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June  2021, 16(2): 187-219. doi: 10.3934/nhm.2021004

## A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function

 1 CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile 2 Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 78035 Versailles, France 3 GIMNAP-Departamento de Matemáticas, Universidad del Bío-Bío, Concepción, Chile, CI2MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile

* Corresponding author: R. Ordoñez

Date: January 11, 2021.

Received  October 2020 Revised  December 2020 Published  June 2021 Early access  January 2021

The well-known Lighthill-Whitham-Richards (LWR) kinematic model of traffic flow models the evolution of the local density of cars by a nonlinear scalar conservation law. The transition between free and congested flow regimes can be described by a flux or velocity function that has a discontinuity at a determined density. A numerical scheme to handle the resulting LWR model with discontinuous velocity was proposed in [J.D. Towers, A splitting algorithm for LWR traffic models with flux discontinuities in the unknown, J. Comput. Phys., 421 (2020), article 109722]. A similar scheme is constructed by decomposing the discontinuous velocity function into a Lipschitz continuous function plus a Heaviside function and designing a corresponding splitting scheme. The part of the scheme related to the discontinuous flux is handled by a semi-implicit step that does, however, not involve the solution of systems of linear or nonlinear equations. It is proved that the whole scheme converges to a weak solution in the scalar case. The scheme can in a straightforward manner be extended to the multiclass LWR (MCLWR) model, which is defined by a hyperbolic system of $N$ conservation laws for $N$ driver classes that are distinguished by their preferential velocities. It is shown that the multiclass scheme satisfies an invariant region principle, that is, all densities are nonnegative and their sum does not exceed a maximum value. In the scalar and multiclass cases no flux regularization or Riemann solver is involved, and the CFL condition is not more restrictive than for an explicit scheme for the continuous part of the flux. Numerical tests for the scalar and multiclass cases are presented.

Citation: Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada. A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks and Heterogeneous Media, 2021, 16 (2) : 187-219. doi: 10.3934/nhm.2021004
##### References:
 [1] S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, Eur. J. Appl. Math., 14 (2003), 587-612.  doi: 10.1017/S0956792503005266. [2] S. Benzoni-Gavage, R. M. Colombo and P. Gwiazda, Measure valued solutions to conservation laws motivated by traffic modelling, Proc. Royal Soc. A, 462 (2006), 1791-1803.  doi: 10.1098/rspa.2005.1649. [3] M. Bulíček, P. Gwiazda, J. Málek and A. Świerczewska-Gwiazda, On scalar hyperbolic conservation laws with a discontinuous flux, Math. Models Methods Appl. Sci., 21 (2011), 89-113.  doi: 10.1142/S021820251100499X. [4] M. Bulíček, P. Gwiazda and A. Świerczewska-Gwiazda, Multi-dimensional scalar conservation laws with fluxes discontinuous in the unknown and the spatial variable, Math. Models Methods Appl. Sci., 23 (2013), 407-439.  doi: 10.1142/S0218202512500510. [5] R. Bürger, C. Chalons and L. M. Villada, Anti-diffusive and random-sampling Lagrangian-remap schemes for the multi-class Lighthill-Whitham-Richards traffic model, SIAM J. Sci. Comput., 35 (2013), B1341–B1368. doi: 10.1137/130923877. [6] R. Bürger, A. García, K. H. Karlsen and J. D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Eng. Math., 60 (2008), 387-425.  doi: 10.1007/s10665-007-9148-4. [7] R. Bürger, A. García, K. H. Karlsen and J. D. Towers, Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model, Netw. Heterog. Media, 3 (2008), 1-41.  doi: 10.3934/nhm.2008.3.1. [8] R. Bürger, K. H. Karlsen, H. Torres and J. D. Towers, Second-order schemes for conservation laws with discontinuous flux modelling clarifier-thickener units, Numer. Math., 116 (2010), 579-617.  doi: 10.1007/s00211-010-0325-4. [9] R. Bürger, K. H. Karlsen and J. D. Towers, On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux, Netw. Heterog. Media, 5 (2010), 461-485.  doi: 10.3934/nhm.2010.5.461. [10] R. Bürger, H. Torres and C. A. Vega, An entropy stable scheme for the multiclass Lighthill-Whitham-Richards traffic model, Adv. Appl. Math. Mech., 11 (2019), 1022-1047.  doi: 10.4208/aamm.OA-2018-0189. [11] R. Bürger, P. Mulet and L. M. Villada, A diffusively corrected multiclass Lighthill-Whitham-Richards traffic model with anticipation lengths and reaction times, Adv. Appl. Math. Mech., 5 (2013), 728-758.  doi: 10.4208/aamm.2013.m135. [12] J. Carrillo, Conservation law with discontinuous flux function and boundary condition, J. Evol. Equ., 3 (2003), 283-301.  doi: 10.1007/s00028-003-0095-x. [13] C. Chalons and P. Goatin, Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling, Interf. Free Bound., 10 (2008), 197-221.  doi: 10.4171/IFB/186. [14] R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721.  doi: 10.1137/S0036139901393184. [15] J. P. Dias and M. Figueira, On the Riemann problem for some discontinuous systems of conservation laws describing phase transitions, Commun. Pure Appl. Anal., 3 (2004), 53-58.  doi: 10.3934/cpaa.2004.3.53. [16] J. P. Dias and M. Figueira, On the approximation of the solutions of the Riemann problem for a discontinuous conservation law, Bull. Braz. Math. Soc. New Ser., 36 (2005), 115-125.  doi: 10.1007/s00574-005-0031-5. [17] J. P. Dias, M. Figueira and J. F. Rodrigues, Solutions to a scalar discontinuous conservation law in a limit case of phase transitions, J. Math. Fluid Mech., 7 (2005), 153-163.  doi: 10.1007/s00021-004-0113-y. [18] S. Diehl, A conservation law with point source and discontinuous flux function, SIAM J. Math. Anal., 56 (1996), 388-419.  doi: 10.1137/S0036139994242425. [19] R. Donat and P. Mulet, Characteristic-based schemes for multi-class Lighthill-Whitham-Richards traffic models, J. Sci. Comput., 37 (2008), 233-250.  doi: 10.1007/s10915-008-9209-5. [20] R. Donat and P. Mulet, A secular equation for the Jacobian matrix of certain multi-species kinematic flow models, Numer. Methods Partial Differential Equations, 26 (2010), 159-175.  doi: 10.1002/num.20423. [21] T. Gimse, Conservation laws with discontinuous flux functions, SIAM J. Numer. Anal., 24 (1993), 279-289.  doi: 10.1137/0524018. [22] T. Gimse and N. H. Risebro, Solution to the Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. Math. Anal., 23 (1992), 635-648.  doi: 10.1137/0523032. [23] M. Hilliges and W. Weidlich, A phenomenological model for dynamic traffic flow in networks, Transp. Res. B, 29 (1995), 407-431.  doi: 10.1016/0191-2615(95)00018-9. [24] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2015. doi: 10.1007/978-3-662-47507-2. [25] M. J. Lighthill and G. B. Whitham, On kinematic waves: II. A theory of traffic flow on long crowded roads, Proc. Royal Soc. A, 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089. [26] Y. Lu, S. Wong, M. Zhang and C.-W. Shu, The entropy solutions for the Lighthill-Whitham-Richards traffic flow model with a discontinuous flow-density relationship, Transp. Sci., 43 (2009), 511-530. [27] S. Martin and J. Vovelle, Convergence of the finite volume method for scalar conservation law with discontinuous flux function, ESAIM Math. Model. Numer. Anal., 42 (2008), 699-727.  doi: 10.1051/m2an:2008023. [28] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42. [29] J. D. Towers, A splitting algorithm for LWR traffic models with flux discontinuities in the unknown, J. Comput. Phys., 421 (2020), 109722, 30 pp. doi: 10.1016/j.jcp.2020.109722. [30] J. K. Wiens, J. M. Stockie and J. F. Williams, Riemann solver for a kinematic wave traffic model with discontinuous flux, J. Comput. Phys., 242 (2013), 1-23.  doi: 10.1016/j.jcp.2013.02.024. [31] G. C. K. Wong and S. C. Wong, A multi-class traffic flow model–-an extension of LWR model with heterogeneous drivers, Transp. Res. A, 36 (2002), 827-841.  doi: 10.1016/S0965-8564(01)00042-8. [32] P. Zhang, R. X. Liu, S. C. Wong and D. Q. Dai, Hyperbolicity and kinematic waves of a class of multi-population partial differential equations, Eur. J. Appl. Math., 17 (2006), 171-200.  doi: 10.1017/S095679250500642X. [33] M. Zhang, C.-W. Shu, G. C. K. Wong and S. C. Wong, A weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill-Whitham-Richards traffic flow model, J. Comput. Phys., 191 (2003), 639-659.  doi: 10.1016/j.jcp.2005.07.019. [34] P. Zhang, S. C. Wong and C.-W. Shu, A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway, J. Comput. Phys., 212 (2006), 739-756.  doi: 10.1016/j.jcp.2005.07.019. [35] P. Zhang, R.-X. Liu, S. C. Wong and S. Q. Dai, Hyperbolicity and kinematic waves of a class of multi-population partial differential equations, Eur. J. Appl. Math., 17 (2006), 171-200.  doi: 10.1017/S095679250500642X. [36] P. Zhang, S. C. Wong and S. Q. Dai, A note on the weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill-Whitham-Richards traffic flow model, Commun. Numer. Meth. Eng., 25 (2009), 1120-1126.  doi: 10.1002/cnm.1277. [37] P. Zhang, S. C. Wong and Z. Xu, A hybrid scheme for solving a multi-class traffic flow model with complex wave breaking, Comput. Methods Appl. Mech. Engrg., 197 (2008), 3816-3827.  doi: 10.1016/j.cma.2008.03.003.

show all references

##### References:
 [1] S. Benzoni-Gavage and R. M. Colombo, An $n$-populations model for traffic flow, Eur. J. Appl. Math., 14 (2003), 587-612.  doi: 10.1017/S0956792503005266. [2] S. Benzoni-Gavage, R. M. Colombo and P. Gwiazda, Measure valued solutions to conservation laws motivated by traffic modelling, Proc. Royal Soc. A, 462 (2006), 1791-1803.  doi: 10.1098/rspa.2005.1649. [3] M. Bulíček, P. Gwiazda, J. Málek and A. Świerczewska-Gwiazda, On scalar hyperbolic conservation laws with a discontinuous flux, Math. Models Methods Appl. Sci., 21 (2011), 89-113.  doi: 10.1142/S021820251100499X. [4] M. Bulíček, P. Gwiazda and A. Świerczewska-Gwiazda, Multi-dimensional scalar conservation laws with fluxes discontinuous in the unknown and the spatial variable, Math. Models Methods Appl. Sci., 23 (2013), 407-439.  doi: 10.1142/S0218202512500510. [5] R. Bürger, C. Chalons and L. M. Villada, Anti-diffusive and random-sampling Lagrangian-remap schemes for the multi-class Lighthill-Whitham-Richards traffic model, SIAM J. Sci. Comput., 35 (2013), B1341–B1368. doi: 10.1137/130923877. [6] R. Bürger, A. García, K. H. Karlsen and J. D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Eng. Math., 60 (2008), 387-425.  doi: 10.1007/s10665-007-9148-4. [7] R. Bürger, A. García, K. H. Karlsen and J. D. Towers, Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model, Netw. Heterog. Media, 3 (2008), 1-41.  doi: 10.3934/nhm.2008.3.1. [8] R. Bürger, K. H. Karlsen, H. Torres and J. D. Towers, Second-order schemes for conservation laws with discontinuous flux modelling clarifier-thickener units, Numer. Math., 116 (2010), 579-617.  doi: 10.1007/s00211-010-0325-4. [9] R. Bürger, K. H. Karlsen and J. D. Towers, On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux, Netw. Heterog. Media, 5 (2010), 461-485.  doi: 10.3934/nhm.2010.5.461. [10] R. Bürger, H. Torres and C. A. Vega, An entropy stable scheme for the multiclass Lighthill-Whitham-Richards traffic model, Adv. Appl. Math. Mech., 11 (2019), 1022-1047.  doi: 10.4208/aamm.OA-2018-0189. [11] R. Bürger, P. Mulet and L. M. Villada, A diffusively corrected multiclass Lighthill-Whitham-Richards traffic model with anticipation lengths and reaction times, Adv. Appl. Math. Mech., 5 (2013), 728-758.  doi: 10.4208/aamm.2013.m135. [12] J. Carrillo, Conservation law with discontinuous flux function and boundary condition, J. Evol. Equ., 3 (2003), 283-301.  doi: 10.1007/s00028-003-0095-x. [13] C. Chalons and P. Goatin, Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling, Interf. Free Bound., 10 (2008), 197-221.  doi: 10.4171/IFB/186. [14] R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721.  doi: 10.1137/S0036139901393184. [15] J. P. Dias and M. Figueira, On the Riemann problem for some discontinuous systems of conservation laws describing phase transitions, Commun. Pure Appl. Anal., 3 (2004), 53-58.  doi: 10.3934/cpaa.2004.3.53. [16] J. P. Dias and M. Figueira, On the approximation of the solutions of the Riemann problem for a discontinuous conservation law, Bull. Braz. Math. Soc. New Ser., 36 (2005), 115-125.  doi: 10.1007/s00574-005-0031-5. [17] J. P. Dias, M. Figueira and J. F. Rodrigues, Solutions to a scalar discontinuous conservation law in a limit case of phase transitions, J. Math. Fluid Mech., 7 (2005), 153-163.  doi: 10.1007/s00021-004-0113-y. [18] S. Diehl, A conservation law with point source and discontinuous flux function, SIAM J. Math. Anal., 56 (1996), 388-419.  doi: 10.1137/S0036139994242425. [19] R. Donat and P. Mulet, Characteristic-based schemes for multi-class Lighthill-Whitham-Richards traffic models, J. Sci. Comput., 37 (2008), 233-250.  doi: 10.1007/s10915-008-9209-5. [20] R. Donat and P. Mulet, A secular equation for the Jacobian matrix of certain multi-species kinematic flow models, Numer. Methods Partial Differential Equations, 26 (2010), 159-175.  doi: 10.1002/num.20423. [21] T. Gimse, Conservation laws with discontinuous flux functions, SIAM J. Numer. Anal., 24 (1993), 279-289.  doi: 10.1137/0524018. [22] T. Gimse and N. H. Risebro, Solution to the Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. Math. Anal., 23 (1992), 635-648.  doi: 10.1137/0523032. [23] M. Hilliges and W. Weidlich, A phenomenological model for dynamic traffic flow in networks, Transp. Res. B, 29 (1995), 407-431.  doi: 10.1016/0191-2615(95)00018-9. [24] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2015. doi: 10.1007/978-3-662-47507-2. [25] M. J. Lighthill and G. B. Whitham, On kinematic waves: II. A theory of traffic flow on long crowded roads, Proc. Royal Soc. A, 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089. [26] Y. Lu, S. Wong, M. Zhang and C.-W. Shu, The entropy solutions for the Lighthill-Whitham-Richards traffic flow model with a discontinuous flow-density relationship, Transp. Sci., 43 (2009), 511-530. [27] S. Martin and J. Vovelle, Convergence of the finite volume method for scalar conservation law with discontinuous flux function, ESAIM Math. Model. Numer. Anal., 42 (2008), 699-727.  doi: 10.1051/m2an:2008023. [28] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42. [29] J. D. Towers, A splitting algorithm for LWR traffic models with flux discontinuities in the unknown, J. Comput. Phys., 421 (2020), 109722, 30 pp. doi: 10.1016/j.jcp.2020.109722. [30] J. K. Wiens, J. M. Stockie and J. F. Williams, Riemann solver for a kinematic wave traffic model with discontinuous flux, J. Comput. Phys., 242 (2013), 1-23.  doi: 10.1016/j.jcp.2013.02.024. [31] G. C. K. Wong and S. C. Wong, A multi-class traffic flow model–-an extension of LWR model with heterogeneous drivers, Transp. Res. A, 36 (2002), 827-841.  doi: 10.1016/S0965-8564(01)00042-8. [32] P. Zhang, R. X. Liu, S. C. Wong and D. Q. Dai, Hyperbolicity and kinematic waves of a class of multi-population partial differential equations, Eur. J. Appl. Math., 17 (2006), 171-200.  doi: 10.1017/S095679250500642X. [33] M. Zhang, C.-W. Shu, G. C. K. Wong and S. C. Wong, A weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill-Whitham-Richards traffic flow model, J. Comput. Phys., 191 (2003), 639-659.  doi: 10.1016/j.jcp.2005.07.019. [34] P. Zhang, S. C. Wong and C.-W. Shu, A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway, J. Comput. Phys., 212 (2006), 739-756.  doi: 10.1016/j.jcp.2005.07.019. [35] P. Zhang, R.-X. Liu, S. C. Wong and S. Q. Dai, Hyperbolicity and kinematic waves of a class of multi-population partial differential equations, Eur. J. Appl. Math., 17 (2006), 171-200.  doi: 10.1017/S095679250500642X. [36] P. Zhang, S. C. Wong and S. Q. Dai, A note on the weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill-Whitham-Richards traffic flow model, Commun. Numer. Meth. Eng., 25 (2009), 1120-1126.  doi: 10.1002/cnm.1277. [37] P. Zhang, S. C. Wong and Z. Xu, A hybrid scheme for solving a multi-class traffic flow model with complex wave breaking, Comput. Methods Appl. Mech. Engrg., 197 (2008), 3816-3827.  doi: 10.1016/j.cma.2008.03.003.
(a) Piecewise continuous velocity function $V(\phi)$ with discontinuity at $\phi = \phi^*$, (b) continuous and discontinuous portions $p_V(\phi)$ (solid line) and $g_V(\phi)$ (dashed line)
(a) function $z \mapsto \smash{\tilde{G}_V}(z;\phi)$ given by (2.9a) with $\lambda v^{\max} = 1/2$, $\alpha_V = 0.3$, and $\phi = 0.8$, (b) its inverse $z \mapsto \smash{\tilde{G}_V^{-1}} (z;\phi)$ given by (2.9b)
Example 1: numerical solution with $M = 800$ and comparison with the exact solution of the Riemann problem (a) with $\phi_{\mathrm{L}} = 0.3$ and $\phi_{\mathrm{R}} = 0.9$ at simulated time $T = 1.8$, (b) with $\phi_{\mathrm{L}} = 0.9$ and $\phi_{\mathrm{R}} = 0.3$ at simulated time $T = 1.5$. Here and in Figures 4 and 5 we label with 'Towers scheme' the scheme (1.7) proposed in [29] and by 'BCOV scheme' the scheme of Algorithm 2.1 advanced in the present work
Example 2: numerical solutions for $M = 100$ at simulated times (a) $T = 0.1$, (b) $T = 0.3$
Example 3: numerical solutions depending on the boundary conditions $\mathcal{F}(t)\in\smash{\tilde{f}}(\phi^*)$ with $M = 1600$ at simulated time $T = 0.5$, with (a) $\smash{\mathcal{F}(t)\in\tilde{f}}(\phi^*-)$ (free flow), (b) $\mathcal{F}(t)\in\smash{\tilde{f}} (\phi^*+)$ (congested flow)
Example 4: density profiles simulated with $M = 1600$ at (a) $T = 0.2$, (b) $T = 0.4$, (c) $T = 0.6$
Example 5: numerical solution for a free-flow regime ($\mathcal{G}(t) = \alpha_V$): (a) initial condition, (b, c) density profiles with $M = 1600$ at simulated times (b) $T = 0.1,$ (c) $T = 0.2$
Example 5: simulated total density computed with BCOV scheme with $N = 3$ and $M = 1600$: (a) free flow ($\mathcal{G}(t) = \alpha_V$), (b) congested flow ($\mathcal{G}(t) = 0$)
Example 5: numerical solution for a congested flow regime ($\mathcal{G}(t) = 0$): density profiles with $M = 1600$ at simulated times (a) $T = 0.1,$ (b) $T = 0.2$. The initial condition is the same as in Figure 7(a)
Example 6: numerical solutions obtained with BCOV scheme with $N = 5$ and $M = 1600$ at simulated times (a) $T = 0.02$, (b) $T = 0.12$
Example 6: simulated total density obtained with BCOV scheme with $N = 5$ and $M = 1600$: (a) discontinuous problem, (b) continuous problem
Example 6: comparison of reference solution ($M_{\text{ref}} = 12800$) with approximate solutions computed by BCOV scheme with $M = 100$ at simulated time $T = 0.02$
Example 6: comparison of reference solution ($M_{\text{ref}} = 12800$) with approximate solutions computed by BCOV scheme with $M = 100$ at simulated time $T = 0.02$
Example 7: numerical solution computed with BCOV scheme with $N = 5$ and $M = 12800$ at simulated times (a) $T = 0.1$, (b) $T = 0.2$ and (c) $T = 0.3$
Example 7: simulated total density computed with BCOV scheme with $N = 5$ and $M = 1600$
Example 2: approximate $L^1$ errors $e_{M}(u)$ with $\Delta x = 2/M$
 $T=0.1$ $T=0.3$ Towers BCOV Towers BCOV $M$ $e_{M}(\phi^{\Delta})$ $e_{M}(\phi^{\Delta})$ $e_{M}(\phi^{\Delta})$ $e_{M}(\phi^{\Delta})$ 100 1.32e-2 1.76e-2 1.63e-2 2.39e-2 200 6.55e-3 9.22e-3 8.59e-3 1.31e-2 400 3.29e-3 4.46e-3 4.25e-3 6.46e-3 800 1.72e-3 2.403-3 2.12e-3 3.31e-3 1600 8.00e-4 1.18e-3 9.29e-4 1.563-3
 $T=0.1$ $T=0.3$ Towers BCOV Towers BCOV $M$ $e_{M}(\phi^{\Delta})$ $e_{M}(\phi^{\Delta})$ $e_{M}(\phi^{\Delta})$ $e_{M}(\phi^{\Delta})$ 100 1.32e-2 1.76e-2 1.63e-2 2.39e-2 200 6.55e-3 9.22e-3 8.59e-3 1.31e-2 400 3.29e-3 4.46e-3 4.25e-3 6.46e-3 800 1.72e-3 2.403-3 2.12e-3 3.31e-3 1600 8.00e-4 1.18e-3 9.29e-4 1.563-3
Example 6: approximate $L^1$ errors $e_{M}(u)$ with $\Delta x = 2/M$
 $T=0.02$ $T=0.12$ $M$ $e_{M}(\phi^{\Delta})$ $e_{M}(\phi^{\Delta})$ 100 1.39e-2 3.87e-2 200 7.90e-3 2.47e-2 400 4.20e-3 1.55e-2 800 2.00e-3 9.20e-3 1600 1.00e-3 5.10e-3
 $T=0.02$ $T=0.12$ $M$ $e_{M}(\phi^{\Delta})$ $e_{M}(\phi^{\Delta})$ 100 1.39e-2 3.87e-2 200 7.90e-3 2.47e-2 400 4.20e-3 1.55e-2 800 2.00e-3 9.20e-3 1600 1.00e-3 5.10e-3
Example 7: Approximate $L^1$ errors $e_{M}(u)$ with $\Delta x = 5/M$
 $T=0.1$ $T=0.2$ $T=0.3$ $M$ $e_{M}(\phi^{\Delta})$ $e_{M}(\phi^{\Delta})$ $e_{M}(\phi^{\Delta})$ 100 7.42e-2 9.50e-2 1.06e-1 200 4.12e-2 5.50e-2 6.49e-2 400 2.27e-2 3.34e-2 3.88e-2 800 1.24e-2 1.97-2 2.35e-2 1600 6.50e-3 1.10e-2 1.35e-2
 $T=0.1$ $T=0.2$ $T=0.3$ $M$ $e_{M}(\phi^{\Delta})$ $e_{M}(\phi^{\Delta})$ $e_{M}(\phi^{\Delta})$ 100 7.42e-2 9.50e-2 1.06e-1 200 4.12e-2 5.50e-2 6.49e-2 400 2.27e-2 3.34e-2 3.88e-2 800 1.24e-2 1.97-2 2.35e-2 1600 6.50e-3 1.10e-2 1.35e-2
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