 2.84e-02 | 1.07 | 2.87e-04 | 2.90 | 1.90e-04 | 3.09 | ||
3.00e-02 | 1.31e-02 | 1.12 | 3.72e-05 | 2.95 | 2.28e-05 | 3.06 | ||
1.50e-02 | 5.53e-03 | 1.24 | 4.73e-06 | 2.97 | 2.79e-06 | 3.03 | ||
7.50e-03 | 1.89e-03 | 1.55 | 5.98e-07 | 2.99 | 3.44e-07 | 3.02 | ||
Global error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
2.40e-01 | 3.73e-01 | - | 5.46e-02 | - | 4.65e-02 | - | ||
1.20e-01 | 9.07e-02 | 2.04 | 2.04e-02 | 1.42 | 1.08e-02 | 2.10 | ||
6.00e-02 | 2.84e-02 | 1.67 | 6.25e-03 | 1.70 | 2.57e-03 | 2.08 | ||
3.00e-02 | 1.31e-02 | 1.12 | 1.73e-03 | 1.85 | 6.24e-04 | 2.05 | ||
1.50e-02 | 5.54e-03 | 1.24 | 4.55e-04 | 1.93 | 1.53e-04 | 2.03 | ||
7.50e-03 | 1.94e-03 | 1.51 | 1.17e-04 | 1.96 | 3.79e-05 | 2.01 |
Local error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
2.40e-01 | 1.25e-01 | - | 1.51e-02 | - | 1.39e-02 | - | ||
1.20e-01 | 5.98e-02 | 1.07 | 2.14e-03 | 2.82 | 1.61e-03 | 3.11 | ||
6.00e-02 | 2.84e-02 | 1.07 | 2.87e-04 | 2.90 | 1.90e-04 | 3.09 | ||
3.00e-02 | 1.31e-02 | 1.12 | 3.72e-05 | 2.95 | 2.28e-05 | 3.06 | ||
1.50e-02 | 5.53e-03 | 1.24 | 4.73e-06 | 2.97 | 2.79e-06 | 3.03 | ||
7.50e-03 | 1.89e-03 | 1.55 | 5.98e-07 | 2.99 | 3.44e-07 | 3.02 | ||
Global error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
2.40e-01 | 3.73e-01 | - | 5.46e-02 | - | 4.65e-02 | - | ||
1.20e-01 | 9.07e-02 | 2.04 | 2.04e-02 | 1.42 | 1.08e-02 | 2.10 | ||
6.00e-02 | 2.84e-02 | 1.67 | 6.25e-03 | 1.70 | 2.57e-03 | 2.08 | ||
3.00e-02 | 1.31e-02 | 1.12 | 1.73e-03 | 1.85 | 6.24e-04 | 2.05 | ||
1.50e-02 | 5.54e-03 | 1.24 | 4.55e-04 | 1.93 | 1.53e-04 | 2.03 | ||
7.50e-03 | 1.94e-03 | 1.51 | 1.17e-04 | 1.96 | 3.79e-05 | 2.01 |
Local error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
2.40e-01 | 1.44e-02 | - | 1.44e-02 | - | 7.19e-03 | - | ||
1.20e-01 | 2.05e-03 | 2.81 | 2.05e-03 | 2.81 | 9.33e-04 | 2.95 | ||
6.00e-02 | 2.75e-04 | 2.90 | 2.75e-04 | 2.90 | 1.24e-04 | 2.91 | ||
3.00e-02 | 3.58e-05 | 2.95 | 3.58e-05 | 2.95 | 1.62e-05 | 2.94 | ||
1.50e-02 | 4.56e-06 | 2.97 | 4.56e-06 | 2.97 | 2.08e-06 | 2.96 | ||
7.50e-03 | 5.76e-07 | 2.99 | 5.76e-07 | 2.99 | 2.63e-07 | 2.98 | ||
Global error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
2.40e-01 | 3.73e-01 | - | 1.02e-01 | - | 2.59e-02 | - | ||
1.20e-01 | 9.07e-02 | 2.04 | 3.31e-02 | 1.63 | 4.72e-03 | 2.46 | ||
6.00e-02 | 1.26e-02 | 2.85 | 9.46e-03 | 1.81 | 9.71e-04 | 2.28 | ||
3.00e-02 | 1.87e-03 | 2.75 | 2.52e-03 | 1.91 | 3.16e-04 | 1.62 | ||
1.50e-02 | 4.88e-04 | 1.94 | 6.51e-04 | 1.95 | 8.99e-05 | 1.81 | ||
7.50e-03 | 1.25e-04 | 1.97 | 1.65e-04 | 1.98 | 2.39e-05 | 1.91 |
Local error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
2.40e-01 | 1.44e-02 | - | 1.44e-02 | - | 7.19e-03 | - | ||
1.20e-01 | 2.05e-03 | 2.81 | 2.05e-03 | 2.81 | 9.33e-04 | 2.95 | ||
6.00e-02 | 2.75e-04 | 2.90 | 2.75e-04 | 2.90 | 1.24e-04 | 2.91 | ||
3.00e-02 | 3.58e-05 | 2.95 | 3.58e-05 | 2.95 | 1.62e-05 | 2.94 | ||
1.50e-02 | 4.56e-06 | 2.97 | 4.56e-06 | 2.97 | 2.08e-06 | 2.96 | ||
7.50e-03 | 5.76e-07 | 2.99 | 5.76e-07 | 2.99 | 2.63e-07 | 2.98 | ||
Global error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
2.40e-01 | 3.73e-01 | - | 1.02e-01 | - | 2.59e-02 | - | ||
1.20e-01 | 9.07e-02 | 2.04 | 3.31e-02 | 1.63 | 4.72e-03 | 2.46 | ||
6.00e-02 | 1.26e-02 | 2.85 | 9.46e-03 | 1.81 | 9.71e-04 | 2.28 | ||
3.00e-02 | 1.87e-03 | 2.75 | 2.52e-03 | 1.91 | 3.16e-04 | 1.62 | ||
1.50e-02 | 4.88e-04 | 1.94 | 6.51e-04 | 1.95 | 8.99e-05 | 1.81 | ||
7.50e-03 | 1.25e-04 | 1.97 | 1.65e-04 | 1.98 | 2.39e-05 | 1.91 |
t=0.5 | |||||
|
27.6(6.9, 0.8) | 4.5(1.2, 0.9) | 27.3(23, 1.5) | 14.1(7.2, 2.2) | 4.0(1.0, 0.7) |
18.2(1.5, 0.9) | 15.4(1.3, 1.0) | 14.1(7.2, 2.2) | 9.4(0.2, 1.0) | 8.2(0.9, 0.7) | |
22.7(5.7, 0.8) | 4.6(1.3, 0.9) | 15.8(13.7, 1.5) | 4.0(0.3, 1.0) | 4.2(1.1, 0.7) | |
5.7(3.1, 0.6) | 2.2(1.4, 0.7) | 1.5(3.3, 0.6) | 1.7(0.4, 1.0) | 2.7(1.4, 0.6) | |
2.4(1.0, 0.7) | 2.4(1.0, 0.9) | 2.5(1.0, 1.7) | 3.7(1.0, 1.0) | 3.4(1.0, 0.7) | |
|
21.6(2.9, 0.6) | 5.7(0.8, 0.4) | 35.3(24.7, 1.3) | 2.6(0.8, 0.5) | 3.9(1.9, 0.4) |
11.1(1.4, 0.5) | 19.3(3.8, 0.3) | 15.5(7.1, 2.0) | 0.9(0.9, 0.2) | 6.0(1.6, 0.5) | |
18.6(2.6, 0.6) | 5.7(0.8, 0.4) | 19.3(13.9, 1.3) | 2.5(1.1, 0.4) | 3.8(1.9, 0.4) | |
6.8(1.4, 0.6) | 8.8(1.8, 0.5) | 1.2(29, 0.4) | 4.0(2.7, 0.2) | 2.1(1.5, 0.4) | |
1.7(1.0, 0.4) | 1.0(1.0, 0.2) | 2.6(1.0, 1.6) | 0.8(1.0, 0.5) | 2.1(1.0, 0.5) |
t=0.5 | |||||
|
27.6(6.9, 0.8) | 4.5(1.2, 0.9) | 27.3(23, 1.5) | 14.1(7.2, 2.2) | 4.0(1.0, 0.7) |
18.2(1.5, 0.9) | 15.4(1.3, 1.0) | 14.1(7.2, 2.2) | 9.4(0.2, 1.0) | 8.2(0.9, 0.7) | |
22.7(5.7, 0.8) | 4.6(1.3, 0.9) | 15.8(13.7, 1.5) | 4.0(0.3, 1.0) | 4.2(1.1, 0.7) | |
5.7(3.1, 0.6) | 2.2(1.4, 0.7) | 1.5(3.3, 0.6) | 1.7(0.4, 1.0) | 2.7(1.4, 0.6) | |
2.4(1.0, 0.7) | 2.4(1.0, 0.9) | 2.5(1.0, 1.7) | 3.7(1.0, 1.0) | 3.4(1.0, 0.7) | |
|
21.6(2.9, 0.6) | 5.7(0.8, 0.4) | 35.3(24.7, 1.3) | 2.6(0.8, 0.5) | 3.9(1.9, 0.4) |
11.1(1.4, 0.5) | 19.3(3.8, 0.3) | 15.5(7.1, 2.0) | 0.9(0.9, 0.2) | 6.0(1.6, 0.5) | |
18.6(2.6, 0.6) | 5.7(0.8, 0.4) | 19.3(13.9, 1.3) | 2.5(1.1, 0.4) | 3.8(1.9, 0.4) | |
6.8(1.4, 0.6) | 8.8(1.8, 0.5) | 1.2(29, 0.4) | 4.0(2.7, 0.2) | 2.1(1.5, 0.4) | |
1.7(1.0, 0.4) | 1.0(1.0, 0.2) | 2.6(1.0, 1.6) | 0.8(1.0, 0.5) | 2.1(1.0, 0.5) |
Local error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
1.20e-02 | 5.84e-03 | - | 1.48e-03 | - | 1.50e-03 | - | ||
6.00e-03 | 2.79e-03 | 1.07 | 2.72e-04 | 2.45 | 2.70e-04 | 2.47 | ||
3.00e-03 | 1.23e-03 | 1.19 | 3.49e-05 | 2.96 | 3.47e-05 | 2.96 | ||
1.50e-03 | 6.38e-04 | 0.94 | 8.77e-06 | 1.99 | 8.65e-06 | 2.00 | ||
7.50e-04 | 2.95e-04 | 1.11 | 2.11e-06 | 2.05 | 2.08e-06 | 2.05 | ||
3.75e-04 | 1.30e-04 | 1.18 | 5.15e-07 | 2.04 | 5.07e-07 | 2.04 | ||
Global error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
1.20e-02 | 2.02e-02 | - | 2.35e-03 | 1.89 | 2.33e-03 | - | ||
6.00e-03 | 1.18e-02 | 0.78 | 5.32e-04 | 2.14 | 5.25e-04 | 2.15 | ||
3.00e-03 | 4.77e-03 | 1.30 | 1.09e-04 | 2.28 | 1.08e-04 | 2.29 | ||
1.50e-03 | 1.04e-03 | 2.20 | 4.85e-05 | 1.17 | 4.82e-05 | 1.16 | ||
7.50e-04 | 6.09e-04 | 0.77 | 1.82e-05 | 1.41 | 1.82e-05 | 1.41 | ||
3.75e-04 | 2.20e-04 | 1.47 | 1.44e-05 | 0.34 | 1.44e-05 | 0.34 | ||
1.88e-04 | 9.37e-05 | 1.23 | 4.70e-07 | 4.94 | 4.66e-07 | 4.95 |
Local error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
1.20e-02 | 5.84e-03 | - | 1.48e-03 | - | 1.50e-03 | - | ||
6.00e-03 | 2.79e-03 | 1.07 | 2.72e-04 | 2.45 | 2.70e-04 | 2.47 | ||
3.00e-03 | 1.23e-03 | 1.19 | 3.49e-05 | 2.96 | 3.47e-05 | 2.96 | ||
1.50e-03 | 6.38e-04 | 0.94 | 8.77e-06 | 1.99 | 8.65e-06 | 2.00 | ||
7.50e-04 | 2.95e-04 | 1.11 | 2.11e-06 | 2.05 | 2.08e-06 | 2.05 | ||
3.75e-04 | 1.30e-04 | 1.18 | 5.15e-07 | 2.04 | 5.07e-07 | 2.04 | ||
Global error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
1.20e-02 | 2.02e-02 | - | 2.35e-03 | 1.89 | 2.33e-03 | - | ||
6.00e-03 | 1.18e-02 | 0.78 | 5.32e-04 | 2.14 | 5.25e-04 | 2.15 | ||
3.00e-03 | 4.77e-03 | 1.30 | 1.09e-04 | 2.28 | 1.08e-04 | 2.29 | ||
1.50e-03 | 1.04e-03 | 2.20 | 4.85e-05 | 1.17 | 4.82e-05 | 1.16 | ||
7.50e-04 | 6.09e-04 | 0.77 | 1.82e-05 | 1.41 | 1.82e-05 | 1.41 | ||
3.75e-04 | 2.20e-04 | 1.47 | 1.44e-05 | 0.34 | 1.44e-05 | 0.34 | ||
1.88e-04 | 9.37e-05 | 1.23 | 4.70e-07 | 4.94 | 4.66e-07 | 4.95 |
Local error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
1.20e-02 | 5.84e-03 | - | 1.51e-03 | - | 1.53e-03 | - | ||
6.00e-03 | 2.79e-03 | 1.07 | 2.48e-04 | 2.61 | 2.38e-04 | 2.69 | ||
3.00e-03 | 1.23e-03 | 1.19 | 2.92e-05 | 3.09 | 2.81e-05 | 3.08 | ||
1.50e-03 | 6.38e-04 | 0.94 | 3.21e-06 | 3.18 | 3.14e-06 | 3.16 | ||
7.50e-04 | 2.95e-04 | 1.11 | 3.82e-07 | 3.07 | 3.72e-07 | 3.08 | ||
3.75e-04 | 1.30e-04 | 1.18 | 4.65e-08 | 3.04 | 4.64e-08 | 3.00 | ||
Global error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
1.20e-02 | 2.02e-02 | - | 9.02e-03 | 2.09 | 7.91e-03 | - | ||
6.00e-03 | 1.18e-02 | 0.78 | 1.84e-03 | 2.29 | 1.73e-03 | 2.19 | ||
3.00e-03 | 4.77e-03 | 1.30 | 4.16e-04 | 2.15 | 4.12e-04 | 2.07 | ||
1.50e-03 | 1.04e-03 | 2.20 | 9.85e-05 | 2.08 | 9.97e-05 | 2.05 | ||
7.50e-04 | 6.09e-04 | 0.77 | 2.42e-05 | 2.03 | 2.44e-05 | 2.03 | ||
3.75e-04 | 2.20e-04 | 1.47 | 6.01e-06 | 2.01 | 6.01e-06 | 2.02 | ||
1.88e-04 | 9.37e-05 | 1.23 | 1.45e-06 | 2.06 | 1.50e-06 | 2.00 |
Local error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
1.20e-02 | 5.84e-03 | - | 1.51e-03 | - | 1.53e-03 | - | ||
6.00e-03 | 2.79e-03 | 1.07 | 2.48e-04 | 2.61 | 2.38e-04 | 2.69 | ||
3.00e-03 | 1.23e-03 | 1.19 | 2.92e-05 | 3.09 | 2.81e-05 | 3.08 | ||
1.50e-03 | 6.38e-04 | 0.94 | 3.21e-06 | 3.18 | 3.14e-06 | 3.16 | ||
7.50e-04 | 2.95e-04 | 1.11 | 3.82e-07 | 3.07 | 3.72e-07 | 3.08 | ||
3.75e-04 | 1.30e-04 | 1.18 | 4.65e-08 | 3.04 | 4.64e-08 | 3.00 | ||
Global error | ||||||||
unmodified | TDBC | CEC | ||||||
step size | order | order | order | |||||
1.20e-02 | 2.02e-02 | - | 9.02e-03 | 2.09 | 7.91e-03 | - | ||
6.00e-03 | 1.18e-02 | 0.78 | 1.84e-03 | 2.29 | 1.73e-03 | 2.19 | ||
3.00e-03 | 4.77e-03 | 1.30 | 4.16e-04 | 2.15 | 4.12e-04 | 2.07 | ||
1.50e-03 | 1.04e-03 | 2.20 | 9.85e-05 | 2.08 | 9.97e-05 | 2.05 | ||
7.50e-04 | 6.09e-04 | 0.77 | 2.42e-05 | 2.03 | 2.44e-05 | 2.03 | ||
3.75e-04 | 2.20e-04 | 1.47 | 6.01e-06 | 2.01 | 6.01e-06 | 2.02 | ||
1.88e-04 | 9.37e-05 | 1.23 | 1.45e-06 | 2.06 | 1.50e-06 | 2.00 |
[1] |
Joseph A. Connolly, Neville J. Ford. Comparison of numerical methods for fractional differential equations. Communications on Pure and Applied Analysis, 2006, 5 (2) : 289-307. doi: 10.3934/cpaa.2006.5.289 |
[2] |
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