American Institute of Mathematical Sciences

Advanced
September  2021, 26(9): 4907-4926. doi: 10.3934/dcdsb.2020319

A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation

Leilei Wei 1,, and  Yinnian He 2,

1. 

College of Science, Henan University of Technology, Zhengzhou 450001, China
2. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

* Corresponding author: Leilei Wei

Received  March 2020 Revised  August 2020 Published  September 2021 Early access  November 2020

Fund Project: Supported by the Fundamental Research Funds for the Henan Provincial Colleges and Universities in Henan University of Technology(2018RCJH10), the Training Plan of Young Backbone Teachers in Henan University of Technology(21420049), the Training Plan of Young Backbone Teachers in Colleges and Universities of Henan Province (2019GGJS094), the Innovative Funds Plan of Henan University of Technology, Foundation of Henan Educational Committee(19A110005) and the National Natural Science Foundation of China (11771348, 11861068)

The tempered fractional diffusion equation could be recognized as the generalization of the classic fractional diffusion equation that the truncation effects are included in the bounded domains. This paper focuses on designing the high order fully discrete local discontinuous Galerkin (LDG) method based on the generalized alternating numerical fluxes for the tempered fractional diffusion equation. From a practical point of view, the generalized alternating numerical flux which is different from the purely alternating numerical flux has a broader range of applications. We first design an efficient finite difference scheme to approximate the tempered fractional derivatives and then a fully discrete LDG method for the tempered fractional diffusion equation. We prove that the scheme is unconditionally stable and convergent with the order $ O(h^{k+1}+\tau^{2-\alpha}) $, where $ h, \tau $ and $ k $ are the step size in space, time and the degree of piecewise polynomials, respectively. Finally numerical experimets are performed to show the effectiveness and testify the accuracy of the method.

Keywords: Tempered fractional diffusion equations, local discontinuous Galerkin method, stability, error estimates.
Mathematics Subject Classification: Primary: 65M12, 65M06; Secondary: 35S10.
Citation: Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4907-4926. doi: 10.3934/dcdsb.2020319
References:
[1]

B. Baeumer and M. M. Meerschaert, Tempered stable Levy motion and transient superdiffusion, J. Comput. Appl. Math., 233 (2010), 2438-2448.  doi: 10.1016/j.cam.2009.10.027.

[2]

A. Cartea and D. del-Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Phys. A, 374 (2007), 749-763.  doi: 10.1016/j.physa.2006.08.071.

[3]

S. Chen, J. Shen and L.-L. Wang, Generalized Jacobi functions and their applications to fractional differential equations, Math. Comp., 85 (2016), 1603-1638.  doi: 10.1090/mcom3035.

[4]

Y. Chen, X. Wang and W. Deng, Tempered fractional Langevin-Brownian motion with inverse $\beta$-stable subordinator, J. Phys. A: Math. Theor., 51 (2018), 495001. doi: 10.1088/1751-8121/aae8b3.

[5]

Y. Cheng, X. Meng and Q. Zhang, Application of generalized Gauss-Radau projections for the local discontinuous Galerkin method for linear convection-diffusion equations, Math. Comp., 86 (2017), 1233-1267.  doi: 10.1090/mcom/3141.

[6]

M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228 (2009), 7792-7804.  doi: 10.1016/j.jcp.2009.07.021.

[7]

V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, umer. Methods Partial Differential Eq., 22 (2006), 558-576.  doi: 10.1002/num.20112.

[8]

L. B. Feng, P. Zhuang, F. Liu, I. Turner and Y. T. Gu, Finite element method for space-time fractional diffusion equation, Numer. Algor., 72 (2016), 749-767.  doi: 10.1007/s11075-015-0065-8.

[9]

J. L. Gracia and M. Stynes, Central difference approximation of convection in Caputo fractional derivative two-point boundary value problems, J. Comput. Appl. Math., 273 (2015), 103-115.  doi: 10.1016/j.cam.2014.05.025.

[10]

E. Hanert and C. Piret, A Chebyshev pseudospectral method to solve the space-time tempered fractional diffusion equation, SIAM J. Sci. Comput., 36 (2014), 1797-1812.  doi: 10.1137/130927292.

[11]

Z. Hao, W. Cao and G. Lin, A second-order difference scheme for the time fractional substantial diffusion equation, J. Comput. Appl. Math., 313 (2017), 54-69.  doi: 10.1016/j.cam.2016.09.006.

[12]

Y. Jiang and J. Ma, High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math., 235 (2011), 3285-3290.  doi: 10.1016/j.cam.2011.01.011.

[13]

B. Jin, R. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., 36 (2016), 197-221.  doi: 10.1093/imanum/dru063.

[14]

I. Koponen, Analytic approach to the problem of convergence of truncated Levy flights towards the Gaussian stochastic process, Phys. Rev. E, 52 (1995), 1197-1199.  doi: 10.1103/PhysRevE.52.1197.

[15]

T. A. M. Langlands and B. I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys., 205 (2005), 719-736.  doi: 10.1016/j.jcp.2004.11.025.

[16]

M. Li, X.-M. Gu, C. Huang, M. Fei and G. Zhang, A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schr$\ddot{o}$dinger equations, J. Compu. Phys., 358 (2018), 256-282.  doi: 10.1016/j.jcp.2017.12.044.

[17]

C. Li and F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Boca Raton, FL, 2015.

[18]

C. Li and W. Deng, High order schemes for the tempered fractional diffusion equations, Adv. Comput. Math., 42 (2016), 543-572.  doi: 10.1007/s10444-015-9434-z.

[19]

X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), 2108-2131.  doi: 10.1137/080718942.

[20]

S. Liao, Notes on the homotopy analysis method: some definitions and theorems, Commun Nonlinear Sci Numer Simul., 14 (2009), 983-997.  doi: 10.1016/j.cnsns.2008.04.013.

[21]

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.

[22] F. W. Liu, P. H. Zhuang and Q. X. Liu, The Applications and Numerical Methods of Fractional Differential Equations, Science Press, Beijing, 2015. 
[23]

R. N. Mantegna and H. E. Stanley, Stochastic process with ultraslow convergence to a Gaussian: The truncated Levy flight, Phys. Rev. Lett., 73 (1994), 2946-2949.  doi: 10.1103/PhysRevLett.73.2946.

[24]

M. M. Meerschaert, Y. Zhang and B. Baeumer, Tempered anomalous diffusion in heterogeneous systems, Geophys., Res. Lett., 35 (2008), 190201.

[25]

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion ow equations, J. Comput. Appl. Math., 172 (2004), 65-77.  doi: 10.1016/j.cam.2004.01.033.

[26]

X. Meng, C.-W. Shu and B. Wu, Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations, Math. Comp., 85 (2016), 1225-1261.  doi: 10.1090/mcom/3022.

[27]

S. Momani and Z. Odibat, Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations, Comput. Math. Appl., 54 (2007), 910-919.  doi: 10.1016/j.camwa.2006.12.037.

[28]

J. Q. Murillo and S. B. Yuste, On three explicit difference schemes for fractional diffusion and diffusion-wave equations, Phys. Scr., 136 (2009), 14025-14030. 

[29]

J. Rosinski, Tempering stable processes, Stochastic Process. Appl., 117 (2007), 677-707.  doi: 10.1016/j.spa.2006.10.003.

[30]

E. Sousa and C. Li, A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative, Appl. Numer. Math., 90 (2015), 22-37.  doi: 10.1016/j.apnum.2014.11.007.

[31]

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.

[32]

Z.-Z. Sun and X. N. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), 193-209.  doi: 10.1016/j.apnum.2005.03.003.

[33]

H. Wang, D. Yang and S. Zhu, Inhomogeneous Dirichlet boundary-value problems of space-fractional diffusion equations and their finite element approximations, SIAM J. Numer. Anal., 52 (2014), 1292-1310.  doi: 10.1137/130932776.

[34]

L. Wei, X. Zhang and Y. He, Analysis of a local discontinuous Galerkin method for time-fractional advection-diffusion equations, Int. J. Heat. Fluid. Fl., 23 (2013), 634-648.  doi: 10.1108/09615531311323782.

[35]

L. Wei and Y. He, Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems, Appl. Math. Model., 38 (2014), 1511-1522.  doi: 10.1016/j.apm.2013.07.040.

[36]

X. Wu, W. Deng and E. Barkai, Tempered fractional Feynman-Kac equation: Theory and examples, Phys. Rev. E, 93 (2016), 032151. doi: 10.1103/physreve.93.032151.

[37]

Y. Xia, Y. Xu and C.-W. Shu, Application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system, Commun. Comput. Phys., 5 (2009), 821-835. 

[38]

Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Comm. Comput. Phys., 7 (2010), 1-46.  doi: 10.4208/cicp.2009.09.023.

[39]

Y. Yu, W. Deng and J. Wu, Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations, Appl. Numer. Math., 112 (2017), 126-145.  doi: 10.1016/j.apnum.2016.10.011.

[40]

H. Zhang, F. Liu, I. Turner and S. Chen, The numerical simulation of the tempered fractional Black-Scholes equation for European double barrier option, Appl. Math. Model., 40 (2016), 5819-5834.  doi: 10.1016/j.apm.2016.01.027.

[41]

Q. Zhang and F.-Z. Gao, A Fully-Discrete Local Discontinuous Galerkin Method for Convection-Dominated Sobolev Equation, J. Sci. Comput., 51 (2012), 107-134. 

[42]

Q. Zhang and C.-W. Shu, Error estimates for the third order explicit Runge-Kutta discontinuous Galerkin method for a linear hyperbolic equation in one-dimension with discontinuous initial data, Numer. Math., 126 (2014), – 703-740. doi: 10.1007/s00211-013-0573-1.

[43]

Y. Zhao, Y. Zhang, F. Liu, I. Turner, Y. Tang and V. Anh, Convergence and superconvergence of a fully-discrete scheme for multi-term time fractional diffusion equations, Comput. Math. Appl., 73 (2017), 1087-1099. doi: 10.1016/j.camwa.2016.05.005.

show all references

References:
[1]

B. Baeumer and M. M. Meerschaert, Tempered stable Levy motion and transient superdiffusion, J. Comput. Appl. Math., 233 (2010), 2438-2448.  doi: 10.1016/j.cam.2009.10.027.

[2]

A. Cartea and D. del-Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Phys. A, 374 (2007), 749-763.  doi: 10.1016/j.physa.2006.08.071.

[3]

S. Chen, J. Shen and L.-L. Wang, Generalized Jacobi functions and their applications to fractional differential equations, Math. Comp., 85 (2016), 1603-1638.  doi: 10.1090/mcom3035.

[4]

Y. Chen, X. Wang and W. Deng, Tempered fractional Langevin-Brownian motion with inverse $\beta$-stable subordinator, J. Phys. A: Math. Theor., 51 (2018), 495001. doi: 10.1088/1751-8121/aae8b3.

[5]

Y. Cheng, X. Meng and Q. Zhang, Application of generalized Gauss-Radau projections for the local discontinuous Galerkin method for linear convection-diffusion equations, Math. Comp., 86 (2017), 1233-1267.  doi: 10.1090/mcom/3141.

[6]

M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228 (2009), 7792-7804.  doi: 10.1016/j.jcp.2009.07.021.

[7]

V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, umer. Methods Partial Differential Eq., 22 (2006), 558-576.  doi: 10.1002/num.20112.

[8]

L. B. Feng, P. Zhuang, F. Liu, I. Turner and Y. T. Gu, Finite element method for space-time fractional diffusion equation, Numer. Algor., 72 (2016), 749-767.  doi: 10.1007/s11075-015-0065-8.

[9]

J. L. Gracia and M. Stynes, Central difference approximation of convection in Caputo fractional derivative two-point boundary value problems, J. Comput. Appl. Math., 273 (2015), 103-115.  doi: 10.1016/j.cam.2014.05.025.

[10]

E. Hanert and C. Piret, A Chebyshev pseudospectral method to solve the space-time tempered fractional diffusion equation, SIAM J. Sci. Comput., 36 (2014), 1797-1812.  doi: 10.1137/130927292.

[11]

Z. Hao, W. Cao and G. Lin, A second-order difference scheme for the time fractional substantial diffusion equation, J. Comput. Appl. Math., 313 (2017), 54-69.  doi: 10.1016/j.cam.2016.09.006.

[12]

Y. Jiang and J. Ma, High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math., 235 (2011), 3285-3290.  doi: 10.1016/j.cam.2011.01.011.

[13]

B. Jin, R. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., 36 (2016), 197-221.  doi: 10.1093/imanum/dru063.

[14]

I. Koponen, Analytic approach to the problem of convergence of truncated Levy flights towards the Gaussian stochastic process, Phys. Rev. E, 52 (1995), 1197-1199.  doi: 10.1103/PhysRevE.52.1197.

[15]

T. A. M. Langlands and B. I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys., 205 (2005), 719-736.  doi: 10.1016/j.jcp.2004.11.025.

[16]

M. Li, X.-M. Gu, C. Huang, M. Fei and G. Zhang, A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schr$\ddot{o}$dinger equations, J. Compu. Phys., 358 (2018), 256-282.  doi: 10.1016/j.jcp.2017.12.044.

[17]

C. Li and F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Boca Raton, FL, 2015.

[18]

C. Li and W. Deng, High order schemes for the tempered fractional diffusion equations, Adv. Comput. Math., 42 (2016), 543-572.  doi: 10.1007/s10444-015-9434-z.

[19]

X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), 2108-2131.  doi: 10.1137/080718942.

[20]

S. Liao, Notes on the homotopy analysis method: some definitions and theorems, Commun Nonlinear Sci Numer Simul., 14 (2009), 983-997.  doi: 10.1016/j.cnsns.2008.04.013.

[21]

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.

[22] F. W. Liu, P. H. Zhuang and Q. X. Liu, The Applications and Numerical Methods of Fractional Differential Equations, Science Press, Beijing, 2015. 
[23]

R. N. Mantegna and H. E. Stanley, Stochastic process with ultraslow convergence to a Gaussian: The truncated Levy flight, Phys. Rev. Lett., 73 (1994), 2946-2949.  doi: 10.1103/PhysRevLett.73.2946.

[24]

M. M. Meerschaert, Y. Zhang and B. Baeumer, Tempered anomalous diffusion in heterogeneous systems, Geophys., Res. Lett., 35 (2008), 190201.

[25]

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion ow equations, J. Comput. Appl. Math., 172 (2004), 65-77.  doi: 10.1016/j.cam.2004.01.033.

[26]

X. Meng, C.-W. Shu and B. Wu, Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations, Math. Comp., 85 (2016), 1225-1261.  doi: 10.1090/mcom/3022.

[27]

S. Momani and Z. Odibat, Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations, Comput. Math. Appl., 54 (2007), 910-919.  doi: 10.1016/j.camwa.2006.12.037.

[28]

J. Q. Murillo and S. B. Yuste, On three explicit difference schemes for fractional diffusion and diffusion-wave equations, Phys. Scr., 136 (2009), 14025-14030. 

[29]

J. Rosinski, Tempering stable processes, Stochastic Process. Appl., 117 (2007), 677-707.  doi: 10.1016/j.spa.2006.10.003.

[30]

E. Sousa and C. Li, A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative, Appl. Numer. Math., 90 (2015), 22-37.  doi: 10.1016/j.apnum.2014.11.007.

[31]

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.

[32]

Z.-Z. Sun and X. N. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), 193-209.  doi: 10.1016/j.apnum.2005.03.003.

[33]

H. Wang, D. Yang and S. Zhu, Inhomogeneous Dirichlet boundary-value problems of space-fractional diffusion equations and their finite element approximations, SIAM J. Numer. Anal., 52 (2014), 1292-1310.  doi: 10.1137/130932776.

[34]

L. Wei, X. Zhang and Y. He, Analysis of a local discontinuous Galerkin method for time-fractional advection-diffusion equations, Int. J. Heat. Fluid. Fl., 23 (2013), 634-648.  doi: 10.1108/09615531311323782.

[35]

L. Wei and Y. He, Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems, Appl. Math. Model., 38 (2014), 1511-1522.  doi: 10.1016/j.apm.2013.07.040.

[36]

X. Wu, W. Deng and E. Barkai, Tempered fractional Feynman-Kac equation: Theory and examples, Phys. Rev. E, 93 (2016), 032151. doi: 10.1103/physreve.93.032151.

[37]

Y. Xia, Y. Xu and C.-W. Shu, Application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system, Commun. Comput. Phys., 5 (2009), 821-835. 

[38]

Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Comm. Comput. Phys., 7 (2010), 1-46.  doi: 10.4208/cicp.2009.09.023.

[39]

Y. Yu, W. Deng and J. Wu, Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations, Appl. Numer. Math., 112 (2017), 126-145.  doi: 10.1016/j.apnum.2016.10.011.

[40]

H. Zhang, F. Liu, I. Turner and S. Chen, The numerical simulation of the tempered fractional Black-Scholes equation for European double barrier option, Appl. Math. Model., 40 (2016), 5819-5834.  doi: 10.1016/j.apm.2016.01.027.

[41]

Q. Zhang and F.-Z. Gao, A Fully-Discrete Local Discontinuous Galerkin Method for Convection-Dominated Sobolev Equation, J. Sci. Comput., 51 (2012), 107-134. 

[42]

Q. Zhang and C.-W. Shu, Error estimates for the third order explicit Runge-Kutta discontinuous Galerkin method for a linear hyperbolic equation in one-dimension with discontinuous initial data, Numer. Math., 126 (2014), – 703-740. doi: 10.1007/s00211-013-0573-1.

[43]

Y. Zhao, Y. Zhang, F. Liu, I. Turner, Y. Tang and V. Anh, Convergence and superconvergence of a fully-discrete scheme for multi-term time fractional diffusion equations, Comput. Math. Appl., 73 (2017), 1087-1099. doi: 10.1016/j.camwa.2016.05.005.

Figure 1.  The evolution of the solution for $ \alpha = 0.3 $. $ \tau = 0.001, h = 0.01, k = 2. $
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Figure 2.  The evolution of the solution for $ \alpha = 0.8 $. $ \tau = 0.001, h = 0.01, k = 2. $
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Table 1.  Spatial accuracy test on uniform meshes with generalized alternating numerical fluxes when $ \delta = 0.3, \gamma = 2, M = 10^3, T = 1 $
$\delta$ $\alpha$ $P^k$ $N$ $L^2$-error order $L^\infty$-error order
$\delta=0.3$ $\alpha=0.1$ $P^0$ 5 3.600997655347402E-002 - 8.470765811325168E-002 -
10 1.751495701129877E-002 1.04 4.247840062543977E-002 1.00
20 8.698273697461307E-003 1.01 2.125369555415509E-002 1.00
40 4.341798450317296E-003 1.00 1.062863299729018E-002 1.00
$P^1$ 5 9.943585411573410E-003 - 2.757075846380118E-002 -
10 3.566273983250412E-003 1.48 1.080003816776576E-002 1.35
20 1.086919816367809E-003 1.71 3.420222759716512E-003 1.66
40 2.902709222939700E-004 1.90 9.193407604523585E-004 1.90
$P^2$ 5 7.990640423350242E-004 - 3.417434847568336E-003 -
10 8.626266947702263E-005 3.21 3.694158436061931E-004 3.20
20 1.044514634698983E-005 3.04 4.454609652087860E-005 3.05
40 1.296172202016491E-006 3.01 5.508580340890919E-006 3.01
$\alpha=0.6$ $P^0$ 5 3.594827779072097E-002 - 8.455751510071857E-002 -
10 1.750788942725992E-002 1.04 4.246067080980019E-002 1.00
20 8.697412259699867E-003 1.01 2.125151852856288E-002 1.00
40 4.341693080827747E-003 1.00 1.062836627483681E-002 1.00
$P^1$ 5 9.921671652323443E-003 - 2.759417252018699E-002 -
10 3.563738012224470E-003 1.48 1.080391242567882E-002 1.35
20 1.086707146130006E-003 1.71 3.420472591014495E-003 1.66
40 2.902558645272467E-004 1.90 9.194266256405403E-004 1.90
$P^2$ 5 7.988664990181788E-004 - 3.416510378879402E-003 -
10 8.626077461730029E-005 3.21 3.694001433403462E-004 3.20
20 1.044545573669924E-005 3.04 4.454547869921856E-005 3.05
40 1.298706242251523E-006 3.01 5.508452148315928E-006 3.01
$\alpha=0.8$ $P^0$ 5 3.592329974624144E-002 - 8.449616965675322E-002 -
10 1.750510512702476E-002 1.04 4.245362255057922E-002 1.00
20 8.697106637946338E-003 1.01 2.125073937587364E-002 1.00
40 4.341672465121832E-003 1.00 1.062831369241851E-002 1.00
$P^1$ 5 9.912664929559693E-003 - 2.760460772303402E-002 -
10 3.562659912073956E-003 1.48 1.080630043834668E-002 1.35
20 1.086605203698557E-003 1.71 3.421287359547415E-003 1.66
40 2.902460763410707E-004 1.90 9.201811924158254E-004 1.90
$P^2$ 5 7.988141577793738E-004 - 3.416151504204671E-003 -
10 8.626566626691957E-005 3.21 3.693913312302985E-004 3.21
20 1.046396713324052E-005 3.04 4.454319799217823E-005 3.05
40 1.436914611732205E-006 2.86 5.619072907782352E-006 2.99
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$\delta$ $\alpha$ $P^k$ $N$ $L^2$-error order $L^\infty$-error order
$\delta=0.3$ $\alpha=0.1$ $P^0$ 5 3.600997655347402E-002 - 8.470765811325168E-002 -
10 1.751495701129877E-002 1.04 4.247840062543977E-002 1.00
20 8.698273697461307E-003 1.01 2.125369555415509E-002 1.00
40 4.341798450317296E-003 1.00 1.062863299729018E-002 1.00
$P^1$ 5 9.943585411573410E-003 - 2.757075846380118E-002 -
10 3.566273983250412E-003 1.48 1.080003816776576E-002 1.35
20 1.086919816367809E-003 1.71 3.420222759716512E-003 1.66
40 2.902709222939700E-004 1.90 9.193407604523585E-004 1.90
$P^2$ 5 7.990640423350242E-004 - 3.417434847568336E-003 -
10 8.626266947702263E-005 3.21 3.694158436061931E-004 3.20
20 1.044514634698983E-005 3.04 4.454609652087860E-005 3.05
40 1.296172202016491E-006 3.01 5.508580340890919E-006 3.01
$\alpha=0.6$ $P^0$ 5 3.594827779072097E-002 - 8.455751510071857E-002 -
10 1.750788942725992E-002 1.04 4.246067080980019E-002 1.00
20 8.697412259699867E-003 1.01 2.125151852856288E-002 1.00
40 4.341693080827747E-003 1.00 1.062836627483681E-002 1.00
$P^1$ 5 9.921671652323443E-003 - 2.759417252018699E-002 -
10 3.563738012224470E-003 1.48 1.080391242567882E-002 1.35
20 1.086707146130006E-003 1.71 3.420472591014495E-003 1.66
40 2.902558645272467E-004 1.90 9.194266256405403E-004 1.90
$P^2$ 5 7.988664990181788E-004 - 3.416510378879402E-003 -
10 8.626077461730029E-005 3.21 3.694001433403462E-004 3.20
20 1.044545573669924E-005 3.04 4.454547869921856E-005 3.05
40 1.298706242251523E-006 3.01 5.508452148315928E-006 3.01
$\alpha=0.8$ $P^0$ 5 3.592329974624144E-002 - 8.449616965675322E-002 -
10 1.750510512702476E-002 1.04 4.245362255057922E-002 1.00
20 8.697106637946338E-003 1.01 2.125073937587364E-002 1.00
40 4.341672465121832E-003 1.00 1.062831369241851E-002 1.00
$P^1$ 5 9.912664929559693E-003 - 2.760460772303402E-002 -
10 3.562659912073956E-003 1.48 1.080630043834668E-002 1.35
20 1.086605203698557E-003 1.71 3.421287359547415E-003 1.66
40 2.902460763410707E-004 1.90 9.201811924158254E-004 1.90
$P^2$ 5 7.988141577793738E-004 - 3.416151504204671E-003 -
10 8.626566626691957E-005 3.21 3.693913312302985E-004 3.21
20 1.046396713324052E-005 3.04 4.454319799217823E-005 3.05
40 1.436914611732205E-006 2.86 5.619072907782352E-006 2.99
Table 2.  Spatial accuracy test on uniform meshes with generalized alternating numerical fluxes when $\delta = 0.1$, $\gamma = 2, M = 10^3, T = 1$
$\delta$ $\alpha$ $P^k$ $N$ $L^2$-error order $L^\infty$-error order
$\delta=0.1$ $\alpha=0.1$ $P^0$ 5 3.600997655347402E-002 - 8.470765811325168E-002 -
10 1.751495701129877E-002 1.04 4.247840062543977E-002 1.00
20 8.698273697461307E-003 1.01 2.125369555415509E-002 1.00
40 4.341798450317296E-003 1.00 1.062863299729018E-002 1.00
$P^1$ 5 9.125860752324633E-003 - 3.386112146946083E-002 -
10 2.295048237777114E-003 1.99 8.756550512034528E-003 1.95
20 5.745423502445809E-004 2.00 2.207822490277067E-003 1.98
40 1.436832777579926E-004 2.00 5.554382129445423E-004 1.99
$P^2$ 5 9.050669939275690E-004 - 4.298666679244556E-003 -
10 1.151488108051345E-004 2.97 5.375579180586712E-004 3.00
20 1.445721437410244E-005 2.99 6.924567827167210E-005 2.96
40 1.809466716341844E-006 3.00 8.720509756160153E-006 2.99
$\alpha=0.6$ $P^0$ 5 3.594827779072097E-002 - 8.455751510071857E-002 -
10 1.750788942725992E-002 1.04 4.246067080980019E-002 1.00
20 8.697412259699867E-003 1.01 2.125151852856288E-002 1.00
40 4.341693080827747E-003 1.00 1.062836627483681E-002 1.00
$P^1$ 5 9.121793113875693E-003 - 3.384039722034468E-002 -
10 2.294831526335838E-003 1.99 8.755484555290433E-003 1.95
20 5.745294690001247E-004 2.00 2.207828785762922E-003 1.98
40 1.436825323167711E-004 2.00 5.555100933167800E-004 1.99
$P^2$ 5 9.047609877074052E-004 - 4.297245174444950E-003 -
10 1.151394966919664E-004 2.97 5.375058641304305E-004 3.00
20 1.445715309857497E-005 2.99 6.924406859557769E-005 2.96
40 1.811270951788605E-006 3.00 8.720463021409203E-006 2.99
$\alpha=0.8$ $P^0$ 5 3.592329974624144E-002 - 8.449616965675322E-002 -
10 1.750510512702476E-002 1.04 4.245362255057922E-002 1.00
20 8.697106637946338E-003 1.01 2.125073937587364E-002 1.00
40 4.341672465121832E-003 1.00 1.062831369241851E-002 1.00
$P^1$ 5 9.120195825181981E-003 - 3.383289362730243E-002 -
10 2.294751340921533E-003 1.99 8.755789596454649E-003 1.95
20 5.745261460924934E-004 2.00 2.208566384755528E-003 1.98
40 1.436839091653192E-004 2.00 5.562652869576801E-004 1.99
$P^2$ 5 9.046409059470311E-004 - 4.297126983902264E-003 -
10 1.151376194022218E-004 2.97 5.374880423776833E-004 3.00
20 1.447012828175748E-005 2.99 6.924379906001369E-005 2.96
40 1.912757116440075E-006 2.99 8.835769827388040E-006 2.99
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$\delta$ $\alpha$ $P^k$ $N$ $L^2$-error order $L^\infty$-error order
$\delta=0.1$ $\alpha=0.1$ $P^0$ 5 3.600997655347402E-002 - 8.470765811325168E-002 -
10 1.751495701129877E-002 1.04 4.247840062543977E-002 1.00
20 8.698273697461307E-003 1.01 2.125369555415509E-002 1.00
40 4.341798450317296E-003 1.00 1.062863299729018E-002 1.00
$P^1$ 5 9.125860752324633E-003 - 3.386112146946083E-002 -
10 2.295048237777114E-003 1.99 8.756550512034528E-003 1.95
20 5.745423502445809E-004 2.00 2.207822490277067E-003 1.98
40 1.436832777579926E-004 2.00 5.554382129445423E-004 1.99
$P^2$ 5 9.050669939275690E-004 - 4.298666679244556E-003 -
10 1.151488108051345E-004 2.97 5.375579180586712E-004 3.00
20 1.445721437410244E-005 2.99 6.924567827167210E-005 2.96
40 1.809466716341844E-006 3.00 8.720509756160153E-006 2.99
$\alpha=0.6$ $P^0$ 5 3.594827779072097E-002 - 8.455751510071857E-002 -
10 1.750788942725992E-002 1.04 4.246067080980019E-002 1.00
20 8.697412259699867E-003 1.01 2.125151852856288E-002 1.00
40 4.341693080827747E-003 1.00 1.062836627483681E-002 1.00
$P^1$ 5 9.121793113875693E-003 - 3.384039722034468E-002 -
10 2.294831526335838E-003 1.99 8.755484555290433E-003 1.95
20 5.745294690001247E-004 2.00 2.207828785762922E-003 1.98
40 1.436825323167711E-004 2.00 5.555100933167800E-004 1.99
$P^2$ 5 9.047609877074052E-004 - 4.297245174444950E-003 -
10 1.151394966919664E-004 2.97 5.375058641304305E-004 3.00
20 1.445715309857497E-005 2.99 6.924406859557769E-005 2.96
40 1.811270951788605E-006 3.00 8.720463021409203E-006 2.99
$\alpha=0.8$ $P^0$ 5 3.592329974624144E-002 - 8.449616965675322E-002 -
10 1.750510512702476E-002 1.04 4.245362255057922E-002 1.00
20 8.697106637946338E-003 1.01 2.125073937587364E-002 1.00
40 4.341672465121832E-003 1.00 1.062831369241851E-002 1.00
$P^1$ 5 9.120195825181981E-003 - 3.383289362730243E-002 -
10 2.294751340921533E-003 1.99 8.755789596454649E-003 1.95
20 5.745261460924934E-004 2.00 2.208566384755528E-003 1.98
40 1.436839091653192E-004 2.00 5.562652869576801E-004 1.99
$P^2$ 5 9.046409059470311E-004 - 4.297126983902264E-003 -
10 1.151376194022218E-004 2.97 5.374880423776833E-004 3.00
20 1.447012828175748E-005 2.99 6.924379906001369E-005 2.96
40 1.912757116440075E-006 2.99 8.835769827388040E-006 2.99
Table 3.  Spatial accuracy test on nonuniform meshes with generalized alternating numerical fluxes when $\delta = 0.3$, $\gamma = 2, M = 10^3, T = 1$
$\delta$ $\alpha$ $P^k$ $N$ $L^2$-error order $L^\infty$-error order
$\delta=0.3$ $\alpha=0.1$ $P^0$ 5 6.559598407976026E-002 - 0.141942066666751 -
10 2.141331033721447E-002 1.61 5.734028406382948E-002 1.30
20 9.793316970413680E-003 1.12 2.815183989101257E-002 1.02
40 4.809485138977877E-003 1.02 1.407535025139766E-002 1.00
$P^1$ 5 9.334237440748634E-003 - 3.145978397121429E-002 -
10 3.638112329117496E-003 1.36 1.339194646885009E-002 1.23
20 1.106145242636678E-003 1.72 4.300814221832788E-003 1.64
40 2.952217379902808E-004 1.91 1.195193749833484E-003 1.85
$P^2$ 5 1.010581921645606E-003 - 5.126434823465115E-003 -
10 1.068201999084790E-004 3.24 5.971277580741609E-004 3.10
20 1.389860346772117E-005 2.94 8.309553534868077E-005 2.85
40 1.770704698351656E-006 2.97 1.076235396982232E-005 2.95
$\alpha=0.6$ $P^0$ 5 6.519959540296516E-002 - 0.141364976011094 -
10 2.137597683554834E-002 1.60 5.728856602677920E-002 1.31
20 9.788419793457237E-003 1.12 2.814456788448857E-002 1.02
40 4.808850888477931E-003 1.01 1.407438548005000E-002 1.00
$P^1$ 5 9.324055653453555E-003 - 3.148494472151778E-002 -
10 3.637084108871119E-003 1.36 1.339218081832141E-002 1.23
20 1.106059577072876E-003 1.72 4.300878545495546E-003 1.64
40 2.952158819276407E-004 1.91 1.195178864173474E-003 1.85
$P^2$ 5 1.010440003914728E-003 - 5.125652210367676E-003 -
10 1.068162274307204E-004 3.23 5.971411497308199E-004 3.10
20 1.389837000772890E-005 2.94 8.309427296208810E-005 2.85
40 1.770454293076160E-006 2.97 1.076230665963618E-005 2.95
$\alpha=0.8$ $P^0$ 5 6.458019587494233E-002 - 0.140458685661883 -
10 2.131839260620296E-002 1.60 5.720848587853570E-002 1.31
20 9.780967395542408E-003 1.12 2.813343523512223E-002 1.01
40 4.807925152620669E-003 1.02 1.407295053444370E-002 1.00
$P^1$ 5 9.308221701165681E-003 - 3.152477270193341E-002 -
10 3.635461850529788E-003 1.36 1.339294605022746E-002 1.24
20 1.105916081128332E-003 1.72 4.301517739581207E-003 1.64
40 2.952039890812090E-004 1.91 1.195672409974230E-003 1.85
$P^2$ 5 1.010240608178436E-003 - 5.124773588667655E-003 -
10 1.068123075153382E-004 3.23 5.971640034306184E-004 3.10
20 1.390323465425386E-005 2.94 8.309265950586455E-005 2.85
40 1.809513967454312E-006 2.98 1.076227541254043E-005 2.95
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$\delta$ $\alpha$ $P^k$ $N$ $L^2$-error order $L^\infty$-error order
$\delta=0.3$ $\alpha=0.1$ $P^0$ 5 6.559598407976026E-002 - 0.141942066666751 -
10 2.141331033721447E-002 1.61 5.734028406382948E-002 1.30
20 9.793316970413680E-003 1.12 2.815183989101257E-002 1.02
40 4.809485138977877E-003 1.02 1.407535025139766E-002 1.00
$P^1$ 5 9.334237440748634E-003 - 3.145978397121429E-002 -
10 3.638112329117496E-003 1.36 1.339194646885009E-002 1.23
20 1.106145242636678E-003 1.72 4.300814221832788E-003 1.64
40 2.952217379902808E-004 1.91 1.195193749833484E-003 1.85
$P^2$ 5 1.010581921645606E-003 - 5.126434823465115E-003 -
10 1.068201999084790E-004 3.24 5.971277580741609E-004 3.10
20 1.389860346772117E-005 2.94 8.309553534868077E-005 2.85
40 1.770704698351656E-006 2.97 1.076235396982232E-005 2.95
$\alpha=0.6$ $P^0$ 5 6.519959540296516E-002 - 0.141364976011094 -
10 2.137597683554834E-002 1.60 5.728856602677920E-002 1.31
20 9.788419793457237E-003 1.12 2.814456788448857E-002 1.02
40 4.808850888477931E-003 1.01 1.407438548005000E-002 1.00
$P^1$ 5 9.324055653453555E-003 - 3.148494472151778E-002 -
10 3.637084108871119E-003 1.36 1.339218081832141E-002 1.23
20 1.106059577072876E-003 1.72 4.300878545495546E-003 1.64
40 2.952158819276407E-004 1.91 1.195178864173474E-003 1.85
$P^2$ 5 1.010440003914728E-003 - 5.125652210367676E-003 -
10 1.068162274307204E-004 3.23 5.971411497308199E-004 3.10
20 1.389837000772890E-005 2.94 8.309427296208810E-005 2.85
40 1.770454293076160E-006 2.97 1.076230665963618E-005 2.95
$\alpha=0.8$ $P^0$ 5 6.458019587494233E-002 - 0.140458685661883 -
10 2.131839260620296E-002 1.60 5.720848587853570E-002 1.31
20 9.780967395542408E-003 1.12 2.813343523512223E-002 1.01
40 4.807925152620669E-003 1.02 1.407295053444370E-002 1.00
$P^1$ 5 9.308221701165681E-003 - 3.152477270193341E-002 -
10 3.635461850529788E-003 1.36 1.339294605022746E-002 1.24
20 1.105916081128332E-003 1.72 4.301517739581207E-003 1.64
40 2.952039890812090E-004 1.91 1.195672409974230E-003 1.85
$P^2$ 5 1.010240608178436E-003 - 5.124773588667655E-003 -
10 1.068123075153382E-004 3.23 5.971640034306184E-004 3.10
20 1.390323465425386E-005 2.94 8.309265950586455E-005 2.85
40 1.809513967454312E-006 2.98 1.076227541254043E-005 2.95
Table 4.  Spatial accuracy test on nonuniform meshes with generalized alternating numerical fluxes when $\delta = 0.1$, $\gamma = 2, M = 10^3, T = 1$
$\delta$ $\alpha$ $P^k$ $N$ $L^2$-error order $L^\infty$-error order
$\delta=0.1$ $\alpha=0.1$ $P^0$ 5 4.554426032174388E-002 - 0.115718177138195 -
10 1.907614555969082E-002 1.25 5.320421957088832E-002 1.12
20 9.138240767269679E-003 1.06 2.598259449662434E-002 1.03
40 4.518863802087745E-003 1.01 1.291377789093143E-002 1.00
$P^1$ 5 8.849830180737654E-003 - 3.969655576082662E-002 -
10 2.670888776568979E-003 1.73 1.117674633593155E-002 1.83
20 6.856209305073176E-004 1.96 2.993783590210047E-003 1.90
40 1.726263868565285E-004 1.99 7.624154117570892E-004 1.97
$P^2$ 5 1.082053891229540E-003 - 5.632531555489165E-003 -
10 1.177827066011148E-004 3.20 7.864352387506149E-004 2.84
20 1.464825722091804E-005 3.01 9.550948635494300E-005 3.04
40 1.829209283100811E-006 3.00 1.207411590455412E-005 2.98
$\alpha=0.6$ $P^0$ 5 4.543056490078846E-002 - 0.115474889448670 -
10 1.906295169538061E-002 1.25 5.318082033209643E-002 1.12
20 9.136635333555037E-003 1.06 2.597982073437711E-002 1.03
40 4.518663596834055E-003 1.01 1.291343451385024E-002 1.00
$P^1$ 5 8.846862605510398E-003 - 3.970373991829590E-002 -
10 2.670694860720131E-003 1.73 1.117614841378017E-002 1.83
20 6.856086070180152E-004 1.96 2.993777705012829E-003 1.90
40 1.726256192263683E-004 1.99 7.623984725640964E-004 1.97
$P^2$ 5 1.081894553798753E-003 - 5.631794818602251E-003 -
10 1.177789297367418E-004 3.20 7.864212743115771E-004 2.84
20 1.464811122465675E-005 3.01 9.550366030672275E-005 3.04
40 1.828970584365630E-006 3.00 1.207133596128895E-005 2.99
$\alpha=0.8$ $P^0$ 5 4.525430019654393E-002 - 0.115095429944861 -
10 1.904269469568265E-002 1.25 5.314471217556128E-002 1.12
20 9.134222951010450E-003 1.06 2.597562042326319E-002 1.03
40 4.518387505390618E-003 1.01 1.291295179596993E-002 1.00
$P^1$ 5 8.842271383522345E-003 - 3.971560941693522E-002 -
10 2.670392762456530E-003 1.73 1.117577463774522E-002 1.83
20 6.855884088997946E-004 1.96 2.994298251659394E-003 1.90
40 1.726244848834557E-004 1.99 7.628868021073432E-004 1.97
$P^2$ 5 1.081660399859714E-003 - 5.630991561418289E-003 -
10 1.177745623909475E-004 3.20 7.864030096010258E-004 2.84
20 1.465280637729603E-005 3.01 9.565506409087154E-005 3.04
40 1.866808245534035E-006 3.00 1.214811102974098E-005 2.99
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$\delta$ $\alpha$ $P^k$ $N$ $L^2$-error order $L^\infty$-error order
$\delta=0.1$ $\alpha=0.1$ $P^0$ 5 4.554426032174388E-002 - 0.115718177138195 -
10 1.907614555969082E-002 1.25 5.320421957088832E-002 1.12
20 9.138240767269679E-003 1.06 2.598259449662434E-002 1.03
40 4.518863802087745E-003 1.01 1.291377789093143E-002 1.00
$P^1$ 5 8.849830180737654E-003 - 3.969655576082662E-002 -
10 2.670888776568979E-003 1.73 1.117674633593155E-002 1.83
20 6.856209305073176E-004 1.96 2.993783590210047E-003 1.90
40 1.726263868565285E-004 1.99 7.624154117570892E-004 1.97
$P^2$ 5 1.082053891229540E-003 - 5.632531555489165E-003 -
10 1.177827066011148E-004 3.20 7.864352387506149E-004 2.84
20 1.464825722091804E-005 3.01 9.550948635494300E-005 3.04
40 1.829209283100811E-006 3.00 1.207411590455412E-005 2.98
$\alpha=0.6$ $P^0$ 5 4.543056490078846E-002 - 0.115474889448670 -
10 1.906295169538061E-002 1.25 5.318082033209643E-002 1.12
20 9.136635333555037E-003 1.06 2.597982073437711E-002 1.03
40 4.518663596834055E-003 1.01 1.291343451385024E-002 1.00
$P^1$ 5 8.846862605510398E-003 - 3.970373991829590E-002 -
10 2.670694860720131E-003 1.73 1.117614841378017E-002 1.83
20 6.856086070180152E-004 1.96 2.993777705012829E-003 1.90
40 1.726256192263683E-004 1.99 7.623984725640964E-004 1.97
$P^2$ 5 1.081894553798753E-003 - 5.631794818602251E-003 -
10 1.177789297367418E-004 3.20 7.864212743115771E-004 2.84
20 1.464811122465675E-005 3.01 9.550366030672275E-005 3.04
40 1.828970584365630E-006 3.00 1.207133596128895E-005 2.99
$\alpha=0.8$ $P^0$ 5 4.525430019654393E-002 - 0.115095429944861 -
10 1.904269469568265E-002 1.25 5.314471217556128E-002 1.12
20 9.134222951010450E-003 1.06 2.597562042326319E-002 1.03
40 4.518387505390618E-003 1.01 1.291295179596993E-002 1.00
$P^1$ 5 8.842271383522345E-003 - 3.971560941693522E-002 -
10 2.670392762456530E-003 1.73 1.117577463774522E-002 1.83
20 6.855884088997946E-004 1.96 2.994298251659394E-003 1.90
40 1.726244848834557E-004 1.99 7.628868021073432E-004 1.97
$P^2$ 5 1.081660399859714E-003 - 5.630991561418289E-003 -
10 1.177745623909475E-004 3.20 7.864030096010258E-004 2.84
20 1.465280637729603E-005 3.01 9.565506409087154E-005 3.04
40 1.866808245534035E-006 3.00 1.214811102974098E-005 2.99
Table 5.  Temporal accuracy test using piecewise $P^2$ polynomials for the scheme (9) with generalized alternating numerical fluxes when $N = 100, T = 1.$
$\delta$ $\alpha$ $\tau$ $L^2$-error order $L^\infty$-error order
$\delta=0.1$ $\alpha=0.5$ 0.04 8.608763604447880E-006 - 1.219684581693636E-005 -
0.02 3.086574970641430E-006 1.48 4.409647689024299E-006 1.47
0.01 1.109311333958263E-006 1.48 1.667659212722938E-006 1.40
0.005 4.122054755449973E-007 1.43 6.985320445991206E-007 1.26
$\alpha=0.7$ 0.04 2.391687146347450E-005 - 3.383563665176892E-005 -
0.02 9.853827361471093E-006 1.28 1.395429884359922E-005 1.28
0.01 4.116933701786636E-006 1.26 5.856072709420346E-006 1.26
0.005 1.782609365838843E-006 1.21 2.587548893207003E-006 1.18
$\delta=0.3$ $\alpha=0.5$ 0.04 8.608382726939050E-006 - 1.217526939467639E-005 -
0.02 3.085511558119693E-006 1.48 4.377333736760303E-006 1.48
0.01 1.106348076087462E-006 1.48 1.601411114160456E-006 1.45
0.005 4.041621399151297E-007 1.45 6.654877294648420E-007 1.27
$\alpha=0.7$ 0.04 2.391673470774867E-005 - 3.382490687975359E-005 -
0.02 9.853494675524166E-006 1.28 1.393619582018557E-005 1.28
0.01 4.116136624036466E-006 1.26 5.832318129950220E-006 1.26
0.005 1.780767052630476E-006 1.21 2.542037339958725E-006 1.20
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$\delta$ $\alpha$ $\tau$ $L^2$-error order $L^\infty$-error order
$\delta=0.1$ $\alpha=0.5$ 0.04 8.608763604447880E-006 - 1.219684581693636E-005 -
0.02 3.086574970641430E-006 1.48 4.409647689024299E-006 1.47
0.01 1.109311333958263E-006 1.48 1.667659212722938E-006 1.40
0.005 4.122054755449973E-007 1.43 6.985320445991206E-007 1.26
$\alpha=0.7$ 0.04 2.391687146347450E-005 - 3.383563665176892E-005 -
0.02 9.853827361471093E-006 1.28 1.395429884359922E-005 1.28
0.01 4.116933701786636E-006 1.26 5.856072709420346E-006 1.26
0.005 1.782609365838843E-006 1.21 2.587548893207003E-006 1.18
$\delta=0.3$ $\alpha=0.5$ 0.04 8.608382726939050E-006 - 1.217526939467639E-005 -
0.02 3.085511558119693E-006 1.48 4.377333736760303E-006 1.48
0.01 1.106348076087462E-006 1.48 1.601411114160456E-006 1.45
0.005 4.041621399151297E-007 1.45 6.654877294648420E-007 1.27
$\alpha=0.7$ 0.04 2.391673470774867E-005 - 3.382490687975359E-005 -
0.02 9.853494675524166E-006 1.28 1.393619582018557E-005 1.28
0.01 4.116136624036466E-006 1.26 5.832318129950220E-006 1.26
0.005 1.780767052630476E-006 1.21 2.542037339958725E-006 1.20
Table 6.  Spatial accuracy test on uniform meshes with generalized alternating numerical fluxes when $\rho = 1, \delta = 0.2$, $\gamma = 2, M = 10^3, T = 1$
$\delta$ $\alpha$ $P^k$ $N$ $L^2$-error order $L^\infty$-error order
$\delta=0.2$ $\alpha=0.3$ $P^0$ 5 1.582340814827774E-003 - 3.587747814566711E-003 -
10 6.072920572041250E-004 1.38 1.361992544606309E-003 1.39
20 2.781722995839050E-004 1.12 6.593363827866452E-004 1.05
40 1.358688098753038E-004 1.03 3.262244021583708E-004 1.01
$P^1$ 5 2.976903263874860E-004 - 8.888076108539667E-004 -
10 9.380682880588326E-005 1.67 3.747599073017630E-004 1.25
20 2.570892164256014E-005 1.87 1.197734627369952E-004 1.65
40 6.616980217734994E-006 1.96 3.372607623089679E-005 1.83
$P^2$ 5 3.744225938986966E-005 - 2.066750403204734E-004 -
10 4.242629022151605E-006 3.14 2.905747077063567E-005 2.83
20 5.108240307324746E-007 3.05 3.733062412131737E-006 2.96
40 1.404276778729993E-007 1.86 4.548930764244681E-007 3.03
$\alpha=0.5$ $P^0$ 5 1.576014035107718E-003 - 3.577323483172182E-003 -
10 6.065883866688050E-004 1.38 1.361430273966585E-003 1.39
20 2.780870829316098E-004 1.12 6.589782886991372E-004 1.05
40 1.358586702912018E-004 1.03 3.262537733518698E-004 1.01
$P^1$ 5 2.975626899907474E-004 - 8.888043631958508E-004 -
10 9.379482889619085E-005 1.68 3.746166818088215E-004 1.25
20 2.570772139874646E-005 1.87 1.196172349767182E-004 1.65
40 6.615791862540715E-006 1.96 3.356898366286719E-005 1.83
$P^2$ 5 3.743605733712328E-005 - 2.068108840347286E-004 -
10 4.240769769090305E-006 3.14 2.921541320911186E-005 2.82
20 4.961774292224783E-007 3.09 3.890639954380062E-006 2.90
40 7.051223319124203E-008 2.81 5.312153408062071E-007 2.87
$\alpha=0.7$ $P^0$ 5 1.569573150535817E-003 - 3.566532563472458E-003 -
10 6.058781928822529E-004 1.37 1.361570566765813E-003 1.39
20 2.780022433132069E-004 1.12 6.584717897009694E-004 1.04
40 1.358494586510882E-004 1.03 3.265674113640243E-004 1.01
$P^1$ 5 2.974342179950080E-004 - 8.886664664762281E-004 -
10 9.378333026032094E-005 1.67 3.743331911596685E-004 1.25
20 2.570912149705274E-005 1.87 1.193198577654686E-004 1.65
40 6.624911739109129E-006 1.96 3.327059953717988E-005 1.84
$P^2$ 5 3.743175301428064E-005 - 2.070928579500273E-004 -
10 4.255010460194694E-006 3.13 2.951598305808386E-005 2.81
20 6.067267031675130E-007 2.81 4.189879914070192E-006 2.82
40 8.591886309964037E-008 2.82 6.269178180106001E-007 2.74
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$\delta$ $\alpha$ $P^k$ $N$ $L^2$-error order $L^\infty$-error order
$\delta=0.2$ $\alpha=0.3$ $P^0$ 5 1.582340814827774E-003 - 3.587747814566711E-003 -
10 6.072920572041250E-004 1.38 1.361992544606309E-003 1.39
20 2.781722995839050E-004 1.12 6.593363827866452E-004 1.05
40 1.358688098753038E-004 1.03 3.262244021583708E-004 1.01
$P^1$ 5 2.976903263874860E-004 - 8.888076108539667E-004 -
10 9.380682880588326E-005 1.67 3.747599073017630E-004 1.25
20 2.570892164256014E-005 1.87 1.197734627369952E-004 1.65
40 6.616980217734994E-006 1.96 3.372607623089679E-005 1.83
$P^2$ 5 3.744225938986966E-005 - 2.066750403204734E-004 -
10 4.242629022151605E-006 3.14 2.905747077063567E-005 2.83
20 5.108240307324746E-007 3.05 3.733062412131737E-006 2.96
40 1.404276778729993E-007 1.86 4.548930764244681E-007 3.03
$\alpha=0.5$ $P^0$ 5 1.576014035107718E-003 - 3.577323483172182E-003 -
10 6.065883866688050E-004 1.38 1.361430273966585E-003 1.39
20 2.780870829316098E-004 1.12 6.589782886991372E-004 1.05
40 1.358586702912018E-004 1.03 3.262537733518698E-004 1.01
$P^1$ 5 2.975626899907474E-004 - 8.888043631958508E-004 -
10 9.379482889619085E-005 1.68 3.746166818088215E-004 1.25
20 2.570772139874646E-005 1.87 1.196172349767182E-004 1.65
40 6.615791862540715E-006 1.96 3.356898366286719E-005 1.83
$P^2$ 5 3.743605733712328E-005 - 2.068108840347286E-004 -
10 4.240769769090305E-006 3.14 2.921541320911186E-005 2.82
20 4.961774292224783E-007 3.09 3.890639954380062E-006 2.90
40 7.051223319124203E-008 2.81 5.312153408062071E-007 2.87
$\alpha=0.7$ $P^0$ 5 1.569573150535817E-003 - 3.566532563472458E-003 -
10 6.058781928822529E-004 1.37 1.361570566765813E-003 1.39
20 2.780022433132069E-004 1.12 6.584717897009694E-004 1.04
40 1.358494586510882E-004 1.03 3.265674113640243E-004 1.01
$P^1$ 5 2.974342179950080E-004 - 8.886664664762281E-004 -
10 9.378333026032094E-005 1.67 3.743331911596685E-004 1.25
20 2.570912149705274E-005 1.87 1.193198577654686E-004 1.65
40 6.624911739109129E-006 1.96 3.327059953717988E-005 1.84
$P^2$ 5 3.743175301428064E-005 - 2.070928579500273E-004 -
10 4.255010460194694E-006 3.13 2.951598305808386E-005 2.81
20 6.067267031675130E-007 2.81 4.189879914070192E-006 2.82
40 8.591886309964037E-008 2.82 6.269178180106001E-007 2.74
Table 7.  Spatial accuracy test on uniform meshes with generalized alternating numerical fluxes when $\rho = 1, \delta = 0.6$, $\gamma = 2, M = 10^3, T = 1$
$\delta$ $\alpha$ $P^k$ $N$ $L^2$-error order $L^\infty$-error order
$\delta=0.6$ $\alpha=0.3$ $P^0$ 5 2.263070303841627E-003 - 4.577360163679048E-003 -
10 6.707774222704496E-004 1.75 1.436684775454216E-003 1.67
20 2.853444715298516E-004 1.23 6.606848412090876E-004 1.12
40 1.367407768126421E-004 1.06 3.273628226099778E-004 1.01
$P^1$ 5 2.974509312553489E-004 - 7.150301084008492E-004 -
10 1.271245501318707E-004 1.22 3.981226597552988E-004 0.84
20 5.095203810569959E-005 1.32 1.756826753133757E-004 1.18
40 1.628155726102348E-005 1.65 6.113921628532708E-005 1.52
$P^2$ 5 4.741117314917393E-005 - 2.322474249148421E-004 -
10 5.251072284968953E-006 3.17 3.482052702302946E-005 2.73
20 5.487061550738519E-007 3.26 4.391297775017012E-006 2.99
40 1.398475833157788E-007 1.97 5.076183265380021E-007 3.11
$\alpha=0.5$ $P^0$ 5 2.244975493970608E-003 - 4.553595493445027E-003 -
10 6.694913740987843E-004 1.75 1.433071024540420E-003 1.66
20 2.851933816998400E-004 1.23 6.606969556207020E-004 1.12
40 1.367226188180968E-004 1.06 3.275293007699463E-004 1.01
$P^1$ 5 2.970077416293948E-004 - 7.150618409357242E-004 -
10 1.270399125198733E-004 1.23 3.980425972542222E-004 0.85
20 5.093888725451710E-005 1.32 1.755280846267336E-004 1.18
40 1.627971177773421E-005 1.65 6.098150580908531E-005 1.53
$P^2$ 5 4.738891552206912E-005 - 2.323268736959109E-004 -
10 5.248488428614583E-006 3.17 3.497357272207592E-005 2.73
20 5.350808403013323E-007 3.29 4.548739501698698E-006 2.94
40 6.934968715889372E-008 2.95 6.122329160500268E-007 2.89
$\alpha=0.7$ $P^0$ 5 2.226364987023399E-003 - 4.528968446397434E-003 -
10 6.681909572138881E-004 1.73 1.429260382731581E-003 1.66
20 2.850422396157703E-004 1.23 6.608559171080138E-004 1.11
40 1.367054729489756E-004 1.06 3.278432462824107E-004 1.01
$P^1$ 5 2.965586452297552E-004 - 7.149566598926194E-004 -
10 1.269543669818585E-004 1.22 3.978234263835360E-004 0.85
20 5.092673020318355E-005 1.32 1.752325021227329E-004 1.18
40 1.628197579228578E-005 1.65 6.068249513503552E-005 1.53
$P^2$ 5 4.736816730385127E-005 - 2.325522068324162E-004 -
10 5.258924978488712E-006 3.17 3.526923646273578E-005 2.72
20 6.389205641380220E-007 3.04 4.847842978104736E-006 2.86
40 8.501389966985875E-008 2.91 6.585375965448209E-007 2.88
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$\delta$ $\alpha$ $P^k$ $N$ $L^2$-error order $L^\infty$-error order
$\delta=0.6$ $\alpha=0.3$ $P^0$ 5 2.263070303841627E-003 - 4.577360163679048E-003 -
10 6.707774222704496E-004 1.75 1.436684775454216E-003 1.67
20 2.853444715298516E-004 1.23 6.606848412090876E-004 1.12
40 1.367407768126421E-004 1.06 3.273628226099778E-004 1.01
$P^1$ 5 2.974509312553489E-004 - 7.150301084008492E-004 -
10 1.271245501318707E-004 1.22 3.981226597552988E-004 0.84
20 5.095203810569959E-005 1.32 1.756826753133757E-004 1.18
40 1.628155726102348E-005 1.65 6.113921628532708E-005 1.52
$P^2$ 5 4.741117314917393E-005 - 2.322474249148421E-004 -
10 5.251072284968953E-006 3.17 3.482052702302946E-005 2.73
20 5.487061550738519E-007 3.26 4.391297775017012E-006 2.99
40 1.398475833157788E-007 1.97 5.076183265380021E-007 3.11
$\alpha=0.5$ $P^0$ 5 2.244975493970608E-003 - 4.553595493445027E-003 -
10 6.694913740987843E-004 1.75 1.433071024540420E-003 1.66
20 2.851933816998400E-004 1.23 6.606969556207020E-004 1.12
40 1.367226188180968E-004 1.06 3.275293007699463E-004 1.01
$P^1$ 5 2.970077416293948E-004 - 7.150618409357242E-004 -
10 1.270399125198733E-004 1.23 3.980425972542222E-004 0.85
20 5.093888725451710E-005 1.32 1.755280846267336E-004 1.18
40 1.627971177773421E-005 1.65 6.098150580908531E-005 1.53
$P^2$ 5 4.738891552206912E-005 - 2.323268736959109E-004 -
10 5.248488428614583E-006 3.17 3.497357272207592E-005 2.73
20 5.350808403013323E-007 3.29 4.548739501698698E-006 2.94
40 6.934968715889372E-008 2.95 6.122329160500268E-007 2.89
$\alpha=0.7$ $P^0$ 5 2.226364987023399E-003 - 4.528968446397434E-003 -
10 6.681909572138881E-004 1.73 1.429260382731581E-003 1.66
20 2.850422396157703E-004 1.23 6.608559171080138E-004 1.11
40 1.367054729489756E-004 1.06 3.278432462824107E-004 1.01
$P^1$ 5 2.965586452297552E-004 - 7.149566598926194E-004 -
10 1.269543669818585E-004 1.22 3.978234263835360E-004 0.85
20 5.092673020318355E-005 1.32 1.752325021227329E-004 1.18
40 1.628197579228578E-005 1.65 6.068249513503552E-005 1.53
$P^2$ 5 4.736816730385127E-005 - 2.325522068324162E-004 -
10 5.258924978488712E-006 3.17 3.526923646273578E-005 2.72
20 6.389205641380220E-007 3.04 4.847842978104736E-006 2.86
40 8.501389966985875E-008 2.91 6.585375965448209E-007 2.88
Table 8.  Spatial accuracy test on nonuniform meshes with generalized alternating numerical fluxes when $\rho = 1, \delta = 0.2$, $\gamma = 2, M = 10^3, T = 1$
$\delta$ $\alpha$ $P^k$ $N$ $L^2$-error order $L^\infty$-error order
$\delta=0.2$ $\alpha=0.3$ $P^0$ 5 1.517845954000122E-003 - 3.511606734885603E-003 -
10 6.227976716020462E-004 1.28 1.670434180879555E-003 1.07
20 2.917321657674542E-004 1.09 8.185201216153321E-004 1.02
40 1.431224670285340E-004 1.02 4.068120471581059E-004 1.01
$P^1$ 5 3.242733095936642E-004 - 1.031247407943849E-003 -
10 9.642143896521544E-005 1.75 4.664783993685712E-004 1.14
20 2.638283335064153E-005 1.87 1.497589900759262E-004 1.64
40 6.784781440645293E-006 1.96 4.208948155767745E-005 1.83
$P^2$ 5 4.371291470422284E-005 - 2.205191157852509E-004 -
10 4.737332465461207E-006 3.21 5.279340967235548E-005 2.06
20 5.836666419066179E-007 3.02 6.846287374035127E-006 2.95
40 1.451353937788389E-007 2.01 7.721987122251807E-007 3.14
$\alpha=0.5$ $P^0$ 5 1.511832756597273E-003 - 3.502121979914235E-003 -
10 6.220893257475209E-004 1.28 1.670078148163840E-003 1.06
20 2.916417957441209E-004 1.09 8.186795059360371E-004 1.03
40 1.431115461417255E-004 1.03 4.069902921658920E-004 1.01
$P^1$ 5 3.241445818441831E-004 - 1.031010873877333E-003 -
10 9.640915416227526E-005 1.75 4.663237471849166E-004 1.14
20 2.638161337326710E-005 1.87 1.496019341579287E-004 1.64
40 6.783621296347129E-006 1.96 4.193227445160367E-005 1.83
$P^2$ 5 4.370398829079597E-005 - 2.206503138086216E-004 -
10 4.735605756347349E-006 3.21 5.295075267031514E-005 2.06
20 5.708884776768829E-007 3.05 7.003847236448262E-006 2.92
40 7.947531188294052E-008 2.84 9.294866692183973E-007 2.91
$\alpha=0.7$ $P^0$ 5 1.505710253623656E-003 - 3.492298886676585E-003 -
10 6.213745175059577E-004 1.28 1.669866295206499E-003 1.06
20 2.915517673954438E-004 1.09 8.189868083813912E-004 1.03
40 1.431015120142753E-004 1.03 4.073153866903747E-004 1.01
$P^1$ 5 3.240154726586564E-004 - 1.030637501521249E-003 -
10 9.639741117415519E-005 1.75 4.660275194837704E-004 1.15
20 2.638293775032076E-005 1.87 1.493034254027140E-004 1.64
40 6.792517114587242E-006 1.96 4.163370092550497E-005 1.84
$P^2$ 5 4.369666230527429E-005 - 2.209276622101592E-004 -
10 4.748293818147065E-006 3.20 5.325073483141655E-005 2.05
20 6.692039201471664E-007 2.83 7.303070295557957E-006 2.87
40 8.581080963134642E-008 2.96 9.228188670033924E-007 2.98
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$\delta$ $\alpha$ $P^k$ $N$ $L^2$-error order $L^\infty$-error order
$\delta=0.2$ $\alpha=0.3$ $P^0$ 5 1.517845954000122E-003 - 3.511606734885603E-003 -
10 6.227976716020462E-004 1.28 1.670434180879555E-003 1.07
20 2.917321657674542E-004 1.09 8.185201216153321E-004 1.02
40 1.431224670285340E-004 1.02 4.068120471581059E-004 1.01
$P^1$ 5 3.242733095936642E-004 - 1.031247407943849E-003 -
10 9.642143896521544E-005 1.75 4.664783993685712E-004 1.14
20 2.638283335064153E-005 1.87 1.497589900759262E-004 1.64
40 6.784781440645293E-006 1.96 4.208948155767745E-005 1.83
$P^2$ 5 4.371291470422284E-005 - 2.205191157852509E-004 -
10 4.737332465461207E-006 3.21 5.279340967235548E-005 2.06
20 5.836666419066179E-007 3.02 6.846287374035127E-006 2.95
40 1.451353937788389E-007 2.01 7.721987122251807E-007 3.14
$\alpha=0.5$ $P^0$ 5 1.511832756597273E-003 - 3.502121979914235E-003 -
10 6.220893257475209E-004 1.28 1.670078148163840E-003 1.06
20 2.916417957441209E-004 1.09 8.186795059360371E-004 1.03
40 1.431115461417255E-004 1.03 4.069902921658920E-004 1.01
$P^1$ 5 3.241445818441831E-004 - 1.031010873877333E-003 -
10 9.640915416227526E-005 1.75 4.663237471849166E-004 1.14
20 2.638161337326710E-005 1.87 1.496019341579287E-004 1.64
40 6.783621296347129E-006 1.96 4.193227445160367E-005 1.83
$P^2$ 5 4.370398829079597E-005 - 2.206503138086216E-004 -
10 4.735605756347349E-006 3.21 5.295075267031514E-005 2.06
20 5.708884776768829E-007 3.05 7.003847236448262E-006 2.92
40 7.947531188294052E-008 2.84 9.294866692183973E-007 2.91
$\alpha=0.7$ $P^0$ 5 1.505710253623656E-003 - 3.492298886676585E-003 -
10 6.213745175059577E-004 1.28 1.669866295206499E-003 1.06
20 2.915517673954438E-004 1.09 8.189868083813912E-004 1.03
40 1.431015120142753E-004 1.03 4.073153866903747E-004 1.01
$P^1$ 5 3.240154726586564E-004 - 1.030637501521249E-003 -
10 9.639741117415519E-005 1.75 4.660275194837704E-004 1.15
20 2.638293775032076E-005 1.87 1.493034254027140E-004 1.64
40 6.792517114587242E-006 1.96 4.163370092550497E-005 1.84
$P^2$ 5 4.369666230527429E-005 - 2.209276622101592E-004 -
10 4.748293818147065E-006 3.20 5.325073483141655E-005 2.05
20 6.692039201471664E-007 2.83 7.303070295557957E-006 2.87
40 8.581080963134642E-008 2.96 9.228188670033924E-007 2.98
Table 9.  Spatial accuracy test on nonuniform meshes with generalized alternating numerical fluxes when $\rho = 1, \delta = 0.6$ $\gamma = 2, M = 10^3, T = 1$
$\delta$ $\alpha$ $P^k$ $N$ $L^2$-error order $L^\infty$-error order
$\delta=0.6$ $\alpha=0.3$ $P^0$ 5 2.218435560209597E-003 - 4.988165866475595E-003 -
10 7.026515160120938E-004 1.66 1.615537336821858E-003 1.63
20 3.238249412680320E-004 1.12 9.344015225186842E-004 0.79
40 1.711992658914959E-004 0.92 5.144905069215206E-004 0.86
$P^1$ 5 3.213669677505128E-004 - 6.232490424656974E-004 -
10 1.281866059118811E-004 1.33 3.003107660959630E-004 1.05
20 5.143385599212686E-005 1.32 1.603700097095964E-004 0.91
40 1.645991800902195E-005 1.64 5.811184200627087E-005 1.46
$P^2$ 5 5.667492210358004E-005 - 3.505324110815492E-004 -
10 5.578235035051399E-006 3.17 4.414888437790894E-005 2.99
20 7.175666478549729E-007 3.26 5.738400869813071E-006 2.94
40 1.571281022897026E-007 2.19 8.425543243264881E-007 2.77
$\alpha=0.5$ $P^0$ 5 2.200335979338480E-003 - 4.956964974689847E-003 -
10 7.011795560405282E-004 1.64 1.611200638415444E-003 1.62
20 3.235547410371371E-004 1.12 9.339361814586587E-004 0.79
40 1.711432886897817E-004 0.92 5.145417733177721E-004 0.86
$P^1$ 5 3.208864422126183E-004 - 6.227158645046031E-004 -
10 1.280994719241301E-004 1.32 3.001105667075760E-004 1.05
20 5.142050916436299E-005 1.32 1.602085136691192E-004 0.91
40 1.645805875671424E-005 1.64 5.795371024010622E-005 1.47
$P^2$ 5 5.664565316990849E-005 - 3.502258146088582E-004 -
10 5.576080115885499E-006 3.34 4.398437752902102E-005 2.99
20 7.071821450175616E-007 2.98 5.873580067984358E-006 2.90
40 9.970460744355951E-008 2.83 8.509068596339163E-007 2.79
$\alpha=0.7$ $P^0$ 5 2.181711723758149E-003 - 4.924677437389718E-003 -
10 6.996904701082016E-004 1.63 1.606649219131593E-003 1.62
20 3.232837185701581E-004 1.11 9.336145912944857E-004 0.78
40 1.710880587147894E-004 0.92 5.147405087257530E-004 0.86
$P^1$ 5 3.203996862436910E-004 - 6.223220958377807E-004 -
10 1.280113679639955E-004 1.32 2.997662559044798E-004 1.05
20 5.140813415494180E-005 1.32 1.599050123649873E-004 0.91
40 1.646026145542672E-005 1.64 5.765402843328131E-005 1.47
$P^2$ 5 5.661762164027293E-005 - 3.497734484754480E-004 -
10 5.586181252970688E-006 3.34 4.367727087437284E-005 3.00
20 7.886608314227983E-007 2.82 6.172794085018479E-006 2.82
40 9.631333432841155E-008 3.03 1.149606829718052E-006 2.42
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$\delta$ $\alpha$ $P^k$ $N$ $L^2$-error order $L^\infty$-error order
$\delta=0.6$ $\alpha=0.3$ $P^0$ 5 2.218435560209597E-003 - 4.988165866475595E-003 -
10 7.026515160120938E-004 1.66 1.615537336821858E-003 1.63
20 3.238249412680320E-004 1.12 9.344015225186842E-004 0.79
40 1.711992658914959E-004 0.92 5.144905069215206E-004 0.86
$P^1$ 5 3.213669677505128E-004 - 6.232490424656974E-004 -
10 1.281866059118811E-004 1.33 3.003107660959630E-004 1.05
20 5.143385599212686E-005 1.32 1.603700097095964E-004 0.91
40 1.645991800902195E-005 1.64 5.811184200627087E-005 1.46
$P^2$ 5 5.667492210358004E-005 - 3.505324110815492E-004 -
10 5.578235035051399E-006 3.17 4.414888437790894E-005 2.99
20 7.175666478549729E-007 3.26 5.738400869813071E-006 2.94
40 1.571281022897026E-007 2.19 8.425543243264881E-007 2.77
$\alpha=0.5$ $P^0$ 5 2.200335979338480E-003 - 4.956964974689847E-003 -
10 7.011795560405282E-004 1.64 1.611200638415444E-003 1.62
20 3.235547410371371E-004 1.12 9.339361814586587E-004 0.79
40 1.711432886897817E-004 0.92 5.145417733177721E-004 0.86
$P^1$ 5 3.208864422126183E-004 - 6.227158645046031E-004 -
10 1.280994719241301E-004 1.32 3.001105667075760E-004 1.05
20 5.142050916436299E-005 1.32 1.602085136691192E-004 0.91
40 1.645805875671424E-005 1.64 5.795371024010622E-005 1.47
$P^2$ 5 5.664565316990849E-005 - 3.502258146088582E-004 -
10 5.576080115885499E-006 3.34 4.398437752902102E-005 2.99
20 7.071821450175616E-007 2.98 5.873580067984358E-006 2.90
40 9.970460744355951E-008 2.83 8.509068596339163E-007 2.79
$\alpha=0.7$ $P^0$ 5 2.181711723758149E-003 - 4.924677437389718E-003 -
10 6.996904701082016E-004 1.63 1.606649219131593E-003 1.62
20 3.232837185701581E-004 1.11 9.336145912944857E-004 0.78
40 1.710880587147894E-004 0.92 5.147405087257530E-004 0.86
$P^1$ 5 3.203996862436910E-004 - 6.223220958377807E-004 -
10 1.280113679639955E-004 1.32 2.997662559044798E-004 1.05
20 5.140813415494180E-005 1.32 1.599050123649873E-004 0.91
40 1.646026145542672E-005 1.64 5.765402843328131E-005 1.47
$P^2$ 5 5.661762164027293E-005 - 3.497734484754480E-004 -
10 5.586181252970688E-006 3.34 4.367727087437284E-005 3.00
20 7.886608314227983E-007 2.82 6.172794085018479E-006 2.82
40 9.631333432841155E-008 3.03 1.149606829718052E-006 2.42
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