# American Institute of Mathematical Sciences

• Previous Article
Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting
• ERA Home
• This Issue
• Next Article
Solvability of the matrix equation $AX^{2} = B$ with semi-tensor product
doi: 10.3934/era.2020076

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

 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  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.

Citation: Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 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
Numerical solution profile of Example 5.1 with $N = 64$, $M = 40$ and $\varepsilon = 2^{-14}$
Numerical solution of Example 5.1 at different time levels with $N = 64$, $M = 40$ and $\varepsilon = 2^{-14}$
Mesh movement of Example 5.1 for $N = 64$, $M = 40$ and $\varepsilon = 2^{-12}$
Numerical solution of Example 5.2 at different time levels with $N = 64$, $M = 40$ and $\varepsilon = 2^{-8}$
Mesh movement of Example 5.2 for $N = 128$, $M = 80$ and $\varepsilon = 2^{-10}$
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
 $\varepsilon$ 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
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
 $\varepsilon$ 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
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
 $\varepsilon$ 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
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
 $\varepsilon$ 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
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
 $\varepsilon$ 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
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
 $\varepsilon$ 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
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
 $\varepsilon$ 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
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
 $\varepsilon$ 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
 [1] Pierre Fabrie, Cedric Galusinski, A. Miranville, Sergey Zelik. Uniform exponential attractors for a singularly perturbed damped wave equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 211-238. doi: 10.3934/dcds.2004.10.211 [2] Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020402 [3] Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644 [4] Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496 [5] Sondre Tesdal Galtung. A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1243-1268. doi: 10.3934/dcds.2018051 [6] Helge Holden, Xavier Raynaud. A convergent numerical scheme for the Camassa--Holm equation based on multipeakons. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 505-523. doi: 10.3934/dcds.2006.14.505 [7] Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709 [8] Alexandre Mouton. Expansion of a singularly perturbed equation with a two-scale converging convection term. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1447-1473. doi: 10.3934/dcdss.2016058 [9] Ronald E. Mickens. A nonstandard finite difference scheme for the drift-diffusion system. Conference Publications, 2009, 2009 (Special) : 558-563. doi: 10.3934/proc.2009.2009.558 [10] Changling Xu, Tianliang Hou. Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equations. Electronic Research Archive, 2020, 28 (2) : 897-910. doi: 10.3934/era.2020047 [11] Yun Chen, Jiasheng Huang, Si Li, Yao Lu, Yuesheng Xu. A content-adaptive unstructured grid based integral equation method with the TV regularization for SPECT reconstruction. Inverse Problems & Imaging, 2020, 14 (1) : 27-52. doi: 10.3934/ipi.2019062 [12] Andrés Ávila, Louis Jeanjean. A result on singularly perturbed elliptic problems. Communications on Pure & Applied Analysis, 2005, 4 (2) : 341-356. doi: 10.3934/cpaa.2005.4.341 [13] Flaviano Battelli, Ken Palmer. Heteroclinic connections in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 431-461. doi: 10.3934/dcdsb.2008.9.431 [14] Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020025 [15] Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic & Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040 [16] Wei Qu, Siu-Long Lei, Seak-Weng Vong. A note on the stability of a second order finite difference scheme for space fractional diffusion equations. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 317-325. doi: 10.3934/naco.2014.4.317 [17] Hawraa Alsayed, Hussein Fakih, Alain Miranville, Ali Wehbe. Finite difference scheme for 2D parabolic problem modelling electrostatic Micro-Electromechanical Systems. Electronic Research Announcements, 2019, 26: 54-71. doi: 10.3934/era.2019.26.005 [18] Wen Li, Song Wang. Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme. Journal of Industrial & Management Optimization, 2013, 9 (2) : 365-389. doi: 10.3934/jimo.2013.9.365 [19] Navnit Jha. Nonpolynomial spline finite difference scheme for nonlinear singuiar boundary value problems with singular perturbation and its mechanization. Conference Publications, 2013, 2013 (special) : 355-363. doi: 10.3934/proc.2013.2013.355 [20] Michel H. Geoffroy, Alain Piétrus. Regularity properties of a cubically convergent scheme for generalized equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 983-996. doi: 10.3934/cpaa.2007.6.983

Impact Factor: 0.263

## Tools

Article outline

Figures and Tables