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The global attractor for the wave equation with nonlocal strong damping
Numerical analysis of two new finite difference methods for time-fractional telegraph equation
School of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China |
Fractional telegraph equations are an important class of evolution equations and have widely applications in signal analysis such as transmission and propagation of electrical signals. Aiming at the one-dimensional time-fractional telegraph equation, a class of explicit-implicit (E-I) difference methods and implicit-explicit (I-E) difference methods are proposed. The two methods are based on a combination of the classical implicit difference method and the classical explicit difference method. Under the premise of smooth solution, theoretical analysis and numerical experiments show that the E-I and I-E difference schemes are unconditionally stable, with 2nd order spatial accuracy, $ 2-\alpha $ order time accuracy, and have significant time-saving, their calculation efficiency is higher than the classical implicit scheme. The research shows that the E-I and I-E difference methods constructed in this paper are effective for solving the time-fractional telegraph equation.
References:
[1] |
T. Akram, M. Abbas, A. I. Ismail, N. Hj. M. Ali and D. Baleanu, Extended cubic B-splines in the numerical solution of time fractional telegraph equation, Adv. Differ. Equ., (2019), Paper No. 365, 20 pp.
doi: 10.1186/s13662-019-2296-9. |
[2] |
D. J. Arrigo and S. G. Krantz, Analytical Techniques for Solving Nonlinear Partial Differential Equations, Morgan & Claypool Publishers, 2019.
doi: 10.2200/S00907ED1V01Y201903MAS025. |
[3] |
A. Atangana,
On the stability and convergence of the time-fractional variable order telegraph equation, J. Comput. Phys., 293 (2015), 104-114.
doi: 10.1016/j.jcp.2014.12.043. |
[4] |
R. C. Cascaval, E. C. Eckstein, C. L. Frota and J. A. Goldstein,
Fractional telegraph equations, J. Math. Anal. Appl., 276 (2002), 145-159.
doi: 10.1016/S0022-247X(02)00394-3. |
[5] |
J. Chen, F. Liu and V. Anh,
Analytical solution for the time-fractional telegraph equation by the method of separating variables, J. Math. Anal. Appl., 338 (2008), 1364-1377.
doi: 10.1016/j.jmaa.2007.06.023. |
[6] |
S. Das, K. Vishal, P. K. Gupta and A. Yildirim,
An approximate analytical solution of time-fractional telegraph equation, Appl. Math. Comput., 217 (2011), 7405-7411.
doi: 10.1016/j.amc.2011.02.030. |
[7] |
W. Deng and Z. Zhang, High Accuracy Algorithm for the Differentail Equations Governing Anomalous Diffusion, Algorithm and Models for Anomalous Diffusion, World Scientific, Singapore, 2019. |
[8] |
K. Diethelm, The Analysis of Fraction Differential Equations, Springer, 2010.
doi: 10.1007/978-3-642-14574-2. |
[9] |
M. Ferreira, M. M. Rodrigues and N. Vieira, First and second fundamental solutions of the time-fractional telegraph equation with Laplace or Dirac oprators, Adv. Appl. Clifford Algebr, 28 (2018), Art. 42, 14 pp.
doi: 10.1007/s00006-018-0858-7. |
[10] |
N. J. Ford, M. M. Rodrigues, J. Xiao and Y. Yan,
Numerical analysis of a two-parameter fractional telegraph equation, J. Comput. Appl. Math., 249 (2013), 95-106.
doi: 10.1016/j.cam.2013.02.009. |
[11] |
B. Guo, X. Pu and F. Huang, Fractional Partial Differential Equations and Their Numerical Solutions, Science Press, Beijing, 2011.
doi: 10.1142/9543.![]() ![]() |
[12] |
L. Hervé and L. Brigitte, Partial Differential Equations: Modeling, Analysis and Numerical Approximation, Springer International Publishing, Switzerland, 2016. Google Scholar |
[13] |
M. H. Heydari, M. R. Hooshmandasl and F. Mohammadi,
Two-Dimensional legendre wavelets for solving time-fractional telegraph equation, Adv. Appl. Math. Mech., 6 (2014), 247-260.
doi: 10.4208/aamm.12-m12132. |
[14] |
V. R. Hosseini, W. Chen and Z. Avazzadeh,
Numerical solution of fractional telegraph equation by using radial basis functions, Eng. Anal. Bound. Elem, 38 (2014), 31-39.
doi: 10.1016/j.enganabound.2013.10.009. |
[15] |
W. Jiang and Y. Lin,
Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 3639-3645.
doi: 10.1016/j.cnsns.2010.12.019. |
[16] |
K. Kumar, R. K. Pandey and S. Yadav, Finite difference scheme for a fractional telegraph equation with generalized fractional derivative terms, Physica A, 535 (2019), Art. 122271, 15 pp.
doi: 10.1016/j.physa.2019.122271. |
[17] |
C. Li and F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Boca Raton, 2015.
![]() |
[18] |
C. Li, Z. Zhao and Y. Chen,
Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl., 62 (2011), 855-875.
doi: 10.1016/j.camwa.2011.02.045. |
[19] |
Y. Lin and C. Xu,
Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.
doi: 10.1016/j.jcp.2007.02.001. |
[20] |
F. Liu, P. Zhuang, V. Anh, I. Turner and K. Burrage,
Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput, 191 (2007), 17-20.
doi: 10.1016/j.amc.2006.08.162. |
[21] | F. Liu, P. Zhuang and Q. Liu, Numerical Solutions of Fractional Order Partial Differential Equations and its Applications, Science Press, Beijing, 2015. Google Scholar |
[22] |
M. O. Mamchuev,
Solutions of the main boundary value problems for the time-fractional telegraph equation by the green function method, Fract. Calc. Appl. Anal., 20 (2017), 190-211.
doi: 10.1515/fca-2017-0010. |
[23] |
S. Momani,
Analytic and approximate solutions of the space- and time-fractional telegraph equations, Appl. Math. Comput., 170 (2005), 1126-1134.
doi: 10.1016/j.amc.2005.01.009. |
[24] |
E. Orsingher and L. Beghin,
Time-fractional telegraph equations and telegraph processes with brownian time, Probab. Theory Relat. Fields, 128 (2004), 141-160.
doi: 10.1007/s00440-003-0309-8. |
[25] |
A. Saadatmandi and M. Mohabbati,
Numerical solution of fractional telegraph equation via the tau method, Math. Rep., 17 (2015), 155-166.
|
[26] |
J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado (Editors), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, World Book Incorporated Beijing, 2014.
doi: 10.1007/978-1-4020-6042-7. |
[27] |
M. Stynes, E. O' Riordan and J. L. Gracia,
Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057-1079.
doi: 10.1137/16M1082329. |
[28] | Z. Sun and G. Gao, Finite Difference Methods for Fractional Differential Equations, Science Press, Beijing, 2015. Google Scholar |
[29] |
V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers: Volume II Applications, Higher Education Press; Springer, Heidelberg, Beijing, 2013.
doi: 10.1007/978-3-642-33911-0. |
[30] |
S. Vong, P. Lyu and Z. Wang,
A compact difference scheme for fractional sub-diffusion equations with the spatially variable coefficient under neumann boundary conditions, J. Sci. Comput., 66 (2016), 725-739.
doi: 10.1007/s10915-015-0040-5. |
[31] |
H. Wang, A. Cheng and K. Wang,
Fast finite volume methods for space-fractional diffusion equations, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 1427-1441.
doi: 10.3934/dcdsb.2015.20.1427. |
[32] |
L. Wei, H. Dai, D. Zhang and Z. Si,
Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation, Calcolo, 51 (2014), 175-192.
doi: 10.1007/s10092-013-0084-6. |
[33] |
P. Xanthoulos and G. E. Zouraris,
A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves, Discrete Contin. Dyn. Syst. Ser. B., 10 (2008), 239-263.
doi: 10.3934/dcdsb.2008.10.239. |
[34] |
X. Yang, L. Wu, S. Sun and X. Zhang, A universal difference method for time-space fractional Black-Scholes equation, Adv. Differ. Equ., (2016), Paper No. 71, 14 pp.
doi: 10.1186/s13662-016-0792-8. |
[35] |
A. Yildirim,
He's homtopy perturbation method for solving the space and time fractional telegraph equations, Int. J. Comput. Math., 87 (2010), 2998-3006.
doi: 10.1080/00207160902874653. |
[36] |
Z. Zhao and C. Li,
Fractional difference/finite element approximations for the time-space fractional telegraph equation, Appl. Math. Comput., 219 (2012), 2975-2988.
doi: 10.1016/j.amc.2012.09.022. |
show all references
References:
[1] |
T. Akram, M. Abbas, A. I. Ismail, N. Hj. M. Ali and D. Baleanu, Extended cubic B-splines in the numerical solution of time fractional telegraph equation, Adv. Differ. Equ., (2019), Paper No. 365, 20 pp.
doi: 10.1186/s13662-019-2296-9. |
[2] |
D. J. Arrigo and S. G. Krantz, Analytical Techniques for Solving Nonlinear Partial Differential Equations, Morgan & Claypool Publishers, 2019.
doi: 10.2200/S00907ED1V01Y201903MAS025. |
[3] |
A. Atangana,
On the stability and convergence of the time-fractional variable order telegraph equation, J. Comput. Phys., 293 (2015), 104-114.
doi: 10.1016/j.jcp.2014.12.043. |
[4] |
R. C. Cascaval, E. C. Eckstein, C. L. Frota and J. A. Goldstein,
Fractional telegraph equations, J. Math. Anal. Appl., 276 (2002), 145-159.
doi: 10.1016/S0022-247X(02)00394-3. |
[5] |
J. Chen, F. Liu and V. Anh,
Analytical solution for the time-fractional telegraph equation by the method of separating variables, J. Math. Anal. Appl., 338 (2008), 1364-1377.
doi: 10.1016/j.jmaa.2007.06.023. |
[6] |
S. Das, K. Vishal, P. K. Gupta and A. Yildirim,
An approximate analytical solution of time-fractional telegraph equation, Appl. Math. Comput., 217 (2011), 7405-7411.
doi: 10.1016/j.amc.2011.02.030. |
[7] |
W. Deng and Z. Zhang, High Accuracy Algorithm for the Differentail Equations Governing Anomalous Diffusion, Algorithm and Models for Anomalous Diffusion, World Scientific, Singapore, 2019. |
[8] |
K. Diethelm, The Analysis of Fraction Differential Equations, Springer, 2010.
doi: 10.1007/978-3-642-14574-2. |
[9] |
M. Ferreira, M. M. Rodrigues and N. Vieira, First and second fundamental solutions of the time-fractional telegraph equation with Laplace or Dirac oprators, Adv. Appl. Clifford Algebr, 28 (2018), Art. 42, 14 pp.
doi: 10.1007/s00006-018-0858-7. |
[10] |
N. J. Ford, M. M. Rodrigues, J. Xiao and Y. Yan,
Numerical analysis of a two-parameter fractional telegraph equation, J. Comput. Appl. Math., 249 (2013), 95-106.
doi: 10.1016/j.cam.2013.02.009. |
[11] |
B. Guo, X. Pu and F. Huang, Fractional Partial Differential Equations and Their Numerical Solutions, Science Press, Beijing, 2011.
doi: 10.1142/9543.![]() ![]() |
[12] |
L. Hervé and L. Brigitte, Partial Differential Equations: Modeling, Analysis and Numerical Approximation, Springer International Publishing, Switzerland, 2016. Google Scholar |
[13] |
M. H. Heydari, M. R. Hooshmandasl and F. Mohammadi,
Two-Dimensional legendre wavelets for solving time-fractional telegraph equation, Adv. Appl. Math. Mech., 6 (2014), 247-260.
doi: 10.4208/aamm.12-m12132. |
[14] |
V. R. Hosseini, W. Chen and Z. Avazzadeh,
Numerical solution of fractional telegraph equation by using radial basis functions, Eng. Anal. Bound. Elem, 38 (2014), 31-39.
doi: 10.1016/j.enganabound.2013.10.009. |
[15] |
W. Jiang and Y. Lin,
Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 3639-3645.
doi: 10.1016/j.cnsns.2010.12.019. |
[16] |
K. Kumar, R. K. Pandey and S. Yadav, Finite difference scheme for a fractional telegraph equation with generalized fractional derivative terms, Physica A, 535 (2019), Art. 122271, 15 pp.
doi: 10.1016/j.physa.2019.122271. |
[17] |
C. Li and F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Boca Raton, 2015.
![]() |
[18] |
C. Li, Z. Zhao and Y. Chen,
Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl., 62 (2011), 855-875.
doi: 10.1016/j.camwa.2011.02.045. |
[19] |
Y. Lin and C. Xu,
Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.
doi: 10.1016/j.jcp.2007.02.001. |
[20] |
F. Liu, P. Zhuang, V. Anh, I. Turner and K. Burrage,
Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput, 191 (2007), 17-20.
doi: 10.1016/j.amc.2006.08.162. |
[21] | F. Liu, P. Zhuang and Q. Liu, Numerical Solutions of Fractional Order Partial Differential Equations and its Applications, Science Press, Beijing, 2015. Google Scholar |
[22] |
M. O. Mamchuev,
Solutions of the main boundary value problems for the time-fractional telegraph equation by the green function method, Fract. Calc. Appl. Anal., 20 (2017), 190-211.
doi: 10.1515/fca-2017-0010. |
[23] |
S. Momani,
Analytic and approximate solutions of the space- and time-fractional telegraph equations, Appl. Math. Comput., 170 (2005), 1126-1134.
doi: 10.1016/j.amc.2005.01.009. |
[24] |
E. Orsingher and L. Beghin,
Time-fractional telegraph equations and telegraph processes with brownian time, Probab. Theory Relat. Fields, 128 (2004), 141-160.
doi: 10.1007/s00440-003-0309-8. |
[25] |
A. Saadatmandi and M. Mohabbati,
Numerical solution of fractional telegraph equation via the tau method, Math. Rep., 17 (2015), 155-166.
|
[26] |
J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado (Editors), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, World Book Incorporated Beijing, 2014.
doi: 10.1007/978-1-4020-6042-7. |
[27] |
M. Stynes, E. O' Riordan and J. L. Gracia,
Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057-1079.
doi: 10.1137/16M1082329. |
[28] | Z. Sun and G. Gao, Finite Difference Methods for Fractional Differential Equations, Science Press, Beijing, 2015. Google Scholar |
[29] |
V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers: Volume II Applications, Higher Education Press; Springer, Heidelberg, Beijing, 2013.
doi: 10.1007/978-3-642-33911-0. |
[30] |
S. Vong, P. Lyu and Z. Wang,
A compact difference scheme for fractional sub-diffusion equations with the spatially variable coefficient under neumann boundary conditions, J. Sci. Comput., 66 (2016), 725-739.
doi: 10.1007/s10915-015-0040-5. |
[31] |
H. Wang, A. Cheng and K. Wang,
Fast finite volume methods for space-fractional diffusion equations, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 1427-1441.
doi: 10.3934/dcdsb.2015.20.1427. |
[32] |
L. Wei, H. Dai, D. Zhang and Z. Si,
Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation, Calcolo, 51 (2014), 175-192.
doi: 10.1007/s10092-013-0084-6. |
[33] |
P. Xanthoulos and G. E. Zouraris,
A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves, Discrete Contin. Dyn. Syst. Ser. B., 10 (2008), 239-263.
doi: 10.3934/dcdsb.2008.10.239. |
[34] |
X. Yang, L. Wu, S. Sun and X. Zhang, A universal difference method for time-space fractional Black-Scholes equation, Adv. Differ. Equ., (2016), Paper No. 71, 14 pp.
doi: 10.1186/s13662-016-0792-8. |
[35] |
A. Yildirim,
He's homtopy perturbation method for solving the space and time fractional telegraph equations, Int. J. Comput. Math., 87 (2010), 2998-3006.
doi: 10.1080/00207160902874653. |
[36] |
Z. Zhao and C. Li,
Fractional difference/finite element approximations for the time-space fractional telegraph equation, Appl. Math. Comput., 219 (2012), 2975-2988.
doi: 10.1016/j.amc.2012.09.022. |













![]() |
21 | 41 | 61 | 81 |
Exact solution | 0.032816 | 0.038767 | 0.045800 | 0.054104 |
Implicit scheme solution | 0.032727 | 0.038662 | 0.045673 | 0.053957 |
E-I scheme solution | 0.032238 | 0.037960 | 0.044992 | 0.053152 |
I-E scheme solution | 0.032168 | 0.038002 | 0.044895 | 0.053037 |
![]() |
21 | 41 | 61 | 81 |
Exact solution | 0.032816 | 0.038767 | 0.045800 | 0.054104 |
Implicit scheme solution | 0.032727 | 0.038662 | 0.045673 | 0.053957 |
E-I scheme solution | 0.032238 | 0.037960 | 0.044992 | 0.053152 |
I-E scheme solution | 0.032168 | 0.038002 | 0.044895 | 0.053037 |
M | Implicit scheme | E-I scheme | I-E scheme | ||||
Order1 | Order1 | Order1 | |||||
0.6 | 6 | 0.601261438 | — | 0.670908875 | — | 0.676867797 | — |
12 | 0.171331703 | 1.811200301 | 0.185496548 | 1.854724481 | 0.185742355 | 1.865571243 | |
24 | 0.027626239 | 1.861354661 | 0.020497637 | 1.880049324 | 0.056380674 | 1.888092124 | |
0.7 | 6 | 0.572754131 | — | 0.625245364 | — | 0.632403830 | — |
12 | 0.154051855 | 1.894499914 | 0.167207909 | 1.902779364 | 0.167706812 | 1.914904807 | |
24 | 0.043125187 | 1.968989813 | 0.042627245 | 1.954591537 | 0.042269721 | 1.953396613 | |
0.8 | 6 | 0.331546455 | — | 0.384936527 | — | 0.392207783 | — |
12 | 0.093281787 | 1.829543269 | 0.000168779 | 1.942187657 | 0.000592926 | 1.963089311 | |
24 | 0.025790914 | 1.887189262 | 0.042627245 | 1.954591537 | 0.042269721 | 1.953396613 | |
0.9 | 6 | 0.312856198 | — | 0.351172503 | — | 0.354347245 | — |
12 | 0.079460269 | 1.977194091 | 0.084117921 | 2.061694794 | 0.084447694 | 2.069033890 | |
24 | 0.020221416 | 1.978009907 | 0.021169310 | 2.057999271 | 0.021427164 | 2.056110578 |
M | Implicit scheme | E-I scheme | I-E scheme | ||||
Order1 | Order1 | Order1 | |||||
0.6 | 6 | 0.601261438 | — | 0.670908875 | — | 0.676867797 | — |
12 | 0.171331703 | 1.811200301 | 0.185496548 | 1.854724481 | 0.185742355 | 1.865571243 | |
24 | 0.027626239 | 1.861354661 | 0.020497637 | 1.880049324 | 0.056380674 | 1.888092124 | |
0.7 | 6 | 0.572754131 | — | 0.625245364 | — | 0.632403830 | — |
12 | 0.154051855 | 1.894499914 | 0.167207909 | 1.902779364 | 0.167706812 | 1.914904807 | |
24 | 0.043125187 | 1.968989813 | 0.042627245 | 1.954591537 | 0.042269721 | 1.953396613 | |
0.8 | 6 | 0.331546455 | — | 0.384936527 | — | 0.392207783 | — |
12 | 0.093281787 | 1.829543269 | 0.000168779 | 1.942187657 | 0.000592926 | 1.963089311 | |
24 | 0.025790914 | 1.887189262 | 0.042627245 | 1.954591537 | 0.042269721 | 1.953396613 | |
0.9 | 6 | 0.312856198 | — | 0.351172503 | — | 0.354347245 | — |
12 | 0.079460269 | 1.977194091 | 0.084117921 | 2.061694794 | 0.084447694 | 2.069033890 | |
24 | 0.020221416 | 1.978009907 | 0.021169310 | 2.057999271 | 0.021427164 | 2.056110578 |
N | Implicit scheme | E-I scheme | I-E scheme | ||||
Order2 | Order2 | Order2 | |||||
0.6 | 300 | 0.001748225 | — | 0.001257729 | — | 0.001257729 | — |
600 | 0.000647496 | 1.432945906 | 0.000769857 | 1.408158851 | 0.000795404 | 1.415571243 | |
1200 | 0.000238734 | 1.437531875 | 0.000300146 | 1.406435732 | 0.000300015 | 1.406473635 | |
0.7 | 300 | 0.003765612 | — | 0.002220764 | — | 0.002081502 | — |
600 | 0.001462733 | 1.364217584 | 0.000902026 | 1.299814239 | 0.000902020 | 1.206383164 | |
1200 | 0.000662609 | 1.342435234 | 0.000608222 | 1.325685702 | 0.000608231 | 1.322349016 | |
0.8 | 300 | 0.002921061 | — | 0.001592343 | — | 0.001453911 | — |
600 | 0.001224089 | 1.254076687 | 0.000647496 | 1.298205933 | 0.000647496 | 1.236914092 | |
1200 | 0.000532251 | 1.205163582 | 0.000267776 | 1.296425166 | 0.000264108 | 1.293815168 | |
0.9 | 300 | 0.003178454 | — | 0.002285670 | — | 0.002149714 | — |
600 | 0.001481610 | 1.101159256 | 0.000973810 | 1.230904340 | 0.000910832 | 1.238886806 | |
1200 | 0.000791962 | 1.052349016 | 0.000348563 | 1.116311302 | 0.000451147 | 1.116552467 |
N | Implicit scheme | E-I scheme | I-E scheme | ||||
Order2 | Order2 | Order2 | |||||
0.6 | 300 | 0.001748225 | — | 0.001257729 | — | 0.001257729 | — |
600 | 0.000647496 | 1.432945906 | 0.000769857 | 1.408158851 | 0.000795404 | 1.415571243 | |
1200 | 0.000238734 | 1.437531875 | 0.000300146 | 1.406435732 | 0.000300015 | 1.406473635 | |
0.7 | 300 | 0.003765612 | — | 0.002220764 | — | 0.002081502 | — |
600 | 0.001462733 | 1.364217584 | 0.000902026 | 1.299814239 | 0.000902020 | 1.206383164 | |
1200 | 0.000662609 | 1.342435234 | 0.000608222 | 1.325685702 | 0.000608231 | 1.322349016 | |
0.8 | 300 | 0.002921061 | — | 0.001592343 | — | 0.001453911 | — |
600 | 0.001224089 | 1.254076687 | 0.000647496 | 1.298205933 | 0.000647496 | 1.236914092 | |
1200 | 0.000532251 | 1.205163582 | 0.000267776 | 1.296425166 | 0.000264108 | 1.293815168 | |
0.9 | 300 | 0.003178454 | — | 0.002285670 | — | 0.002149714 | — |
600 | 0.001481610 | 1.101159256 | 0.000973810 | 1.230904340 | 0.000910832 | 1.238886806 | |
1200 | 0.000791962 | 1.052349016 | 0.000348563 | 1.116311302 | 0.000451147 | 1.116552467 |
![]() |
21 | 41 | 61 | 81 |
Exact solution | -0.03079 | -0.04618 | -0.04618 | -0.03079 |
Implicit scheme solution | -0.03008 | -0.04505 | -0.04505 | -0.03008 |
E-I scheme solution | -0.03001 | -0.04492 | -0.04492 | -0.03001 |
I-E scheme solution | -0.03001 | -0.04492 | -0.04492 | -0.03001 |
![]() |
21 | 41 | 61 | 81 |
Exact solution | -0.03079 | -0.04618 | -0.04618 | -0.03079 |
Implicit scheme solution | -0.03008 | -0.04505 | -0.04505 | -0.03008 |
E-I scheme solution | -0.03001 | -0.04492 | -0.04492 | -0.03001 |
I-E scheme solution | -0.03001 | -0.04492 | -0.04492 | -0.03001 |
M | Implicit scheme | E-I scheme | |||
Order1 | Order1 | ||||
8 | 0.007395715 | — | 0.006928145 | — | |
0.2 | 16 | 0.001926582 | 1.940537064 | 0.001786161 | 1.955606884 |
32 | 0.000486619 | 1.985178414 | 0.000447580 | 1.996642455 | |
64 | 0.000121970 | 1.996305831 | 0.000110261 | 2.004338515 | |
8 | 0.007399299 | — | 0.006730926 | — | |
0.3 | 16 | 0.001926975 | 1.944911724 | 0.001748227 | 1.944911722 |
32 | 0.000486657 | 1.996642453 | 0.000440517 | 1.988623313 | |
64 | 0.000121970 | 1.996374655 | 0.000110261 | 1.998266445 | |
8 | 0.007406310 | — | 0.006655592 | — | |
0.4 | 16 | 0.001927758 | 1.941830430 | 0.001736738 | 1.938186701 |
32 | 0.000486747 | 1.985359045 | 0.000438647 | 1.988623313 | |
64 | 1.988623313 | 1.996518415 | 0.000109876 | 1.996547989 | |
8 | 0.007418404 | — | 0.006638599 | — | |
0.5 | 16 | 0.001929358 | 1.942987829 | 0.001735124 | 1.935839557 |
32 | 0.000486961 | 1.986241089 | 0.000438424 | 1.984638177 | |
64 | 0.000122001 | 1.996814720 | 0.000109876 | 1.996450948 |
M | Implicit scheme | E-I scheme | |||
Order1 | Order1 | ||||
8 | 0.007395715 | — | 0.006928145 | — | |
0.2 | 16 | 0.001926582 | 1.940537064 | 0.001786161 | 1.955606884 |
32 | 0.000486619 | 1.985178414 | 0.000447580 | 1.996642455 | |
64 | 0.000121970 | 1.996305831 | 0.000110261 | 2.004338515 | |
8 | 0.007399299 | — | 0.006730926 | — | |
0.3 | 16 | 0.001926975 | 1.944911724 | 0.001748227 | 1.944911722 |
32 | 0.000486657 | 1.996642453 | 0.000440517 | 1.988623313 | |
64 | 0.000121970 | 1.996374655 | 0.000110261 | 1.998266445 | |
8 | 0.007406310 | — | 0.006655592 | — | |
0.4 | 16 | 0.001927758 | 1.941830430 | 0.001736738 | 1.938186701 |
32 | 0.000486747 | 1.985359045 | 0.000438647 | 1.988623313 | |
64 | 1.988623313 | 1.996518415 | 0.000109876 | 1.996547989 | |
8 | 0.007418404 | — | 0.006638599 | — | |
0.5 | 16 | 0.001929358 | 1.942987829 | 0.001735124 | 1.935839557 |
32 | 0.000486961 | 1.986241089 | 0.000438424 | 1.984638177 | |
64 | 0.000122001 | 1.996814720 | 0.000109876 | 1.996450948 |
N | Implicit scheme | E-I scheme | |||
Order2 | Order2 | ||||
150 | 0.003247111 | — | 0.003127877 | — | |
0.2 | 300 | 0.0038902781 | 0.260726667 | 0.003593341 | 0.200141963 |
600 | 0.0046878951 | 0.269069826 | 0.000447580 | 0.211580117 | |
150 | 0.003053496 | — | 0.002880601 | — | |
0.3 | 300 | 0.003751789 | 0.297116513 | 0.003603101 | 0.322869023 |
600 | 0.004664256 | 0.314069826 | 0.004535791 | 0.332115801 | |
150 | 0.002847062 | — | 0.002645836 | — | |
0.4 | 300 | 0.003604515 | 0.340331208 | 0.003490906 | 0.399877794 |
600 | 0.004639253 | 0.359191478 | 0.004623847 | 0.405493914 | |
150 | 0.002620013 | — | 0.002391991 | — | |
0.5 | 300 | 0.003442873 | 0.394038915 | 0.003330367 | 0.477464645 |
600 | 0.004561314 | 0.405837085 | 0.004690116 | 0.493943176 |
N | Implicit scheme | E-I scheme | |||
Order2 | Order2 | ||||
150 | 0.003247111 | — | 0.003127877 | — | |
0.2 | 300 | 0.0038902781 | 0.260726667 | 0.003593341 | 0.200141963 |
600 | 0.0046878951 | 0.269069826 | 0.000447580 | 0.211580117 | |
150 | 0.003053496 | — | 0.002880601 | — | |
0.3 | 300 | 0.003751789 | 0.297116513 | 0.003603101 | 0.322869023 |
600 | 0.004664256 | 0.314069826 | 0.004535791 | 0.332115801 | |
150 | 0.002847062 | — | 0.002645836 | — | |
0.4 | 300 | 0.003604515 | 0.340331208 | 0.003490906 | 0.399877794 |
600 | 0.004639253 | 0.359191478 | 0.004623847 | 0.405493914 | |
150 | 0.002620013 | — | 0.002391991 | — | |
0.5 | 300 | 0.003442873 | 0.394038915 | 0.003330367 | 0.477464645 |
600 | 0.004561314 | 0.405837085 | 0.004690116 | 0.493943176 |
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