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December  2020, 28(4): 1439-1457. doi: 10.3934/era.2020076

A robust adaptive grid method for singularly perturbed Burger-Huxley equations

Li-Bin Liu 1,, , Ying Liang 1, , Jian Zhang 1, and  Xiaobing Bao 2,

1. 

School of Mathematics and Statistics, Nanning Normal University, Nanning 530001, China
2. 

School of Big Data and Artificial Intelligence, Chizhou University, Chizhou, Anhui 247000, China

* Corresponding author: liulibin969@163.com

Received  November 2019 Revised  July 2020 Published  December 2020 Early access  July 2020

Fund Project: The first author is supported by National Science Foundation of China (11761015), the Natural Science Foundation of Guangxi(2020GXNSFAA159010), the key project of Natural Science Foundation of Guangxi(2017GXNSFDA198014, 2018GXNSFDA050014), the key project of Natural Science Foundation of Chizhou University(CZ2018ZR06)

In this paper, an adaptive grid method is proposed to solve one-dimensional unsteady singularly perturbed Burger-Huxley equation with appropriate initial and boundary conditions. Firstly, we use the classical backward-Euler scheme on a uniform mesh to approximate time derivative. The resulting nonlinear singularly perturbed semi-discrete problem is linearized by using Newton-Raphson-Kantorovich approximation method which is quadratically convergent. Then, an upwind finite difference scheme on an adaptive nonuniform grid is used for space derivative. The nonuniform grid is generated by equidistribution of a positive monitor function, which is similar to the arc-length function. It is shown that the presented adaptive grid method is first order uniform convergent in the time and spatial directions, respectively. Finally, numerical results are given to validate the theoretical results.

Keywords: Singularly perturbed, Burger-Huxley equation, upwind finite difference scheme, adaptive grid, uniform convergent.
Mathematics Subject Classification: 65M06, 65M12, 65M50.
Citation: Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076
References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[2]

B. Batiha, M. S. M. Noorani and I. Hashim, Numerical simulation of the generalized Huxley equation by He's variational iteration method, Appl. Math. Comput., 186 (2007), 1322-1325.  doi: 10.1016/j.amc.2006.07.166.  Google Scholar

[3]

R. E. Bellman and R. E. Kalaba, Quasilineaization and Nonlinear Boundary-Value Problems, Modern Analytic and Computional Methods in Science and Mathematics, Vol. 3 American Elsevier Publishing Co., Inc., New York 1965.  Google Scholar

[4]

Y. Chen, Uniform pointwise convergence for a singularly perturbed problem using arc-length equidistribution, J. Comput. Appl. Math., 159 (2003), 25-34.  doi: 10.1016/S0377-0427(03)00563-6.  Google Scholar

[5]

Y. Chen, Uniform convergence analysis of finite difference approximations for singular perturbation problems on an adapted grid, Adv. Comput. Math., 24 (2006), 197-212.  doi: 10.1007/s10444-004-7641-0.  Google Scholar

[6]

Y. Chen and L.-B. Liu, An adaptive grid method for singularly perturbed time-dependent convection-diffusion problems, Commum. Comput. Phys., 20 (2016), 1340-1358.  doi: 10.4208/cicp.240315.301215a.  Google Scholar

[7]

C. Clavero, J. C. Jorge and F. Lisbona, A uniformly convergent scheme on a nonuniform mesh for convection-diffusion parabolic problems, J. Comput. Appl. Math., 154 (2003), 415-429.  doi: 10.1016/S0377-0427(02)00861-0.  Google Scholar

[8]

M. T. Darvishi, S. Kheybari and F. Khani, Spectral collocation method and Darvishi's preconditionings to solve the generalized Burgers-Huxley equation, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 2091-2103.  doi: 10.1016/j.cnsns.2007.05.023.  Google Scholar

[9]

L. Duan and Q. Lu, Bursting oscillations near codimension-two bifurcations in the Chay neuron model, Int. J. Nonlinear Sci. Numer. Simul., 7 (2006), 59-63.  doi: 10.1515/IJNSNS.2006.7.1.59.  Google Scholar

[10]

S. Gowrisankar and S. Natesan, The parameter uniform numerical method for singularly perturbed parabolic reaction-diffusion problems on equidistributed grids, Appl. Math. Lett., 26 (2013), 1053-1060.  doi: 10.1016/j.aml.2013.05.017.  Google Scholar

[11]

S. Gowrisankar and S. Natesan, Uniformly convergent numerical method for singularly perturbed parabolic initial-boundary-value problems with equidistributed grids, Int. J. Comput. Math., 91 (2014), 553-577.  doi: 10.1080/00207160.2013.792925.  Google Scholar

[12]

S. Gowrisankar and S. Natesan, Robust numerical scheme for singularly perturbed convection-diffusion parabolic initial-boundary-value problems on equidistributed grids, Comput. Phys. Commun., 185 (2014), 2008-2019.  doi: 10.1016/j.cpc.2014.04.004.  Google Scholar

[13]

V. Gupta and M. K. Kadalbajoo, A singular perturbation approach to solve Burgers-Huxley equation via monotone finite difference scheme on layer-adaptive mesh, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1825-1844.  doi: 10.1016/j.cnsns.2010.07.020.  Google Scholar

[14]

I. Hashim, M. S. M. Noorani and B. Batiha, A note on the Adomian decomposition method for the generalized Huxley equation, Appl. Math. Comput., 181 (2006), 1439-1445.  doi: 10.1016/j.amc.2006.03.011.  Google Scholar

[15]

I. Hashim, M. S. M. Noorani and M. R. Said Al-Hadidi, Solving the generalized Burgers-Huxley equation using the adomian decomposition method, Math. Comput. Model., 43 (2006), 1404-1411.  doi: 10.1016/j.mcm.2005.08.017.  Google Scholar

[16]

H. N. A. Ismail, K. Raslan and A. A. A. Rabboh, Adomian decomposition method for Burgers-Huxley and Burgers-Fisher equations, Appl. Math. Comput., 159 (2004), 291-301.  doi: 10.1016/j.amc.2003.10.050.  Google Scholar

[17]

M. Javidi, A numerical solution of the generalized Burgers-Huxley equation by spectral collocation method, Appl. Math. Comput., 178 (2006), 338-344.  doi: 10.1016/j.amc.2005.11.051.  Google Scholar

[18]

M. Javidi and A. Golbabai, A new domain decomposition algorithm for generalized Burgers-Huxley equation based on Chebyshev polynomials and preconditioning, Chaos Solitons Fractals, 39 (2009), 849-857.  doi: 10.1016/j.chaos.2007.01.099.  Google Scholar

[19]

A. Kaushik and M. D. Sharma, A uniformly convergent numerical method on non-uniform mesh for singularly perturbed unsteady Burger-Huxley equation, Appl. Math. Comput., 195 (2008), 688-706.  doi: 10.1016/j.amc.2007.05.067.  Google Scholar

[20]

A. J. Khattak, A computational meshless method for the generalized Burger's-Huxley equation, Appl. Math. Model., 33 (2009), 3718-3729.  doi: 10.1016/j.apm.2008.12.010.  Google Scholar

[21]

N. Kopteva, Maximum norm a posteriori error estimates for a one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39 (2001), 423-441.  doi: 10.1137/S0036142900368642.  Google Scholar

[22]

N. Kopteva and M. Stynes, A robust adaptive method for quasi-linear one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39 (2001), 1446-1467.  doi: 10.1137/S003614290138471X.  Google Scholar

[23]

L.-B. Liu and Y. Chen, A robust adaptive grid method for a system of two singularly perturbed convection-diffusion equations with weak coupling, J. Sci. Comput., 61 (2014), 1-16.  doi: 10.1007/s10915-013-9814-9.  Google Scholar

[24]

S. Liu, T. Fan and Q. Lu, The spike order of the winnerless competition (WLC) model and its application to the inhibition neural system, Int. J. Nonlin. Sci. Numer. Simul., 6 (2005), 133-138.  doi: 10.1515/IJNSNS.2005.6.2.133.  Google Scholar

[25]

R. C. Mittal and A. Tripathi, Numerical solutions of generalized Burgers-Fisher and generalized Burgers-Huxley equations using collocation of cubic $B$-splines, Int. J. Comput. Math., 92 (2015), 1053-1077.  doi: 10.1080/00207160.2014.920834.  Google Scholar

[26]

R. Mohammadi, B-spline collocation algorithm for numerical solution of the generalized Burger's-Huxley equation, Numer. Methods Partial Differential Equations, 29 (2013), 1173-1191.  doi: 10.1002/num.21750.  Google Scholar

[27]

R. K. Mohanty, W. Dai and D. Liu, Operator compact method of accuracy two in time and four in space for the solution of time dependent Burgers-Huxley equation, Numer. Algorithms, 70 (2015), 591-605.  doi: 10.1007/s11075-015-9963-z.  Google Scholar

[28]

S. Murat, G. Görhan and Z. Asuman, High-order finite difference schemes for numerical solutions of the generalized Burgers-Huxley equation, Numer. Methods Partial Differential Equations, 27 (2011), 1313-1326.   Google Scholar

[29]

H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Convection-diffusion and flow problems. Springer Series in Computational Mathematics, 24. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03206-0.  Google Scholar

[30]

J. Satsuma J, Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, World Scientific, Singapore, 1987. Google Scholar

[31]

X. Y. Wang, Z. S. Zhu and Y. K. Lu, Solitary wave solutions of the generalised Burgers-Huxley equation, J. Phys. A, 23 (1990), 271-274.  doi: 10.1088/0305-4470/23/3/011.  Google Scholar

[32]

A.-M. Wazwaz, Travelling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations, Appl. Math. Comput., 169 (2005), 639-656.  doi: 10.1016/j.amc.2004.09.081.  Google Scholar

[33]

G.-J. Zhang, J.-X. Xu, H. Yao et al., Mechanism of bifurcation-dependent coherence resonance of an excitable neuron model, Int. J. Nonlin. Sci. Numer. Simul., 7 (2006), 447-450. Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[2]

B. Batiha, M. S. M. Noorani and I. Hashim, Numerical simulation of the generalized Huxley equation by He's variational iteration method, Appl. Math. Comput., 186 (2007), 1322-1325.  doi: 10.1016/j.amc.2006.07.166.  Google Scholar

[3]

R. E. Bellman and R. E. Kalaba, Quasilineaization and Nonlinear Boundary-Value Problems, Modern Analytic and Computional Methods in Science and Mathematics, Vol. 3 American Elsevier Publishing Co., Inc., New York 1965.  Google Scholar

[4]

Y. Chen, Uniform pointwise convergence for a singularly perturbed problem using arc-length equidistribution, J. Comput. Appl. Math., 159 (2003), 25-34.  doi: 10.1016/S0377-0427(03)00563-6.  Google Scholar

[5]

Y. Chen, Uniform convergence analysis of finite difference approximations for singular perturbation problems on an adapted grid, Adv. Comput. Math., 24 (2006), 197-212.  doi: 10.1007/s10444-004-7641-0.  Google Scholar

[6]

Y. Chen and L.-B. Liu, An adaptive grid method for singularly perturbed time-dependent convection-diffusion problems, Commum. Comput. Phys., 20 (2016), 1340-1358.  doi: 10.4208/cicp.240315.301215a.  Google Scholar

[7]

C. Clavero, J. C. Jorge and F. Lisbona, A uniformly convergent scheme on a nonuniform mesh for convection-diffusion parabolic problems, J. Comput. Appl. Math., 154 (2003), 415-429.  doi: 10.1016/S0377-0427(02)00861-0.  Google Scholar

[8]

M. T. Darvishi, S. Kheybari and F. Khani, Spectral collocation method and Darvishi's preconditionings to solve the generalized Burgers-Huxley equation, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 2091-2103.  doi: 10.1016/j.cnsns.2007.05.023.  Google Scholar

[9]

L. Duan and Q. Lu, Bursting oscillations near codimension-two bifurcations in the Chay neuron model, Int. J. Nonlinear Sci. Numer. Simul., 7 (2006), 59-63.  doi: 10.1515/IJNSNS.2006.7.1.59.  Google Scholar

[10]

S. Gowrisankar and S. Natesan, The parameter uniform numerical method for singularly perturbed parabolic reaction-diffusion problems on equidistributed grids, Appl. Math. Lett., 26 (2013), 1053-1060.  doi: 10.1016/j.aml.2013.05.017.  Google Scholar

[11]

S. Gowrisankar and S. Natesan, Uniformly convergent numerical method for singularly perturbed parabolic initial-boundary-value problems with equidistributed grids, Int. J. Comput. Math., 91 (2014), 553-577.  doi: 10.1080/00207160.2013.792925.  Google Scholar

[12]

S. Gowrisankar and S. Natesan, Robust numerical scheme for singularly perturbed convection-diffusion parabolic initial-boundary-value problems on equidistributed grids, Comput. Phys. Commun., 185 (2014), 2008-2019.  doi: 10.1016/j.cpc.2014.04.004.  Google Scholar

[13]

V. Gupta and M. K. Kadalbajoo, A singular perturbation approach to solve Burgers-Huxley equation via monotone finite difference scheme on layer-adaptive mesh, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1825-1844.  doi: 10.1016/j.cnsns.2010.07.020.  Google Scholar

[14]

I. Hashim, M. S. M. Noorani and B. Batiha, A note on the Adomian decomposition method for the generalized Huxley equation, Appl. Math. Comput., 181 (2006), 1439-1445.  doi: 10.1016/j.amc.2006.03.011.  Google Scholar

[15]

I. Hashim, M. S. M. Noorani and M. R. Said Al-Hadidi, Solving the generalized Burgers-Huxley equation using the adomian decomposition method, Math. Comput. Model., 43 (2006), 1404-1411.  doi: 10.1016/j.mcm.2005.08.017.  Google Scholar

[16]

H. N. A. Ismail, K. Raslan and A. A. A. Rabboh, Adomian decomposition method for Burgers-Huxley and Burgers-Fisher equations, Appl. Math. Comput., 159 (2004), 291-301.  doi: 10.1016/j.amc.2003.10.050.  Google Scholar

[17]

M. Javidi, A numerical solution of the generalized Burgers-Huxley equation by spectral collocation method, Appl. Math. Comput., 178 (2006), 338-344.  doi: 10.1016/j.amc.2005.11.051.  Google Scholar

[18]

M. Javidi and A. Golbabai, A new domain decomposition algorithm for generalized Burgers-Huxley equation based on Chebyshev polynomials and preconditioning, Chaos Solitons Fractals, 39 (2009), 849-857.  doi: 10.1016/j.chaos.2007.01.099.  Google Scholar

[19]

A. Kaushik and M. D. Sharma, A uniformly convergent numerical method on non-uniform mesh for singularly perturbed unsteady Burger-Huxley equation, Appl. Math. Comput., 195 (2008), 688-706.  doi: 10.1016/j.amc.2007.05.067.  Google Scholar

[20]

A. J. Khattak, A computational meshless method for the generalized Burger's-Huxley equation, Appl. Math. Model., 33 (2009), 3718-3729.  doi: 10.1016/j.apm.2008.12.010.  Google Scholar

[21]

N. Kopteva, Maximum norm a posteriori error estimates for a one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39 (2001), 423-441.  doi: 10.1137/S0036142900368642.  Google Scholar

[22]

N. Kopteva and M. Stynes, A robust adaptive method for quasi-linear one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39 (2001), 1446-1467.  doi: 10.1137/S003614290138471X.  Google Scholar

[23]

L.-B. Liu and Y. Chen, A robust adaptive grid method for a system of two singularly perturbed convection-diffusion equations with weak coupling, J. Sci. Comput., 61 (2014), 1-16.  doi: 10.1007/s10915-013-9814-9.  Google Scholar

[24]

S. Liu, T. Fan and Q. Lu, The spike order of the winnerless competition (WLC) model and its application to the inhibition neural system, Int. J. Nonlin. Sci. Numer. Simul., 6 (2005), 133-138.  doi: 10.1515/IJNSNS.2005.6.2.133.  Google Scholar

[25]

R. C. Mittal and A. Tripathi, Numerical solutions of generalized Burgers-Fisher and generalized Burgers-Huxley equations using collocation of cubic $B$-splines, Int. J. Comput. Math., 92 (2015), 1053-1077.  doi: 10.1080/00207160.2014.920834.  Google Scholar

[26]

R. Mohammadi, B-spline collocation algorithm for numerical solution of the generalized Burger's-Huxley equation, Numer. Methods Partial Differential Equations, 29 (2013), 1173-1191.  doi: 10.1002/num.21750.  Google Scholar

[27]

R. K. Mohanty, W. Dai and D. Liu, Operator compact method of accuracy two in time and four in space for the solution of time dependent Burgers-Huxley equation, Numer. Algorithms, 70 (2015), 591-605.  doi: 10.1007/s11075-015-9963-z.  Google Scholar

[28]

S. Murat, G. Görhan and Z. Asuman, High-order finite difference schemes for numerical solutions of the generalized Burgers-Huxley equation, Numer. Methods Partial Differential Equations, 27 (2011), 1313-1326.   Google Scholar

[29]

H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Convection-diffusion and flow problems. Springer Series in Computational Mathematics, 24. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03206-0.  Google Scholar

[30]

J. Satsuma J, Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, World Scientific, Singapore, 1987. Google Scholar

[31]

X. Y. Wang, Z. S. Zhu and Y. K. Lu, Solitary wave solutions of the generalised Burgers-Huxley equation, J. Phys. A, 23 (1990), 271-274.  doi: 10.1088/0305-4470/23/3/011.  Google Scholar

[32]

A.-M. Wazwaz, Travelling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations, Appl. Math. Comput., 169 (2005), 639-656.  doi: 10.1016/j.amc.2004.09.081.  Google Scholar

[33]

G.-J. Zhang, J.-X. Xu, H. Yao et al., Mechanism of bifurcation-dependent coherence resonance of an excitable neuron model, Int. J. Nonlin. Sci. Numer. Simul., 7 (2006), 447-450. Google Scholar

Figure 1.  Numerical solution profile of Example 5.1 with $ N = 64 $, $ M = 40 $ and $ \varepsilon = 2^{-14} $
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Figure 2.  Numerical solution of Example 5.1 at different time levels with $ N = 64 $, $ M = 40 $ and $ \varepsilon = 2^{-14} $
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Figure 3.  Mesh movement of Example 5.1 for $ N = 64 $, $ M = 40 $ and $ \varepsilon = 2^{-12} $
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Figure 4.  Numerical solution of Example 5.2 at different time levels with $ N = 64 $, $ M = 40 $ and $ \varepsilon = 2^{-8} $
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Figure 5.  Mesh movement of Example 5.2 for $ N = 128 $, $ M = 80 $ and $ \varepsilon = 2^{-10} $
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Table 1.  Maximum error of solution $ E_\varepsilon^{N,M} $ for Example 5.1 using the adaptive grid method
$ \varepsilon $ Number of intervals $ N $/time size $ \Delta t $
32/$ \frac{1}{20} $ 64/$ \frac{1}{40} $ 128/$ \frac{1}{80} $ 256/$ \frac{1}{160} $ 512/$ \frac{1}{320} $ 1024/$ \frac{1}{640} $
$ 2^{0} $ 2.4400e-02 1.4033e-02 7.5889e-03 3.9547e-03 2.0227e-03 1.0233e-03
$ 2^{-2} $ 1.8830e-02 1.1126e-02 6.5407e-03 3.6221e-03 1.9209e-03 1.0074e-03
$ 2^{-4} $ 2.0658e-02 1.2614e-02 7.0567e-03 3.7952e-03 1.9563e-03 1.0231e-03
$ 2^{-6} $ 2.5289e-02 1.7672e-02 9.0066e-03 4.8378e-03 2.5035e-03 1.2731e-03
$ 2^{-8} $ 3.8607e-02 1.9497e-02 1.1221e-02 6.2852e-03 3.3405e-03 1.7233e-03
$ 2^{-10} $ 9.3183e-02 7.0120e-02 4.4773e-02 2.0546e-02 1.0545e-02 5.1608e-03
$ 2^{-12} $ 1.7017e-01 1.0083e-01 6.2216e-02 3.9526e-02 2.0493e-02 1.0027e-02
$ 2^{-14} $ 2.0410e-01 1.6703e-01 9.2110e-02 5.2580e-02 2.8766e-02 1.6523e-02
$ 2^{-16} $ 2.0450e-01 1.5975e-01 1.2612e-01 6.9772e-02 3.8531e-02 2.0526e-02
$ 2^{-18} $ 2.5614e-01 2.1031e-01 1.3406e-01 8.5618e-02 4.8834e-02 2.6219e-02
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$ \varepsilon $ Number of intervals $ N $/time size $ \Delta t $
32/$ \frac{1}{20} $ 64/$ \frac{1}{40} $ 128/$ \frac{1}{80} $ 256/$ \frac{1}{160} $ 512/$ \frac{1}{320} $ 1024/$ \frac{1}{640} $
$ 2^{0} $ 2.4400e-02 1.4033e-02 7.5889e-03 3.9547e-03 2.0227e-03 1.0233e-03
$ 2^{-2} $ 1.8830e-02 1.1126e-02 6.5407e-03 3.6221e-03 1.9209e-03 1.0074e-03
$ 2^{-4} $ 2.0658e-02 1.2614e-02 7.0567e-03 3.7952e-03 1.9563e-03 1.0231e-03
$ 2^{-6} $ 2.5289e-02 1.7672e-02 9.0066e-03 4.8378e-03 2.5035e-03 1.2731e-03
$ 2^{-8} $ 3.8607e-02 1.9497e-02 1.1221e-02 6.2852e-03 3.3405e-03 1.7233e-03
$ 2^{-10} $ 9.3183e-02 7.0120e-02 4.4773e-02 2.0546e-02 1.0545e-02 5.1608e-03
$ 2^{-12} $ 1.7017e-01 1.0083e-01 6.2216e-02 3.9526e-02 2.0493e-02 1.0027e-02
$ 2^{-14} $ 2.0410e-01 1.6703e-01 9.2110e-02 5.2580e-02 2.8766e-02 1.6523e-02
$ 2^{-16} $ 2.0450e-01 1.5975e-01 1.2612e-01 6.9772e-02 3.8531e-02 2.0526e-02
$ 2^{-18} $ 2.5614e-01 2.1031e-01 1.3406e-01 8.5618e-02 4.8834e-02 2.6219e-02
Table 2.  Rate of convergence of solution $ r_\varepsilon^{N,M} $ for Example 5.1 using the adaptive grid method
$ \varepsilon $ Number of intervals $ N $/time size $ \Delta t $
32/$ \frac{1}{20} $ 64/$ \frac{1}{40} $ 128/$ \frac{1}{80} $ 256/$ \frac{1}{160} $ 512/$ \frac{1}{320} $
$ 2^{0} $ 0.7981 0.8869 0.9403 0.9673 0.9831
$ 2^{-2} $ 0.7591 0.7664 0.8526 0.9150 0.9311
$ 2^{-4} $ 0.7117 0.8380 0.8948 0.9560 0.9352
$ 2^{-6} $ 0.5170 0.9724 0.8966 0.9504 0.9756
$ 2^{-8} $ 0.9856 0.7971 0.8362 0.9119 0.9549
$ 2^{-10} $ 0.4102 0.6472 1.1238 0.9623 1.0309
$ 2^{-12} $ 0.7550 0.6966 0.6544 0.9477 1.0312
$ 2^{-14} $ 0.2892 0.8587 0.8088 0.8701 0.7999
$ 2^{-16} $ 0.3563 0.3410 0.8540 0.8566 0.9086
$ 2^{-18} $ 0.2844 0.6496 0.6469 0.8100 0.8973
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$ \varepsilon $ Number of intervals $ N $/time size $ \Delta t $
32/$ \frac{1}{20} $ 64/$ \frac{1}{40} $ 128/$ \frac{1}{80} $ 256/$ \frac{1}{160} $ 512/$ \frac{1}{320} $
$ 2^{0} $ 0.7981 0.8869 0.9403 0.9673 0.9831
$ 2^{-2} $ 0.7591 0.7664 0.8526 0.9150 0.9311
$ 2^{-4} $ 0.7117 0.8380 0.8948 0.9560 0.9352
$ 2^{-6} $ 0.5170 0.9724 0.8966 0.9504 0.9756
$ 2^{-8} $ 0.9856 0.7971 0.8362 0.9119 0.9549
$ 2^{-10} $ 0.4102 0.6472 1.1238 0.9623 1.0309
$ 2^{-12} $ 0.7550 0.6966 0.6544 0.9477 1.0312
$ 2^{-14} $ 0.2892 0.8587 0.8088 0.8701 0.7999
$ 2^{-16} $ 0.3563 0.3410 0.8540 0.8566 0.9086
$ 2^{-18} $ 0.2844 0.6496 0.6469 0.8100 0.8973
Table 3.  Maximum error of solution $ E_\varepsilon^{N,M} $ for Example 5.1 calculated on Shishkin grid
$ \varepsilon $ Number of intervals $ N $/time size $ \Delta t $
32/$ \frac{1}{20} $ 64/$ \frac{1}{40} $ 128/$ \frac{1}{80} $ 256/$ \frac{1}{160} $ 512/$ \frac{1}{320} $ 1024/$ \frac{1}{640} $
$ 2^{0} $ 2.5632e-02 1.6476e-02 9.9288e-03 5.6712e-03 3.1149e-03 1.7000e-03
$ 2^{-2} $ 2.9611e-02 1.8504e-02 1.0796e-02 6.0104e-03 3.2417e-03 1.7105e-03
$ 2^{-4} $ 3.1540e-02 2.0968e-02 1.1706e-02 6.3027e-03 3.3320e-03 1.7024e-03
$ 2^{-6} $ 3.4637e-02 1.7587e-02 7.9159e-03 3.8469e-03 2.2367e-03 2.5185e-03
$ 2^{-8} $ 8.3069e-02 6.3873e-02 3.7579e-02 1.9554e-02 9.7037e-03 3.0087e-03
$ 2^{-10} $ 8.3319e-02 1.1804e-01 8.7025e-02 5.1153e-02 2.7435e-02 1.1428e-02
$ 2^{-12} $ 5.9149e-02 1.2245e-01 1.4419e-01 9.5806e-02 5.5767e-02 2.7152e-02
$ 2^{-14} $ 3.8012e-02 1.2070e-01 1.9830e-01 1.4945e-01 9.1763e-02 5.4821e-02
$ 2^{-16} $ 3.7835e-02 8.8086e-02 1.9473e-01 2.0775e-01 1.3885e-01 9.6989e-02
$ 2^{-18} $ 3.7788e-02 3.6581e-02 1.7951e-01 2.5413e-01 1.9134e-01 1.4338e-01
Table Options

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$ \varepsilon $ Number of intervals $ N $/time size $ \Delta t $
32/$ \frac{1}{20} $ 64/$ \frac{1}{40} $ 128/$ \frac{1}{80} $ 256/$ \frac{1}{160} $ 512/$ \frac{1}{320} $ 1024/$ \frac{1}{640} $
$ 2^{0} $ 2.5632e-02 1.6476e-02 9.9288e-03 5.6712e-03 3.1149e-03 1.7000e-03
$ 2^{-2} $ 2.9611e-02 1.8504e-02 1.0796e-02 6.0104e-03 3.2417e-03 1.7105e-03
$ 2^{-4} $ 3.1540e-02 2.0968e-02 1.1706e-02 6.3027e-03 3.3320e-03 1.7024e-03
$ 2^{-6} $ 3.4637e-02 1.7587e-02 7.9159e-03 3.8469e-03 2.2367e-03 2.5185e-03
$ 2^{-8} $ 8.3069e-02 6.3873e-02 3.7579e-02 1.9554e-02 9.7037e-03 3.0087e-03
$ 2^{-10} $ 8.3319e-02 1.1804e-01 8.7025e-02 5.1153e-02 2.7435e-02 1.1428e-02
$ 2^{-12} $ 5.9149e-02 1.2245e-01 1.4419e-01 9.5806e-02 5.5767e-02 2.7152e-02
$ 2^{-14} $ 3.8012e-02 1.2070e-01 1.9830e-01 1.4945e-01 9.1763e-02 5.4821e-02
$ 2^{-16} $ 3.7835e-02 8.8086e-02 1.9473e-01 2.0775e-01 1.3885e-01 9.6989e-02
$ 2^{-18} $ 3.7788e-02 3.6581e-02 1.7951e-01 2.5413e-01 1.9134e-01 1.4338e-01
Table 4.  Rate of convergence of solution $ r_\varepsilon^{N,M} $ for Example 5.1 calculated on Shishkin grid
$ \varepsilon $ Number of intervals $ N $/time size $ \Delta t $
32/$ \frac{1}{20} $ 64/$ \frac{1}{40} $ 128/$ \frac{1}{80} $ 256/$ \frac{1}{160} $ 512/$ \frac{1}{320} $
$ 2^{0} $ 0.6375 0.7307 0.8079 0.8644 0.8736
$ 2^{-2} $ 0.6783 0.7773 0.8450 0.8907 0.9223
$ 2^{-4} $ 0.5890 0.8409 0.8932 0.9196 0.9688
$ 2^{-6} $ 0.9778 1.1517 1.0411 0.7823 -0.1712
$ 2^{-8} $ 0.3791 0.7653 0.9425 1.0108 1.6893
$ 2^{-10} $ -0.5026 0.4398 0.7666 0.8988 1.2634
$ 2^{-12} $ -1.0498 -0.2357 0.5897 0.7807 1.0384
$ 2^{-14} $ -1.6666 -0.7162 0.4080 0.7037 0.7432
$ 2^{-16} $ -1.2192 -1.1445 -0.0933 0.5814 0.5176
$ 2^{-18} $ 0.0468 -2.2948 -0.5015 0.4093 0.4163
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$ \varepsilon $ Number of intervals $ N $/time size $ \Delta t $
32/$ \frac{1}{20} $ 64/$ \frac{1}{40} $ 128/$ \frac{1}{80} $ 256/$ \frac{1}{160} $ 512/$ \frac{1}{320} $
$ 2^{0} $ 0.6375 0.7307 0.8079 0.8644 0.8736
$ 2^{-2} $ 0.6783 0.7773 0.8450 0.8907 0.9223
$ 2^{-4} $ 0.5890 0.8409 0.8932 0.9196 0.9688
$ 2^{-6} $ 0.9778 1.1517 1.0411 0.7823 -0.1712
$ 2^{-8} $ 0.3791 0.7653 0.9425 1.0108 1.6893
$ 2^{-10} $ -0.5026 0.4398 0.7666 0.8988 1.2634
$ 2^{-12} $ -1.0498 -0.2357 0.5897 0.7807 1.0384
$ 2^{-14} $ -1.6666 -0.7162 0.4080 0.7037 0.7432
$ 2^{-16} $ -1.2192 -1.1445 -0.0933 0.5814 0.5176
$ 2^{-18} $ 0.0468 -2.2948 -0.5015 0.4093 0.4163
Table 5.  Maximum error of solution $ E_\varepsilon^{N,M} $ for Example 5.2 using the adaptive grid method
$ \varepsilon $ Number of intervals $ N $/time size $ \Delta t $
32/$ \frac{1}{20} $ 64/$ \frac{1}{40} $ 128/$ \frac{1}{80} $ 256/$ \frac{1}{160} $ 512/$ \frac{1}{320} $ 1024/$ \frac{1}{640} $
$ 2^{0} $ 2.4502e-02 1.4057e-02 7.5969e-03 3.9589e-03 2.0249e-03 1.0244e-03
$ 2^{-2} $ 1.9156e-02 1.1404e-02 6.5847e-03 3.6356e-03 1.9226e-03 1.0085e-03
$ 2^{-4} $ 2.2788e-02 1.3365e-02 7.2769e-03 3.8441e-03 1.9728e-03 1.0270e-03
$ 2^{-6} $ 3.8767e-02 1.8983e-02 9.6122e-03 5.0867e-03 2.6211e-03 1.3306e-03
$ 2^{-8} $ 4.4450e-02 2.0109e-02 1.0519e-02 5.9653e-02 3.1896e-03 1.6508e-03
$ 2^{-10} $ 8.3339e-02 6.6120e-02 4.0769e-02 1.9284e-02 9.1775e-03 4.9998e-03
$ 2^{-12} $ 1.8762e-01 8.4106e-02 5.7234e-02 3.8309e-02 1.9041e-02 9.2260e-03
$ 2^{-14} $ 1.9755e-01 1.5347e-01 8.3864e-02 4.6945e-02 2.5508e-02 1.4930e-02
$ 2^{-16} $ 2.1340e-01 1.5016e-01 1.1582e-01 6.1958e-02 3.3441e-02 1.7672e-02
$ 2^{-18} $ 2.8299e-01 1.7036e-01 1.1805e-01 7.4251e-02 4.1879e-02 2.2413e-02
Table Options

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$ \varepsilon $ Number of intervals $ N $/time size $ \Delta t $
32/$ \frac{1}{20} $ 64/$ \frac{1}{40} $ 128/$ \frac{1}{80} $ 256/$ \frac{1}{160} $ 512/$ \frac{1}{320} $ 1024/$ \frac{1}{640} $
$ 2^{0} $ 2.4502e-02 1.4057e-02 7.5969e-03 3.9589e-03 2.0249e-03 1.0244e-03
$ 2^{-2} $ 1.9156e-02 1.1404e-02 6.5847e-03 3.6356e-03 1.9226e-03 1.0085e-03
$ 2^{-4} $ 2.2788e-02 1.3365e-02 7.2769e-03 3.8441e-03 1.9728e-03 1.0270e-03
$ 2^{-6} $ 3.8767e-02 1.8983e-02 9.6122e-03 5.0867e-03 2.6211e-03 1.3306e-03
$ 2^{-8} $ 4.4450e-02 2.0109e-02 1.0519e-02 5.9653e-02 3.1896e-03 1.6508e-03
$ 2^{-10} $ 8.3339e-02 6.6120e-02 4.0769e-02 1.9284e-02 9.1775e-03 4.9998e-03
$ 2^{-12} $ 1.8762e-01 8.4106e-02 5.7234e-02 3.8309e-02 1.9041e-02 9.2260e-03
$ 2^{-14} $ 1.9755e-01 1.5347e-01 8.3864e-02 4.6945e-02 2.5508e-02 1.4930e-02
$ 2^{-16} $ 2.1340e-01 1.5016e-01 1.1582e-01 6.1958e-02 3.3441e-02 1.7672e-02
$ 2^{-18} $ 2.8299e-01 1.7036e-01 1.1805e-01 7.4251e-02 4.1879e-02 2.2413e-02
Table 6.  Rate of convergence of solution $ r_\varepsilon^{N,M} $ for Example 5.2 using the adaptive grid method
$ \varepsilon $ Number of intervals $ N $/time size $ \Delta t $
32/$ \frac{1}{20} $ 64/$ \frac{1}{40} $ 128/$ \frac{1}{80} $ 256/$ \frac{1}{160} $ 512/$ \frac{1}{320} $
$ 2^{0} $ 0.8016 0.8878 0.9403 0.9672 1.1059
$ 2^{-2} $ 0.7843 0.7924 0.8864 0.9191 0.9308
$ 2^{-4} $ 0.7698 0.8771 0.9207 0.9624 0.9418
$ 2^{-6} $ 1.0301 0.9818 0.9181 0.9566 0.9781
$ 2^{-8} $ 1.1443 0.9348 0.8183 0.9032 0.9502
$ 2^{-10} $ 0.3339 0.6976 1.0801 1.0712 0.8762
$ 2^{-12} $ 1.1575 0.5553 0.5792 1.0086 1.0453
$ 2^{-14} $ 0.3643 0.8718 0.8371 0.8800 0.7727
$ 2^{-16} $ 0.5076 0.3746 0.9025 0.8897 0.9202
$ 2^{-18} $ 0.7322 0.5292 0.6689 0.8262 0.9019
Table Options

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$ \varepsilon $ Number of intervals $ N $/time size $ \Delta t $
32/$ \frac{1}{20} $ 64/$ \frac{1}{40} $ 128/$ \frac{1}{80} $ 256/$ \frac{1}{160} $ 512/$ \frac{1}{320} $
$ 2^{0} $ 0.8016 0.8878 0.9403 0.9672 1.1059
$ 2^{-2} $ 0.7843 0.7924 0.8864 0.9191 0.9308
$ 2^{-4} $ 0.7698 0.8771 0.9207 0.9624 0.9418
$ 2^{-6} $ 1.0301 0.9818 0.9181 0.9566 0.9781
$ 2^{-8} $ 1.1443 0.9348 0.8183 0.9032 0.9502
$ 2^{-10} $ 0.3339 0.6976 1.0801 1.0712 0.8762
$ 2^{-12} $ 1.1575 0.5553 0.5792 1.0086 1.0453
$ 2^{-14} $ 0.3643 0.8718 0.8371 0.8800 0.7727
$ 2^{-16} $ 0.5076 0.3746 0.9025 0.8897 0.9202
$ 2^{-18} $ 0.7322 0.5292 0.6689 0.8262 0.9019
Table 7.  Maximum error of solution $ E_\varepsilon^{N,M} $ for Example 5.2 calculated on Shishkin grid
$ \varepsilon $ Number of intervals $ N $/time size $ \Delta t $
32/$ \frac{1}{20} $ 64/$ \frac{1}{40} $ 128/$ \frac{1}{80} $ 256/$ \frac{1}{160} $ 512/$ \frac{1}{320} $ 1024/$ \frac{1}{640} $
$ 2^{0} $ 2.4502e-02 1.4057e-02 7.5969e-03 3.9589e-03 2.0249e-03 1.6649e-03
$ 2^{-2} $ 3.0271e-02 1.8725e-02 1.0868e-02 6.0332e-03 3.2486e-03 1.7125e-03
$ 2^{-4} $ 3.7645e-02 2.2072e-02 1.2090e-02 6.4115e-03 3.3584e-03 1.7453e-03
$ 2^{-6} $ 3.6325e-02 2.6153e-02 1.6509e-02 9.7972e-03 5.6201e-03 2.9364e-03
$ 2^{-8} $ 9.9532e-02 5.4894e-02 2.7274e-02 1.2963e-02 6.0337e-03 3.6346e-03
$ 2^{-10} $ 2.1833e-01 1.5551e-01 8.7820e-02 4.5033e-02 2.2291e-02 1.3620e-02
$ 2^{-12} $ 2.2602e-01 2.6665e-01 1.8203e-01 1.0258e-01 5.2694e-02 2.9881e-02
$ 2^{-14} $ 1.2823e-01 3.2914e-01 2.7178e-01 1.8208e-01 1.0356e-01 5.2147e-02
$ 2^{-16} $ 4.4024e-02 3.2475e-01 3.2607e-01 2.5174e-01 1.6644e-01 8.0344e-02
$ 2^{-18} $ 3.3007e-02 2.5627e-01 3.5442e-01 2.9823e-01 2.1963e-01 1.2277e-01
Table Options

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$ \varepsilon $ Number of intervals $ N $/time size $ \Delta t $
32/$ \frac{1}{20} $ 64/$ \frac{1}{40} $ 128/$ \frac{1}{80} $ 256/$ \frac{1}{160} $ 512/$ \frac{1}{320} $ 1024/$ \frac{1}{640} $
$ 2^{0} $ 2.4502e-02 1.4057e-02 7.5969e-03 3.9589e-03 2.0249e-03 1.6649e-03
$ 2^{-2} $ 3.0271e-02 1.8725e-02 1.0868e-02 6.0332e-03 3.2486e-03 1.7125e-03
$ 2^{-4} $ 3.7645e-02 2.2072e-02 1.2090e-02 6.4115e-03 3.3584e-03 1.7453e-03
$ 2^{-6} $ 3.6325e-02 2.6153e-02 1.6509e-02 9.7972e-03 5.6201e-03 2.9364e-03
$ 2^{-8} $ 9.9532e-02 5.4894e-02 2.7274e-02 1.2963e-02 6.0337e-03 3.6346e-03
$ 2^{-10} $ 2.1833e-01 1.5551e-01 8.7820e-02 4.5033e-02 2.2291e-02 1.3620e-02
$ 2^{-12} $ 2.2602e-01 2.6665e-01 1.8203e-01 1.0258e-01 5.2694e-02 2.9881e-02
$ 2^{-14} $ 1.2823e-01 3.2914e-01 2.7178e-01 1.8208e-01 1.0356e-01 5.2147e-02
$ 2^{-16} $ 4.4024e-02 3.2475e-01 3.2607e-01 2.5174e-01 1.6644e-01 8.0344e-02
$ 2^{-18} $ 3.3007e-02 2.5627e-01 3.5442e-01 2.9823e-01 2.1963e-01 1.2277e-01
Table 8.  Rate of convergence of solution $ r_\varepsilon^{N,M} $ for Example 5.2 calculated on Shishkin grid
$ \varepsilon $ Number of intervals $ N $/time size $ \Delta t $
32/$ \frac{1}{20} $ 64/$ \frac{1}{40} $ 128/$ \frac{1}{80} $ 256/$ \frac{1}{160} $ 512/$ \frac{1}{320} $
$ 2^{-0} $ 0.8016 0.8878 0.9403 0.9672 0.2824
$ 2^{-2} $ 0.6930 0.7849 0.8491 0.8931 0.9237
$ 2^{-4} $ 0.7702 0.8684 0.9151 0.9329 0.9443
$ 2^{-6} $ 0.4740 0.6637 0.7528 0.8018 0.9365
$ 2^{-8} $ 0.8585 1.0091 1.0731 1.1033 0.7312
$ 2^{-10} $ 0.4895 0.8244 0.9636 1.0145 0.7107
$ 2^{-12} $ -0.2385 0.5508 0.8274 0.9610 0.8184
$ 2^{-14} $ -1.3600 0.2763 0.5779 0.8141 0.9898
$ 2^{-16} $ -2.8830 -0.0059 0.3732 0.5969 1.0507
$ 2^{-18} $ -2.9568 -0.4678 0.2490 0.4388 0.8391
Table Options

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$ \varepsilon $ Number of intervals $ N $/time size $ \Delta t $
32/$ \frac{1}{20} $ 64/$ \frac{1}{40} $ 128/$ \frac{1}{80} $ 256/$ \frac{1}{160} $ 512/$ \frac{1}{320} $
$ 2^{-0} $ 0.8016 0.8878 0.9403 0.9672 0.2824
$ 2^{-2} $ 0.6930 0.7849 0.8491 0.8931 0.9237
$ 2^{-4} $ 0.7702 0.8684 0.9151 0.9329 0.9443
$ 2^{-6} $ 0.4740 0.6637 0.7528 0.8018 0.9365
$ 2^{-8} $ 0.8585 1.0091 1.0731 1.1033 0.7312
$ 2^{-10} $ 0.4895 0.8244 0.9636 1.0145 0.7107
$ 2^{-12} $ -0.2385 0.5508 0.8274 0.9610 0.8184
$ 2^{-14} $ -1.3600 0.2763 0.5779 0.8141 0.9898
$ 2^{-16} $ -2.8830 -0.0059 0.3732 0.5969 1.0507
$ 2^{-18} $ -2.9568 -0.4678 0.2490 0.4388 0.8391
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