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September  2018, 23(7): 2641-2660. doi: 10.3934/dcdsb.2018081

A comparison of boundary correction methods for Strang splitting

Lukas Einkemmer and  Alexander Ostermann ,

Department of Mathematics, University of Innsbruck, A-6020 Innsbruck, Austria

* Corresponding author: Alexander Ostermann

Received  September 2016 Revised  August 2017 Published  September 2018 Early access  March 2018

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.

Keywords: Splitting methods, Dirichlet boundary condition, order reduction, numerical comparison.
Mathematics Subject Classification: Primary: 65M12, 65J08.
Citation: Lukas Einkemmer, Alexander Ostermann. A comparison of boundary correction methods for Strang splitting. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2641-2660. doi: 10.3934/dcdsb.2018081
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. 

Figure 1.  The global error in the infinity norm as a function of the time step size is shown. The error for TDBC2 and CEC2 is almost identical and therefore only the (erratic) error for TDBC2 is shown in the plot. In addition, the dashed lines are of slope $1$ and $2$, respectively. In all simulations equation (22) with $f(u) = {\rm{e}}^{u-1}$ is employed and the initial value $u(0, x) = 1+\sin \pi x + i \sin 2 \pi x$ is imposed. The space discretization is conducted by using the standard centered finite difference stencil with $200$ grid points and all simulations are conducted until $t = 0.19$
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Figure 2.  The local (full lines) and global errors (dashed lines) in the infinity norm are shown as a function of time for the second order CEC (top) and the third order CEC (bottom) corrections. The following step sizes are used (from top to bottom in this order in both cases): $1.5\cdot10^{-3}$ (yellow), $7.5\cdot10^{-4}$ (magenta), $3.75\cdot10^{-4}$ (cyan), $1.88\cdot10^{-4}$ (blue), $9.38\cdot10^{-5}$ (green), $4.69\cdot10^{-5}$ (red). In all simulations equation (22) with $f(u) = {\rm{e}}^{u-1}$ is employed and the initial value $u(0, x) = 1+\sin \pi x + i \sin 2 \pi x$ is imposed. The space discretization is conducted by using the standard centered finite difference stencil with $200$ grid points
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Table 1.  The local (at $t = 0$) and global errors using the unmodified Strang splitting applied to equation (19) are shown. The three different reaction terms indicated in the text are used. The space discretization is conducted by using the standard centered finite difference stencil with $200$ grid points. All problems are integrated until $t = 0.25$ and use the initial value $u(0, x) = 0$
Local error
$f(u, x)=u+1$ $f(u, x)=u+p(x)$ $f(u, x)=u+q(x)$
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
$f(u, x)=u+1$ $f(u, x)=u+p(x)$ $f(u, x)=u+q(x)$
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
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Local error
$f(u, x)=u+1$ $f(u, x)=u+p(x)$ $f(u, x)=u+q(x)$
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
$f(u, x)=u+1$ $f(u, x)=u+p(x)$ $f(u, x)=u+q(x)$
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
Table 2.  The local (at $t = 0$) and global errors for the unmodified Strang splitting as well as the second order TDBC and CEC corrected Strang splitting applied to equation (19) with $f(u) = {\rm{e}}^{u-1}$ are shown. The space discretization is conducted by using the standard centered finite difference stencil with $200$ grid points. All problems are integrated until $t = 0.25$ and use the initial value $u(0, x) = \sin \pi x$
Local error
unmodified TDBC CEC
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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 $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
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Local error
unmodified TDBC CEC
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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 $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
Table 3.  The local (at $t = 0$) and global errors for the unmodified Strang splitting as well as the third order TDBC and CEC corrected Strang splitting applied to equation (19) with $f(u) = {\rm{e}}^{u-1}$ are shown. The space discretization is conducted by using the standard centered finite difference stencil with $200$ grid points. All problems are integrated until $t = 0.25$ and use the initial value $u(0, x) = \sin \pi x$
Local error
unmodified TDBC CEC
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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 $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
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Local error
unmodified TDBC CEC
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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 $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
Table 4.  The local (at $t = 0$) and global errors using the unmodified Strang splitting applied to equation (20) are shown for the three different reaction terms indicated in the table. The space discretization is conducted by using a second order upwind finite difference stencil with $10^3$ grid points. All problems are integrated until $t = 1.9$ and use the initial value $u(0, x) = 0$
Local error
$f(u, x)=u+1$ $f(u, x)=u+x$ $f(u, x)=u+x^2$
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
$f(u, x)=u+1$ $f(u, x)=u+x$ $f(u, x)=u+x^2$
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
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Local error
$f(u, x)=u+1$ $f(u, x)=u+x$ $f(u, x)=u+x^2$
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
$f(u, x)=u+1$ $f(u, x)=u+x$ $f(u, x)=u+x^2$
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
Table 5.  The local (at $t = 0$) and global errors for the unmodified Strang splitting as well as the second order TDBC and CEC corrected Strang splitting applied to equation (21) with $f(u) = {\rm{e}}^{u-1}$ are shown. The space discretization is conducted by using a second order upwind finite difference stencil with $500$ grid points. All problems are integrated until $t = 1.9$ and use the initial value $u(0, x) = 1+x$
Local error
unmodified TDBC CEC
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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 $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
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Local error
unmodified TDBC CEC
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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 $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
Table 6.  The local (at $t = 0$) and global errors for the unmodified Strang splitting as well as the third order TDBC and CEC corrected Strang splitting applied to equation (21) with $f(u) = {\rm{e}}^{u-1}$ are shown. The space discretization is conducted by using a second order upwind finite difference stencil with $500$ grid points. All problems are integrated until $t = 1.9$ and use the initial value $u(0, x) = 1+x$
Local error
unmodified TDBC CEC
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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 $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
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Local error
unmodified TDBC CEC
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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 $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
Table 7.  The local (at $t = 0$) and global errors computed in $[\tfrac{1}{2}, 1]$ for the unmodified Strang splitting as well as the second order TDBC and CEC corrected Strang splitting applied to equation (21) with $f(u) = {\rm{e}}^{u-1}$ are shown. The space discretization is conducted by using a second order upwind finite difference stencil with $500$ grid points. All problems are integrated until $t = 1.9$ and use the initial value $u(0, x) = 1+x$
Local error
unmodified TDBC CEC
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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 $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
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Local error
unmodified TDBC CEC
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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 $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
Table 8.  The accuracy (at times $t = 0.5$ and $t = 2$) of the best TDBC approach (this can be the second or third order correction) divided by the accuracy of the best CEC approach is shown for five different reactions $f_1 = \sqrt{u+1}$, $f_2 = {\rm{e}}^{u/5}$, $f_3 = \log(2+u)$, $f_4 = 1/2+\text{arsinhpt}{u}$, $f_5 = \cos u$ and five different advection coefficients $a_1 = 1+\sin x$, $a_2 = \sin(\pi x/2)+2/5$, $a_3 = 3/2-x$, $a_4 = 1/5+{\rm{e}}^{-50 (x-1/2)^2}$, $a_5 = 1 + \sin(2\pi x)/5$. The number in parentheses shows the gain in accuracy achieved by going from CEC2 to CEC3 and from TDBC2 to TDBC3, respectively (values larger than one indicate a gain in accuracy, while values smaller than one indicate a loss in accuracy). The space discretization is conducted by using a second order upwind finite difference stencil with $500$ grid points
t=0.5
$a_1$ $a_2$ $a_3$ $a_4$ $a_5$
$f_1$ 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)
$f_2$ 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)
$f_3$ 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)
$f_4$ 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)
$f_5$ 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)
$t=2$
$a_1$ $a_2$ $a_3$ $a_4$ $a_5$
$f_1$ 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)
$f_2$ 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)
$f_3$ 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)
$f_4$ 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)
$f_5$ 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)
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t=0.5
$a_1$ $a_2$ $a_3$ $a_4$ $a_5$
$f_1$ 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)
$f_2$ 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)
$f_3$ 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)
$f_4$ 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)
$f_5$ 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)
$t=2$
$a_1$ $a_2$ $a_3$ $a_4$ $a_5$
$f_1$ 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)
$f_2$ 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)
$f_3$ 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)
$f_4$ 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)
$f_5$ 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)
Table 9.  The local (at $t = 0$) and global errors for the unmodified Strang splitting as well as the second order TDBC and CEC corrected Strang splitting applied to equation (22) with $f(u) = {\rm{e}}^{u-1}$ are shown. The space discretization is conducted by using the standard centered finite difference stencil with $200$ grid points. All problems are integrated until $t = 0.19$ and use the initial value $u(0, x) = 1+\sin \pi x + i \sin 2 \pi x$
Local error
unmodified TDBC CEC
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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 $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
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Local error
unmodified TDBC CEC
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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 $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
Table 10.  The local (at $t = 0$) and global errors for the unmodified Strang splitting as well as the third order TDBC and CEC corrected Strang splitting applied to equation (22) with $f(u) = {\rm{e}}^{u-1}$ are shown. The space discretization is conducted by using the standard centered finite difference stencil with $200$ grid points. All problems are integrated until $t = 0.19$ and use the initial value $u(0, x) = 1+\sin \pi x + i \sin 2 \pi x$
Local error
unmodified TDBC CEC
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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 $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
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Local error
unmodified TDBC CEC
step size $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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 $l^{\infty}$ error order $l^{\infty}$ error order $l^{\infty}$ error 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
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