American Institute of Mathematical Sciences

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

 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  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.

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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020319
References:
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Lett., 35 (2008), 190201. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [29] J. 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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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] C. Li and F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Boca Raton, FL, 2015.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] F. W. Liu, P. H. Zhuang and Q. X. Liu, The Applications and Numerical Methods of Fractional Differential Equations, Science Press, Beijing, 2015.   Google Scholar [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.  Google Scholar [24] M. M. Meerschaert, Y. Zhang and B. Baeumer, Tempered anomalous diffusion in heterogeneous systems, Geophys., Res. Lett., 35 (2008), 190201. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [29] J. Rosinski, Tempering stable processes, Stochastic Process. Appl., 117 (2007), 677-707.  doi: 10.1016/j.spa.2006.10.003.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar
The evolution of the solution for $\alpha = 0.3$. $\tau = 0.001, h = 0.01, k = 2.$
The evolution of the solution for $\alpha = 0.8$. $\tau = 0.001, h = 0.01, k = 2.$
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
 $\delta$ $\alpha$ $P^k$ $N$ $L^2$-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
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
 $\delta$ $\alpha$ $P^k$ $N$ $L^2$-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
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
 $\delta$ $\alpha$ $P^k$ $N$ $L^2$-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
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
 $\delta$ $\alpha$ $P^k$ $N$ $L^2$-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
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
 $\delta$ $\alpha$ $\tau$ $L^2$-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
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
 $\delta$ $\alpha$ $P^k$ $N$ $L^2$-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
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
 $\delta$ $\alpha$ $P^k$ $N$ $L^2$-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
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
 $\delta$ $\alpha$ $P^k$ $N$ $L^2$-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
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
 $\delta$ $\alpha$ $P^k$ $N$ $L^2$-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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