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August  2021, 14(8): 2805-2821. doi: 10.3934/dcdss.2020411

Numerical simulations of parity–time symmetric nonlinear Schrödinger equations in critical case

Edès Destyl , , Jacques Laminie , Paul Nuiro and  Pascal Poullet

Laboratoire de Mathématiques Informatique et Applications, Université des Antilles, BP 250, F-97157 Pointe à Pitre cedex, Guadeloupe FWI

* Corresponding author: E. Destyl, edes.destyl@univ-antilles.fr

Received  April 2020 Revised  May 2020 Published  July 2020

In this paper, we study the solution behavior of two coupled non–linear Schrödinger equations (CNLS) in the critical case, where one equation includes gain, while the other includes losses. Next, we present two numerical methods for solving the CNLS equations, for which we have made a comparison. These numerical experiments permit to illustrate other theoretical results proven by the authors [11]. We also obtain several numerical results for different non–linearities and investigate on the value of the blow up time relatively to some parameters.

Keywords: Numerical simulations, Blow up solution, Parity–Time symmetry, Nonlinear Schrödinger equations, Crank–Nicolson scheme, Finite difference method.
Mathematics Subject Classification: 35B40, 35B44, 35J10, 35Q41, 65M06, 68N15, 65N05.
Citation: Edès Destyl, Jacques Laminie, Paul Nuiro, Pascal Poullet. Numerical simulations of parity–time symmetric nonlinear Schrödinger equations in critical case. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2805-2821. doi: 10.3934/dcdss.2020411
References:
[1]

G. P. Agrawal, Applications of Nonlinear Fiber Optics, Optics and Photonics Series 2nd edition, Academic Press, Elsevier, (2008). Google Scholar

[2]

X. Antoine, C. Besse and S. Descombes, Artificial boundary conditions for one–dimensional cubic nonlinear Schrödinger equations, SIAM J. Numer. Anal., 43 (2006), 2272-2293.  doi: 10.1137/040606983.  Google Scholar

[3]

V. A. Baskakov and A. V. Popov, Implementation of transparent boundaries for numerical solution of the Schrödinger equation, Wave Motion, 14 (1991), 123-128.  doi: 10.1016/0165-2125(91)90053-Q.  Google Scholar

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C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys., 70 (2007), 947-1018.  doi: 10.1088/0034-4885/70/6/R03.  Google Scholar

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C. M. Bender, B. Berntson, D. Parker and E. Samuel, Observation of PT phase transition in a simple mechanical system, Am. J. Phys., 81 (2013), 173. doi: 10.1119/1.4789549.  Google Scholar

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C. M. Bender and S. Borttcher, Real spectra in non-Hermitian Hamiltonians having $\mathcal{PT}-$symmetry, Phys. Rev. Lett., 80 (1998), 5243-5246.  doi: 10.1103/PhysRevLett.80.5243.  Google Scholar

[7]

E. Destyl, S. P. Nuiro and P. Poullet, On the global behavior of solutions of a coupled system of nonlinear Schrödinger equation, Stud. Appl. Math., 138 (2017), 227-244.  doi: 10.1111/sapm.12150.  Google Scholar

[8]

E. Destyl, S. P. Nuiro, D. E. Pelinovsky and P. Poullet, Coupled pendula chains under parametric $\mathcal{PT}$-symmetric driving force, Phys. Lett. A, 381 (2017), 3884-3892.  doi: 10.1016/j.physleta.2017.10.021.  Google Scholar

[9]

E. Destyl, S. P. Nuiro and P. Poullet, Critical blow up in coupled Parity-Time-symmetric nonlinear Schrödinger equations, AIMS Math., 2 (2017), 195-206.   Google Scholar

[10]

E. Destyl, Modélisation et Analyse de Systèmes D'équations de Schrödinger non Linéaire, Thèse de doctorat de l'Universté des Antilles, 28 septembre 2018, Pointe-à-Pitre, Guadeloupe. Google Scholar

[11]

J.-P. Dias, M. Figueira, V. V. Konotop and D. A. Zezyulin, Supercritical blow up in coupled parity-time-symmetric nonlinear Schrödinger equations, Stud. Appl. Math., 133 (2014), 422-440.  doi: 10.1111/sapm.12063.  Google Scholar

[12]

J. P. Dias, M. M. Figueira and V. V. Konotop, The Cauchy problem for coupled nonlinear Schrödinger equations with linear damping: local and global existence and blow up of solutions, Chin. Ann. Math., 37 (2016), 665-682.  doi: 10.1007/s11401-016-1006-0.  Google Scholar

[13]

L. Di Menza, Transparent and absorbing boundary conditions for the Schrödinger equation in a bounded domain, (English summary), Numer. Funct. Anal. Optim., 18 (1997), 759-775.  doi: 10.1080/01630569708816790.  Google Scholar

[14]

R. Driben and B. A. Malomed, Stability of solitons in parity-time-symmetric couplers, Opt. Lett., 36 (2011), 4323-4325.  doi: 10.1364/OL.36.004323.  Google Scholar

[15]

M. S. Ismail and T. R. Taha, Numerical simulation of coupled nonlinear Schödinger equation, Math. Comput. Simul., 56 (2001), 547-562.  doi: 10.1016/S0378-4754(01)00324-X.  Google Scholar

[16]

A. Jüngel and R.-M. Weishäupl, Blow up in two-component nonlinear Schrödinger systems with an external driven field, Math. Models Methods Appl. Sci., 23 (2013), 1699-1727.  doi: 10.1142/S0218202513500206.  Google Scholar

[17]

V. V. Konotop, J. Yang and D. A. Zezyulin, Nonlinear waves in PT–symmetric system, Rev. Mod., 88 (2016), 035002(59). Google Scholar

[18]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Springer, LLC, 2009.  Google Scholar

[19]

D. E. Pelinovsky, D. A. Zezyulin and V. V. Konotop, Global existence of solutions to coupled PT–symmetric nonlinear Schrodiger equations, Int. J. Theor. Phys., 54 (2015), 3920-3931.  doi: 10.1007/s10773-014-2422-0.  Google Scholar

[20]

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Springer, New-York, 1999.  Google Scholar

[21]

T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations II. Numerical nonlinear Schrödinger equation, J. Comp. Phys., 55 (2006), 203-230.  doi: 10.1016/0021-9991(84)90003-2.  Google Scholar

[22]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983), 567-576.   Google Scholar

[23]

C. Zheng, A perfectly matched layer approach to the nonlinear Schrödinger wave equations, J. Comput. Phys., 227 (2007), 537-556.  doi: 10.1016/j.jcp.2007.08.004.  Google Scholar

show all references

References:
[1]

G. P. Agrawal, Applications of Nonlinear Fiber Optics, Optics and Photonics Series 2nd edition, Academic Press, Elsevier, (2008). Google Scholar

[2]

X. Antoine, C. Besse and S. Descombes, Artificial boundary conditions for one–dimensional cubic nonlinear Schrödinger equations, SIAM J. Numer. Anal., 43 (2006), 2272-2293.  doi: 10.1137/040606983.  Google Scholar

[3]

V. A. Baskakov and A. V. Popov, Implementation of transparent boundaries for numerical solution of the Schrödinger equation, Wave Motion, 14 (1991), 123-128.  doi: 10.1016/0165-2125(91)90053-Q.  Google Scholar

[4]

C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys., 70 (2007), 947-1018.  doi: 10.1088/0034-4885/70/6/R03.  Google Scholar

[5]

C. M. Bender, B. Berntson, D. Parker and E. Samuel, Observation of PT phase transition in a simple mechanical system, Am. J. Phys., 81 (2013), 173. doi: 10.1119/1.4789549.  Google Scholar

[6]

C. M. Bender and S. Borttcher, Real spectra in non-Hermitian Hamiltonians having $\mathcal{PT}-$symmetry, Phys. Rev. Lett., 80 (1998), 5243-5246.  doi: 10.1103/PhysRevLett.80.5243.  Google Scholar

[7]

E. Destyl, S. P. Nuiro and P. Poullet, On the global behavior of solutions of a coupled system of nonlinear Schrödinger equation, Stud. Appl. Math., 138 (2017), 227-244.  doi: 10.1111/sapm.12150.  Google Scholar

[8]

E. Destyl, S. P. Nuiro, D. E. Pelinovsky and P. Poullet, Coupled pendula chains under parametric $\mathcal{PT}$-symmetric driving force, Phys. Lett. A, 381 (2017), 3884-3892.  doi: 10.1016/j.physleta.2017.10.021.  Google Scholar

[9]

E. Destyl, S. P. Nuiro and P. Poullet, Critical blow up in coupled Parity-Time-symmetric nonlinear Schrödinger equations, AIMS Math., 2 (2017), 195-206.   Google Scholar

[10]

E. Destyl, Modélisation et Analyse de Systèmes D'équations de Schrödinger non Linéaire, Thèse de doctorat de l'Universté des Antilles, 28 septembre 2018, Pointe-à-Pitre, Guadeloupe. Google Scholar

[11]

J.-P. Dias, M. Figueira, V. V. Konotop and D. A. Zezyulin, Supercritical blow up in coupled parity-time-symmetric nonlinear Schrödinger equations, Stud. Appl. Math., 133 (2014), 422-440.  doi: 10.1111/sapm.12063.  Google Scholar

[12]

J. P. Dias, M. M. Figueira and V. V. Konotop, The Cauchy problem for coupled nonlinear Schrödinger equations with linear damping: local and global existence and blow up of solutions, Chin. Ann. Math., 37 (2016), 665-682.  doi: 10.1007/s11401-016-1006-0.  Google Scholar

[13]

L. Di Menza, Transparent and absorbing boundary conditions for the Schrödinger equation in a bounded domain, (English summary), Numer. Funct. Anal. Optim., 18 (1997), 759-775.  doi: 10.1080/01630569708816790.  Google Scholar

[14]

R. Driben and B. A. Malomed, Stability of solitons in parity-time-symmetric couplers, Opt. Lett., 36 (2011), 4323-4325.  doi: 10.1364/OL.36.004323.  Google Scholar

[15]

M. S. Ismail and T. R. Taha, Numerical simulation of coupled nonlinear Schödinger equation, Math. Comput. Simul., 56 (2001), 547-562.  doi: 10.1016/S0378-4754(01)00324-X.  Google Scholar

[16]

A. Jüngel and R.-M. Weishäupl, Blow up in two-component nonlinear Schrödinger systems with an external driven field, Math. Models Methods Appl. Sci., 23 (2013), 1699-1727.  doi: 10.1142/S0218202513500206.  Google Scholar

[17]

V. V. Konotop, J. Yang and D. A. Zezyulin, Nonlinear waves in PT–symmetric system, Rev. Mod., 88 (2016), 035002(59). Google Scholar

[18]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Springer, LLC, 2009.  Google Scholar

[19]

D. E. Pelinovsky, D. A. Zezyulin and V. V. Konotop, Global existence of solutions to coupled PT–symmetric nonlinear Schrodiger equations, Int. J. Theor. Phys., 54 (2015), 3920-3931.  doi: 10.1007/s10773-014-2422-0.  Google Scholar

[20]

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Springer, New-York, 1999.  Google Scholar

[21]

T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations II. Numerical nonlinear Schrödinger equation, J. Comp. Phys., 55 (2006), 203-230.  doi: 10.1016/0021-9991(84)90003-2.  Google Scholar

[22]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983), 567-576.   Google Scholar

[23]

C. Zheng, A perfectly matched layer approach to the nonlinear Schrödinger wave equations, J. Comput. Phys., 227 (2007), 537-556.  doi: 10.1016/j.jcp.2007.08.004.  Google Scholar

Figure 1.  Convergence for the scheme using the fixed point iteration method
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Figure 2.  Convergence in time of the scheme
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Figure 3.  $ ||u||_{L^2} $ (green), $ ||v||_{L^2} $ (blue), $ ||\nabla u||_{L^2}^2 $ (yellow), $ ||\nabla v||_{L^2}^2 $ (brown), $ ||u||_{L^{\infty}}^2 $ (black) and $ ||v||_{L^{\infty}}^2 $ (red), $ S_1(t) $ (magenta), $ Q(t) $ (purple) et $ D(t) $ (celestial blue) for initial condition (2) such that $ (A, B) = (0.1, 0.2) $, $ (\kappa, \gamma) = (1, 0.5) $ and $ g_1 = g_2 = g = 1 $
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Figure 4.  Modulus squared of the computed density (1st component in red, 2nd one in blue) at several times, in the Manakov case, for an initial condition (2) such that $ (A, B)\! = \!(0.1, 0.2), \, (\kappa, \gamma)\! = \!(1, 0.5) $ and $ g_1 = g_2 = g = 1 $
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Figure 5.  $ ||u||_{L^2} $ (green), $ ||v||_{L^2} $ (blue), $ ||\nabla u||_{L^2}^2 $ (yellow), $ ||\nabla v||_{L^2}^2 $ (brown), $ ||u||_{L^{\infty}}^2 $ (black) and $ ||v||_{L^{\infty}}^2 $ (red), $ S_1(t) $ (magenta), $ Q(t) $ (green) et $ D(t) $ (celestial blue) for an initial condition (2) such that $ (A, B) = (5, 1) $, $ (\kappa, \gamma) = (1, 0.5) $ and $ g_1 = g_2 = g = 1 $
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Figure 6.  Behavior of $ S_1 $ versus time for initial condition (2) such that $ (A, B) = (0.1, 0.2) $, $ (\kappa, \gamma) = (1, 0.5) $ and $ g_1 = g_2 = g = 1 $
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Figure 7.  Surfaces of the position density $ |u(x, t)|^2 $ (red) and $ |v(x, t)|^2 $ (blue) at time $ t = 0.0 $, $ t = 0.032 $ (upper row) and for $ t = 0.05 $, $ t = 0.062 $ (lower row) in the Manakov case, for the initial condition (2) such that $ (A, B) = (5, 1) $, $ (\kappa, \gamma) = (1, 0.5) $ and $ g_1 = g_2 = g = 1 $
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Figure 8.  $ ||u||_{L^2} $ (green), $ ||v||_{L^2} $ (blue), $ ||\nabla u||_{L^2}^2 $ (yellow), $ ||\nabla v||_{L^2}^2 $ (brown), $ ||u||_{L^{\infty}}^2 $ (black) and $ ||v||_{L^{\infty}}^2 $ (red), $ S_1(t) $ (magenta), $ Q(t) $ (purple) and $ D(t) $ celestial blue) for initial condition (2) such that $ (A, B) = (1.5, 1.5) $, $ (\kappa, \gamma) = (3, 0.5) $ and $ g_1 = g_2 = g = 1 $
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Figure 9.  $ ||u||_{L^2} $ (green), $ ||v||_{L^2} $ (blue), $ ||\nabla u||_{L^2}^2 $ (yellow), $ ||\nabla v||_{L^2}^2 $ (brown), $ ||u||_{L^{\infty}}^2 $ (black) and $ ||v||_{L^{\infty}}^2 $ (red), $ S_1(t) $ (magenta), $ Q(t) $ (purple) et $ D(t) $ (celestial blue) for initial condition (2) such that $ (A, B) = (1.5, 1.5) $, $ (\kappa, \gamma) = (1, 0.5) $ and $ g_1 = g_2 = g = 1 $
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Figure 10.  $ ||u||_{L^2} $ (green), $ ||v||_{L^2} $ (blue), $ ||\nabla u||_{L^2}^2 $ (yellow), $ ||\nabla v||_{L^2}^2 $ (brown), $ ||u||_{L^{\infty}}^2 $ (black) and $ ||v||_{L^{\infty}}^2 $ (red), $ S_1(t) $ (magenta), $ Q(t) $ (purple) et $ D(t) $ (celestial blue) for initial condition {(2)} with $ (A, B) = (0.1, 0.2) $, $ (\kappa, \gamma) = (1, 0.5) $ and $ g_1 = g_2 = 1 \neq g = 0.5 $
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Figure 11.  $ ||u||_{L^2} $ (green), $ ||v||_{L^2} $ (blue), $ ||\nabla u||_{L^2}^2 $ (yellow), $ ||\nabla v||_{L^2}^2 $ (brown), $ ||u||_{L^{\infty}}^2 $ (black) and $ ||v||_{L^{\infty}}^2 $ (red), $ S_1(t) $ (magenta), $ Q(t) $ (purple) et $ D(t) $ { (celestial blue)} for initial conditions {(2)} with $ (A, B) = (1, 3) $, $ (\kappa, \gamma) = (1, 0.5) $ and $ g_1 = g_2 = 1 \neq g = 0.5 $
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Figure 12.  Surfaces of the position density $ |u(x, t)|^2 $ (red) and $ |v(x, t)|^2 $ (blue) at time $ t = 0.0 $, $ t = 5.014 $ (upper row) and for $ t = 9.718 $, $ t = 10.0 $ (lower row) in the general model case, for the initial condition (2) with $ (A, B) = (0.1, 0.2) $, $ (\kappa, \gamma) = (1, 0.5) $ and $ g_1 = g_2 = g = 1 $
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Figure 13.  Surfaces of the position density $ |u(x, t)|^2 $ (red) and $ |v(x, t)|^2 $ (blue) at time $ t = 0.0 $, $ t = 3.860 $ (upper row) and for $ t = 4.708 $, $ t = 4.746 $ (lower row) in the general model case, for initial condition { (2)} with $ (A, B) = (1, 3) $, $ (\kappa, \gamma) = (1, 0.5) $ and $ g_1 = g_2 = 1 \neq g = 0.5 $
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Table 1.  Convergence for the fixed point iteration method
$s$ Spatial step Error Order $s$ Spatial step Error Order
$0$ $4.0\times 10^{-3}$ $4.0\times10^{-8}$ $0$ $4.0\times 10^{-3}$ $6.15\times 10^{-8}$
$1$ $8.0\times10^{-3}$ $2.0\times 10^{-7}$ 2.32 $1$ $8.0\times 10^{-3}$ $3.07\times 10^{-7}$ 2.32
$2$ $1.6\times 10^{-2}$ $8.4\times 10^{-7}$ 2.07 $2$ $1.6\times 10^{-2}$ $1.29\times 10^{-6}$ 2.07
$3$ $3.2\times 10^{-2}$ $3.4\times 10^{-6}$ 2.01 $3$ $3.2\times 10^{-2}$ $5.22\times 10^{-6}$ 2.01
(a) Results for the 1st equation (b) Results for the 2nd equation
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$s$ Spatial step Error Order $s$ Spatial step Error Order
$0$ $4.0\times 10^{-3}$ $4.0\times10^{-8}$ $0$ $4.0\times 10^{-3}$ $6.15\times 10^{-8}$
$1$ $8.0\times10^{-3}$ $2.0\times 10^{-7}$ 2.32 $1$ $8.0\times 10^{-3}$ $3.07\times 10^{-7}$ 2.32
$2$ $1.6\times 10^{-2}$ $8.4\times 10^{-7}$ 2.07 $2$ $1.6\times 10^{-2}$ $1.29\times 10^{-6}$ 2.07
$3$ $3.2\times 10^{-2}$ $3.4\times 10^{-6}$ 2.01 $3$ $3.2\times 10^{-2}$ $5.22\times 10^{-6}$ 2.01
(a) Results for the 1st equation (b) Results for the 2nd equation
Table 2.  $\ell^2(]0, T[; \ell^2(I))$-error behavior of the scheme using the fixed point iteration method
$ s $ Time step Error Order $s$ Time step Error Order
$ 0 $ $ 4.0\times10^{-4} $ $ 1.30 \times 10^{-6} $ $0$ $4.0\times 10^{-4}$ $1.04\times 10^{-6}$
$ 1 $ $ 8.0\times10^{-4} $ $ 4.30\times10^{-6} $ 1.62 $1$ $8.0\times 10^{-4}$ $3.22\times 10^{-6}$ 1.62
$ 2 $ $ 1.6\times10^{-3} $ $ 1.06\times10^{-5} $ 1.30 $2$ $1.6\times 10^{-3}$ $7.96\times 10^{-6}$ 1.30
$ 3 $ $ 3.2\times10^{-3} $ $ 1.98\times 10^{-4} $ 4.22 $3$ $3.2\times 10^{-3}$ $2.52\times 10^{-5}$ 1.66
(a) Results for the 1st equation (b) Results for the 2nd equation
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$ s $ Time step Error Order $s$ Time step Error Order
$ 0 $ $ 4.0\times10^{-4} $ $ 1.30 \times 10^{-6} $ $0$ $4.0\times 10^{-4}$ $1.04\times 10^{-6}$
$ 1 $ $ 8.0\times10^{-4} $ $ 4.30\times10^{-6} $ 1.62 $1$ $8.0\times 10^{-4}$ $3.22\times 10^{-6}$ 1.62
$ 2 $ $ 1.6\times10^{-3} $ $ 1.06\times10^{-5} $ 1.30 $2$ $1.6\times 10^{-3}$ $7.96\times 10^{-6}$ 1.30
$ 3 $ $ 3.2\times10^{-3} $ $ 1.98\times 10^{-4} $ 4.22 $3$ $3.2\times 10^{-3}$ $2.52\times 10^{-5}$ 1.66
(a) Results for the 1st equation (b) Results for the 2nd equation
Table 3.  $\ell^2(]0, T[;\ell^2(I))$-error behavior of the scheme with the Newton method
$s$ Time step Error Order $s$ Time step Error Order
$0$ $4.0 \times 10^{-4}$ $9.88\times 10^{-9}$ $0$ $4.0\times10^{-4}$ $1.47\times 10^{-8}$
$1$ $8.0\times 10^{-4}$ $4.14\times 10^{-8}$ 2.06 $1$ $8.0\times 10^{-4}$ $5.75\times 10^{-8}$ 1.97
$2$ $1.6\times 10^{-3}$ $1.73\times 10^{-7}$ 2.06 $2$ $1.6\times 10^{-3}$ $2.28\times 10^{-7}$ 1.99
$3$ $3.2\times 10^{-3}$ $7.79\times 10^{-7}$ 2.17 $3$ $3.2\times 10^{-3}$ $1.01\times 10^{-6}$ 2.140
(a) Results for the 1st equation (b) Results for the 2nd equation
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$s$ Time step Error Order $s$ Time step Error Order
$0$ $4.0 \times 10^{-4}$ $9.88\times 10^{-9}$ $0$ $4.0\times10^{-4}$ $1.47\times 10^{-8}$
$1$ $8.0\times 10^{-4}$ $4.14\times 10^{-8}$ 2.06 $1$ $8.0\times 10^{-4}$ $5.75\times 10^{-8}$ 1.97
$2$ $1.6\times 10^{-3}$ $1.73\times 10^{-7}$ 2.06 $2$ $1.6\times 10^{-3}$ $2.28\times 10^{-7}$ 1.99
$3$ $3.2\times 10^{-3}$ $7.79\times 10^{-7}$ 2.17 $3$ $3.2\times 10^{-3}$ $1.01\times 10^{-6}$ 2.140
(a) Results for the 1st equation (b) Results for the 2nd equation
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