# American Institute of Mathematical Sciences

## Krylov implicit integration factor WENO method for SIR model with directed diffusion

 1 Department of Mathematics and Statistics, Minnesota State University, Mankato, Mankato, MN 56001, USA 2 Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA 3 Department of Mathematical Sciences, Indiana University South Bend, South Bend, IN 46615, USA

* Corresponding author: Ruijun Zhao

Received  June 2018 Revised  September 2018 Published  February 2019

Fund Project: The second author is supported by NSF grant DMS-1620108

SIR models with directed diffusions are important in describing the population movement. However, efficient numerical simulations of such systems of fully nonlinear second order partial differential equations (PDEs) are challenging. They are often mixed type PDEs with ill-posed or degenerate components. The solutions may develop singularities along with time evolution. Stiffness due to nonlinear diffusions in the system gives strict constraints in time step sizes for numerical methods. In this paper, we design efficient Krylov implicit integration factor (IIF) Weighted Essentially Non-Oscillatory (WENO) method to solve SIR models with directed diffusions. Numerical experiments are performed to show the good accuracy and stability of the method. Singularities in the solutions are resolved stably and sharply by the WENO approximations in the scheme. Unlike a usual implicit method for solving stiff nonlinear PDEs, the Krylov IIF WENO method avoids solving large coupled nonlinear algebraic systems at every time step. Large time step size computations are achieved for solving the fully nonlinear second-order PDEs, namely, the time step size is proportional to the spatial grid size as that for solving a pure hyperbolic PDE. Two biologically interesting cases are simulated by the developed scheme to study the finite-time blow-up time and location or discontinuity locations in the solution of the SIR model.

Citation: Ruijun Zhao, Yong-Tao Zhang, Shanqin Chen. Krylov implicit integration factor WENO method for SIR model with directed diffusion. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019041
##### References:
 [1] L. Allen, B. Bolker, Y. Lou and A. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete and Continuous Dynamical Systems Series B, 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1. [2] S. Berres and R. Ruiz-Baier, A fully adaptive numerical approximation for a two-dimensional epidemic model with nonlinear cross-diffusion, Nonlinear Analysis: Real World Applications, 12 (2011), 2888-2903. doi: 10.1016/j.nonrwa.2011.04.014. [3] M. Bertsch and M. E. Gurtin, On predator-prey dispersal, repulsive dispersal, and the presence of shock waves, Quarterly of Applied Mathematics, 44 (1986), 339-351. doi: 10.1090/qam/856189. [4] M. Bertsch, M. E. Gurtin, D. Hilhorst and L. Peletier, On interacting populations that disperse to avoid crowding: Preservation of segregation, J. Math. Biology, 23 (1985), 1-13. doi: 10.1007/BF00276555. [5] M. Bertsch, M. E. Gurtin, D. Hilhorst and L. A. Peletier, On interacting populations that disperse to avoid crowding: The effect of a sedentary colony, Journal of Mathematical Biology, 19 (1984), 1-12. doi: 10.1007/BF00275928. [6] E. A. Carl, Population control in arctic ground squirrels, Ecology, 52 (1971), 395-413. doi: 10.2307/1937623. [7] L. Chang and Z. Jin, Efficient numerical methods for spatially extended population and epidemic models with time delay, Applied Mathematics and Computation, 316 (2018), 138-154. doi: 10.1016/j.amc.2017.08.028. [8] S. Chen and Y.-T. Zhang, Krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes: application to discontinuous Galerkin methods, Journal of Computational Physics, 230 (2011), 4336-4352. doi: 10.1016/j.jcp.2011.01.010. [9] M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Mathematical Biosciences, 33 (1977), 35-49. doi: 10.1016/0025-5564(77)90062-1. [10] C. Hirota and K. Ozawa, Numerical method of estimating the blow-up time and rate of the solution of ordinary differential equations–-an application to the blow-up problems of partial differential equations, Journal of Computational and Applied Mathematics, 193 (2006), 614-637. doi: 10.1016/j.cam.2005.04.069. [11] T. Jiang and Y.-T. Zhang, Krylov implicit integration factor WENO methods for semi-linear and fully nonlinear advection-diffusion-reaction equations, Journal of Computational Physics, 253 (2013), 368-388. doi: 10.1016/j.jcp.2013.07.015. [12] F. Li and N. K. Yip, Long time behavior of some epidemic models, Discrete and Continuous Dynamical Systems Series B, 16 (2011), 867-881. doi: 10.3934/dcdsb.2011.16.867. [13] D. Lu and Y.-T. Zhang, Krylov integration factor method on sparse grids for high spatial dimension convection-diffusion equations, Journal of Scientific Computing, 69 (2016), 736-763. doi: 10.1007/s10915-016-0216-7. [14] D. Lu and Y.-T. Zhang, Computational complexity study on Krylov integration factor WENO method for high spatial dimension convection-diffusion problems, Journal of Scientific Computing, 73 (2017), 980-1027. doi: 10.1007/s10915-017-0398-7. [15] R. C. MacCamy, Simple population models with diffusion, Maths with Appls, 9 (1983), 341-344. doi: 10.1016/0898-1221(83)90021-4. [16] D. B. Meade and F. A. Milner, An S-I-R model for epidemics with diffusion to avoid infection and overcrowding, Proceedings of the 13th IMACS World Congress on Computation and Applied Mathematics, 3 (1991), 1444-1445. [17] D. B. Meade and F. A. Milner, S-I-R epidemic models with directed diffusion., In G. D. Prato, editor, Mathematical Aspects of Human Diseases, 1992. [18] F. A. Milner and R. Zhao, S-I-R model with directed spatial diffusion, Mathematical Population Studies, 15 (2008), 160-181. doi: 10.1080/08898480802221889. [19] Q. Nie, Y.-T. Zhang and R. Zhao, Efficient semi-implicit schemes for stiff systems, Journal of Computational Physics, 214 (2006), 521-537. doi: 10.1016/j.jcp.2005.09.030. [20] Q. Nie, F. Wan, Y.-T. Zhang and X.-F. Liu, Compact integration factor methods in high spatial dimensions, Journal of Computational Physics, 227 (2008), 5238-5255. doi: 10.1016/j.jcp.2008.01.050. [21] S. Osher and C.-W. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal., 28 (1991), 907-922. doi: 10.1137/0728049. [22] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3. [23] C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor (Editor: A. Quarteroni), Lecture Notes in Mathematics, volume 1697, Springer, 1998, 325–432. doi: 10.1007/BFb0096355. [24] J. G. Skellam, Random dispersal in theoretical populations, Biometrika., 38 (1981), 196-218. doi: 10.1093/biomet/38.1-2.196. [25] G.-Q. Sun, Z. Jin, Q.-X. Liu and L. Li, Spatial pattern in an epidemic system with cross-diffusion of the susceptible, Journal of Biological Systems, 17 (2009), 141-152. doi: 10.1142/S0218339009002843. [26] D. Wang, W. Chen and Q. Nie, Semi-implicit integration factor methods on sparse grids for high-dimensional systems, Journal of Computational Physics, 292 (2015), 43-55. doi: 10.1016/j.jcp.2015.03.033. [27] G. F. Webb, A reaction-diffusion model for a deterministic diffusion epidemic, J. Math. Anal. Appl., 84 (1981), 150-161. doi: 10.1016/0022-247X(81)90156-6. [28] K. E. Yong, E. D. Herrera and and C. Castillo-Chavez, From bee species aggregation to models of disease avoidance: The ben-hur effect., Mathematical and Statistical Modeling for Emerging and Re-emerging Infectious Diseases, 12 (2016), 169-185. [29] Y.-T. Zhang and C.-W. Shu, High order WENO schemes for Hamilton-Jacobi equations on triangular meshes, SIAM Journal on Scientific Computing, 24 (2003), 1005-1030. doi: 10.1137/S1064827501396798. [30] Y.-T. Zhang and C.-W. Shu, ENO and WENO schemes, in Handbook of Numerical Analysis, Volume 17, Handbook of Numerical Methods for Hyperbolic Problems: Basic and Fundamental Issues, R. Abgrall and C.-W. Shu, Editors, North-Holland, Elsevier, Amsterdam, (2016), 103–122. [31] Y.-T. Zhang, H.-K. Zhao and J. Qian, High order fast sweeping methods for static Hamilton-Jacobi equations, Journal of Scientific Computing, 29 (2006), 25-56. doi: 10.1007/s10915-005-9014-3.

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##### References:
 [1] L. Allen, B. Bolker, Y. Lou and A. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete and Continuous Dynamical Systems Series B, 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1. [2] S. Berres and R. Ruiz-Baier, A fully adaptive numerical approximation for a two-dimensional epidemic model with nonlinear cross-diffusion, Nonlinear Analysis: Real World Applications, 12 (2011), 2888-2903. doi: 10.1016/j.nonrwa.2011.04.014. [3] M. Bertsch and M. E. Gurtin, On predator-prey dispersal, repulsive dispersal, and the presence of shock waves, Quarterly of Applied Mathematics, 44 (1986), 339-351. doi: 10.1090/qam/856189. [4] M. Bertsch, M. E. Gurtin, D. Hilhorst and L. Peletier, On interacting populations that disperse to avoid crowding: Preservation of segregation, J. Math. Biology, 23 (1985), 1-13. doi: 10.1007/BF00276555. [5] M. Bertsch, M. E. Gurtin, D. Hilhorst and L. A. Peletier, On interacting populations that disperse to avoid crowding: The effect of a sedentary colony, Journal of Mathematical Biology, 19 (1984), 1-12. doi: 10.1007/BF00275928. [6] E. A. Carl, Population control in arctic ground squirrels, Ecology, 52 (1971), 395-413. doi: 10.2307/1937623. [7] L. Chang and Z. Jin, Efficient numerical methods for spatially extended population and epidemic models with time delay, Applied Mathematics and Computation, 316 (2018), 138-154. doi: 10.1016/j.amc.2017.08.028. [8] S. Chen and Y.-T. Zhang, Krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes: application to discontinuous Galerkin methods, Journal of Computational Physics, 230 (2011), 4336-4352. doi: 10.1016/j.jcp.2011.01.010. [9] M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Mathematical Biosciences, 33 (1977), 35-49. doi: 10.1016/0025-5564(77)90062-1. [10] C. Hirota and K. Ozawa, Numerical method of estimating the blow-up time and rate of the solution of ordinary differential equations–-an application to the blow-up problems of partial differential equations, Journal of Computational and Applied Mathematics, 193 (2006), 614-637. doi: 10.1016/j.cam.2005.04.069. [11] T. Jiang and Y.-T. Zhang, Krylov implicit integration factor WENO methods for semi-linear and fully nonlinear advection-diffusion-reaction equations, Journal of Computational Physics, 253 (2013), 368-388. doi: 10.1016/j.jcp.2013.07.015. [12] F. Li and N. K. Yip, Long time behavior of some epidemic models, Discrete and Continuous Dynamical Systems Series B, 16 (2011), 867-881. doi: 10.3934/dcdsb.2011.16.867. [13] D. Lu and Y.-T. Zhang, Krylov integration factor method on sparse grids for high spatial dimension convection-diffusion equations, Journal of Scientific Computing, 69 (2016), 736-763. doi: 10.1007/s10915-016-0216-7. [14] D. Lu and Y.-T. Zhang, Computational complexity study on Krylov integration factor WENO method for high spatial dimension convection-diffusion problems, Journal of Scientific Computing, 73 (2017), 980-1027. doi: 10.1007/s10915-017-0398-7. [15] R. C. MacCamy, Simple population models with diffusion, Maths with Appls, 9 (1983), 341-344. doi: 10.1016/0898-1221(83)90021-4. [16] D. B. Meade and F. A. Milner, An S-I-R model for epidemics with diffusion to avoid infection and overcrowding, Proceedings of the 13th IMACS World Congress on Computation and Applied Mathematics, 3 (1991), 1444-1445. [17] D. B. Meade and F. A. Milner, S-I-R epidemic models with directed diffusion., In G. D. Prato, editor, Mathematical Aspects of Human Diseases, 1992. [18] F. A. Milner and R. Zhao, S-I-R model with directed spatial diffusion, Mathematical Population Studies, 15 (2008), 160-181. doi: 10.1080/08898480802221889. [19] Q. Nie, Y.-T. Zhang and R. Zhao, Efficient semi-implicit schemes for stiff systems, Journal of Computational Physics, 214 (2006), 521-537. doi: 10.1016/j.jcp.2005.09.030. [20] Q. Nie, F. Wan, Y.-T. Zhang and X.-F. Liu, Compact integration factor methods in high spatial dimensions, Journal of Computational Physics, 227 (2008), 5238-5255. doi: 10.1016/j.jcp.2008.01.050. [21] S. Osher and C.-W. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal., 28 (1991), 907-922. doi: 10.1137/0728049. [22] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3. [23] C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor (Editor: A. Quarteroni), Lecture Notes in Mathematics, volume 1697, Springer, 1998, 325–432. doi: 10.1007/BFb0096355. [24] J. G. Skellam, Random dispersal in theoretical populations, Biometrika., 38 (1981), 196-218. doi: 10.1093/biomet/38.1-2.196. [25] G.-Q. Sun, Z. Jin, Q.-X. Liu and L. Li, Spatial pattern in an epidemic system with cross-diffusion of the susceptible, Journal of Biological Systems, 17 (2009), 141-152. doi: 10.1142/S0218339009002843. [26] D. Wang, W. Chen and Q. Nie, Semi-implicit integration factor methods on sparse grids for high-dimensional systems, Journal of Computational Physics, 292 (2015), 43-55. doi: 10.1016/j.jcp.2015.03.033. [27] G. F. Webb, A reaction-diffusion model for a deterministic diffusion epidemic, J. Math. Anal. Appl., 84 (1981), 150-161. doi: 10.1016/0022-247X(81)90156-6. [28] K. E. Yong, E. D. Herrera and and C. Castillo-Chavez, From bee species aggregation to models of disease avoidance: The ben-hur effect., Mathematical and Statistical Modeling for Emerging and Re-emerging Infectious Diseases, 12 (2016), 169-185. [29] Y.-T. Zhang and C.-W. Shu, High order WENO schemes for Hamilton-Jacobi equations on triangular meshes, SIAM Journal on Scientific Computing, 24 (2003), 1005-1030. doi: 10.1137/S1064827501396798. [30] Y.-T. Zhang and C.-W. Shu, ENO and WENO schemes, in Handbook of Numerical Analysis, Volume 17, Handbook of Numerical Methods for Hyperbolic Problems: Basic and Fundamental Issues, R. Abgrall and C.-W. Shu, Editors, North-Holland, Elsevier, Amsterdam, (2016), 103–122. [31] Y.-T. Zhang, H.-K. Zhao and J. Qian, High order fast sweeping methods for static Hamilton-Jacobi equations, Journal of Scientific Computing, 29 (2006), 25-56. doi: 10.1007/s10915-005-9014-3.
Numerical solution of the one-dimensional case of Eq. 1 for the case of avoiding infection ($k_1 = 0$ and $k_2 = 0.1$) at $t = 0.706$. CFL = 0.1
Numerical solution of the one-dimensional case of Eq. 1 for the case of avoiding infection ($k_1 = 0$, $k_2 = 0.1$) at $t = 1.5$. CFL = 0.1
Numerical solution of the two-dimensional case of Eq. 1 for the case of avoiding infection ($k_1 = 0$, $k_2 = 0.1$) at $t = 0.7$. CFL = 0.2
Numerical solution for the one-dimensional case of Eq. 1 for avoiding crowd ($k_1 = 0.1$, $k_2 = 0$). CFL = 0.6
Initial density profiles of population at $t = 0$
Numerical solution of the two-dimensional case of Eq. 1 for avoiding crowd ($k_1 = 0.1$, $k_2 = 0$) at $t = 1$. CFL = 0.2
Numerical solution of the two-dimensional case of Eq. 1 for avoiding crowd ($k_1 = 0.1$, $k_2 = 0$) at $t = 10$. CFL = 0.2
Numerical solution of the two-dimensional case of Eq. 1 for avoiding crowd ($k_1 = 0.1$, $k_2 = 0$) at $t = 25$. CFL = 0.2
Numerical results of the one-dimensional system Eq. 10 for $k_1 = 0.1$ and $k_2 = 0.001$. $\pi/N$ is the mesh size in the spatial direction. Here the constant CFL $= 0.2$
 $N$ $L^\infty$ error $L^\infty$ order $L^1$ error $L^1$ order $L^2$ error $L^2$ order 10 3.85e-02 - 7.48e-03 - 1.07e-02 - 20 1.24e-02 1.64 2.28e-03 1.72 3.53e-03 1.60 40 3.29e-03 1.91 5.56e-04 2.03 8.95e-04 1.98 80 8.30e-04 1.99 1.45e-04 1.94 2.26e-04 1.98 160 2.07e-04 2.00 3.68e-05 1.98 5.67e-05 2.00 320 5.12e-05 2.01 9.21e-06 2.00 1.42e-05 2.00
 $N$ $L^\infty$ error $L^\infty$ order $L^1$ error $L^1$ order $L^2$ error $L^2$ order 10 3.85e-02 - 7.48e-03 - 1.07e-02 - 20 1.24e-02 1.64 2.28e-03 1.72 3.53e-03 1.60 40 3.29e-03 1.91 5.56e-04 2.03 8.95e-04 1.98 80 8.30e-04 1.99 1.45e-04 1.94 2.26e-04 1.98 160 2.07e-04 2.00 3.68e-05 1.98 5.67e-05 2.00 320 5.12e-05 2.01 9.21e-06 2.00 1.42e-05 2.00
Numerical results of the one-dimensional system Eq. 10 for $k_1 = 0.1$ and $k_2 = 0$. CFL $= 0.2$. $\pi/N$ is the mesh size in the spatial direction
 $N$ $L^\infty$ error $L^\infty$ order $L^1$ error $L^1$ order $L^2$ error $L^2$ order 10 3.86e-02 - 7.51e-03 - 1.07e-02 - 20 1.24e-02 1.63 2.29e-03 1.72 3.54e-03 1.60 40 3.30e-03 1.91 5.59e-04 2.03 9.00e-04 1.98 80 8.31e-04 1.99 1.45e-04 1.95 2.27e-04 1.98 160 2.07e-04 2.00 3.69e-05 1.97 5.70e-05 2.00 320 5.15e-05 2.01 9.29e-06 1.99 1.43e-05 2.00
 $N$ $L^\infty$ error $L^\infty$ order $L^1$ error $L^1$ order $L^2$ error $L^2$ order 10 3.86e-02 - 7.51e-03 - 1.07e-02 - 20 1.24e-02 1.63 2.29e-03 1.72 3.54e-03 1.60 40 3.30e-03 1.91 5.59e-04 2.03 9.00e-04 1.98 80 8.31e-04 1.99 1.45e-04 1.95 2.27e-04 1.98 160 2.07e-04 2.00 3.69e-05 1.97 5.70e-05 2.00 320 5.15e-05 2.01 9.29e-06 1.99 1.43e-05 2.00
Numerical results of the one-dimensional system Eq. 10 for $k_1 = 0.1$ and $k_2 = 0$. CFL $= 0.5$. $\pi/N$ is the mesh size in the spatial direction
 $N$ $L^\infty$ error $L^\infty$ order $L^1$ error $L^1$ order $L^2$ error $L^2$ order 10 1.05e-01 - 3.23e-02 - 4.05e-02 - 20 4.17e-02 1.33 8.41e-03 1.94 1.27e-02 1.67 40 1.46e-02 1.51 2.69e-03 1.65 4.04e-03 1.65 80 3.94e-03 1.89 7.25e-04 1.89 1.10e-03 1.88 160 1.05e-03 1.91 1.94e-04 1.91 2.91e-04 1.92 320 2.61e-04 2.00 4.96e-05 1.96 7.35e-05 1.99
 $N$ $L^\infty$ error $L^\infty$ order $L^1$ error $L^1$ order $L^2$ error $L^2$ order 10 1.05e-01 - 3.23e-02 - 4.05e-02 - 20 4.17e-02 1.33 8.41e-03 1.94 1.27e-02 1.67 40 1.46e-02 1.51 2.69e-03 1.65 4.04e-03 1.65 80 3.94e-03 1.89 7.25e-04 1.89 1.10e-03 1.88 160 1.05e-03 1.91 1.94e-04 1.91 2.91e-04 1.92 320 2.61e-04 2.00 4.96e-05 1.96 7.35e-05 1.99
Numerical results of the one-dimensional system Eq. 10 for $k_1 = 0$ and $k_2 = 0.1$. CFL $= 0.1$. $\pi/N$ is the mesh size in the spatial direction
 $N$ $L^\infty$ error $L^\infty$ order $L^1$ error $L^1$ order $L^2$ error $L^2$ order 10 3.38e-02 - 5.95e-03 - 7.33e-03 - 20 1.05e-02 1.68 1.88e-03 1.66 2.27e-03 1.69 40 2.92e-03 1.85 5.53e-04 1.77 6.64e-04 1.78 80 7.93e-04 1.88 1.53e-04 1.85 1.80e-04 1.88 160 2.08e-04 1.93 4.03e-05 1.92 4.71e-05 1.94 320 5.35e-05 1.96 1.04e-05 1.95 1.21e-05 1.96
 $N$ $L^\infty$ error $L^\infty$ order $L^1$ error $L^1$ order $L^2$ error $L^2$ order 10 3.38e-02 - 5.95e-03 - 7.33e-03 - 20 1.05e-02 1.68 1.88e-03 1.66 2.27e-03 1.69 40 2.92e-03 1.85 5.53e-04 1.77 6.64e-04 1.78 80 7.93e-04 1.88 1.53e-04 1.85 1.80e-04 1.88 160 2.08e-04 1.93 4.03e-05 1.92 4.71e-05 1.94 320 5.35e-05 1.96 1.04e-05 1.95 1.21e-05 1.96
Numerical results of the two-dimensional system Eq. 11 for $k_1 = 0.1$ and $k_2 = 0.001$. CFL $= 0.4$. $\pi/N$ is the mesh size in each of the spatial directions
 $N$ $L^\infty$ error $L^\infty$ order $L^1$ error $L^1$ order $L^2$ error $L^2$ order 10 6.21e-02 - 7.34e-03 - 9.75e-03 - 20 1.73e-02 1.84 1.65e-03 2.15 2.39e-03 2.03 40 2.69e-03 2.69 4.01e-04 2.04 6.37e-04 1.91 80 5.99e-04 2.16 9.09e-05 2.14 1.45e-04 2.13 160 1.39e-04 2.10 2.15e-05 2.08 3.43e-05 2.08 320 3.35e-05 2.05 5.18e-06 2.06 8.25e-06 2.06
 $N$ $L^\infty$ error $L^\infty$ order $L^1$ error $L^1$ order $L^2$ error $L^2$ order 10 6.21e-02 - 7.34e-03 - 9.75e-03 - 20 1.73e-02 1.84 1.65e-03 2.15 2.39e-03 2.03 40 2.69e-03 2.69 4.01e-04 2.04 6.37e-04 1.91 80 5.99e-04 2.16 9.09e-05 2.14 1.45e-04 2.13 160 1.39e-04 2.10 2.15e-05 2.08 3.43e-05 2.08 320 3.35e-05 2.05 5.18e-06 2.06 8.25e-06 2.06
Numerical results of the two-dimensional system Eq. 11 for $k_1 = 0.1$ and $k_2 = 0$. CFL $= 0.6$. $\pi/N$ is the mesh size in each of the spatial directions
 $N$ $L^\infty$ error $L^\infty$ order $L^1$ error $L^1$ order $L^2$ error $L^2$ order 10 7.30e-02 - 1.02e-02 - 1.32e-02 - 20 2.12e-02 1.78 2.54e-03 2.00 3.76e-03 1.82 40 4.15e-03 2.35 6.54e-04 1.96 1.03e-03 1.88 80 1.03e-03 2.01 1.56e-04 2.07 2.47e-04 2.05 160 2.77e-04 1.90 3.86e-05 2.02 6.05e-05 2.03 320 6.97e-05 1.99 7.92e-06 2.28 1.28e-05 2.25
 $N$ $L^\infty$ error $L^\infty$ order $L^1$ error $L^1$ order $L^2$ error $L^2$ order 10 7.30e-02 - 1.02e-02 - 1.32e-02 - 20 2.12e-02 1.78 2.54e-03 2.00 3.76e-03 1.82 40 4.15e-03 2.35 6.54e-04 1.96 1.03e-03 1.88 80 1.03e-03 2.01 1.56e-04 2.07 2.47e-04 2.05 160 2.77e-04 1.90 3.86e-05 2.02 6.05e-05 2.03 320 6.97e-05 1.99 7.92e-06 2.28 1.28e-05 2.25
Numerical results of the two-dimensional system Eq. 11 for $k_1 = 0$ and $k_2 = 0.1$. CFL $= 0.2$. $\pi/N$ is the mesh size in each of the spatial directions
 $N$ $L^\infty$ error $L^\infty$ order $L^1$ error $L^1$ order $L^2$ error $L^2$ order 10 3.08e-02 - 2.91e-03 - 3.72e-03 - 20 7.74e-03 1.99 8.37e-04 1.80 1.17e-03 1.67 40 1.68e-03 2.20 2.37e-04 1.82 3.32e-04 1.82 80 4.18e-04 2.01 6.24e-05 1.93 8.68e-05 1.94 160 1.07e-04 1.97 1.64e-05 1.93 2.24e-05 1.96 320 2.69e-05 1.99 4.13e-06 1.98 5.65e-06 1.99
 $N$ $L^\infty$ error $L^\infty$ order $L^1$ error $L^1$ order $L^2$ error $L^2$ order 10 3.08e-02 - 2.91e-03 - 3.72e-03 - 20 7.74e-03 1.99 8.37e-04 1.80 1.17e-03 1.67 40 1.68e-03 2.20 2.37e-04 1.82 3.32e-04 1.82 80 4.18e-04 2.01 6.24e-05 1.93 8.68e-05 1.94 160 1.07e-04 1.97 1.64e-05 1.93 2.24e-05 1.96 320 2.69e-05 1.99 4.13e-06 1.98 5.65e-06 1.99
Numerical solution of $S$ at $t = 0.7$ and $t = 0.706$ and $x = 0.8 - 1/M$ for different $M$. All values are computed using the uniform mesh $\Delta x = 1/1280$
 M 10 20 40 80 160 320 640 1280 S (at T = 0.7) 7.05 9.72 13.39 18.49 25.56 36.15 66.84 80.74 r (at T = 0.7) - 1.37 1.38 1.38 1.38 1.41 1.85 1.21 S (at T = 0.706) 6.97 9.56 13.09 17.88 24.31 33.71 54.05 89.15 r (at T = 0.706) - 1.37 1.37 1.37 1.36 1.39 1.6 1.65
 M 10 20 40 80 160 320 640 1280 S (at T = 0.7) 7.05 9.72 13.39 18.49 25.56 36.15 66.84 80.74 r (at T = 0.7) - 1.37 1.38 1.38 1.38 1.41 1.85 1.21 S (at T = 0.706) 6.97 9.56 13.09 17.88 24.31 33.71 54.05 89.15 r (at T = 0.706) - 1.37 1.37 1.37 1.36 1.39 1.6 1.65
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