Figure 1.
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
-
Previous Article
Nonlocal time-porous medium equation: Weak solutions and finite speed of propagation
- DCDS-B Home
- This Issue
-
Next Article
On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium
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 |
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.
Keywords:
SIR model,
directed diffusion,
Krylov implicit integration factor methods,
weighted essentially non-oscillatory schemes.
Mathematics Subject Classification:
Primary: 92D30, 65M06; Secondary: 92-08.
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. |
show all references
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. |
Figure 2.
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
Figure 3.
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
Figure 4.
Numerical solution for the one-dimensional case of Eq. 1 for avoiding crowd ($ k_1 = 0.1 $ , $ k_2 = 0 $ ). CFL = 0.6
Figure 5.
Initial density profiles of population at $ t = 0 $
Figure 6.
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
Figure 7.
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
Figure 8.
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
Table 1.
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 $
| 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 |
| 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 |
Table 2.
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
| 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 |
| 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 |
Table 3.
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
| 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 |
| 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 |
Table 4.
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
| 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 |
| 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 |
Table 5.
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
| 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 |
| 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 |
Table 6.
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
| 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 |
| 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 |
Table 7.
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
| 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 |
| 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 |
Table 8.
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 | 1.60 | 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 | 1.60 | 1.65 |
| [1] |
Alexander Kurganov, Anthony Polizzi. Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics. Networks & Heterogeneous Media, 2009, 4 (3) : 431-451. doi: 10.3934/nhm.2009.4.431 |
| [2] |
Lili Ju, Xinfeng Liu, Wei Leng. Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1667-1687. doi: 10.3934/dcdsb.2014.19.1667 |
| [3] |
Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039 |
| [4] |
Keng Deng. Asymptotic behavior of an SIR reaction-diffusion model with a linear source. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-13. doi: 10.3934/dcdsb.2019114 |
| [5] |
Michael Gekhtman, Michael Shapiro, Serge Tabachnikov, Alek Vainshtein. Higher pentagram maps, weighted directed networks, and cluster dynamics. Electronic Research Announcements, 2012, 19: 1-17. doi: 10.3934/era.2012.19.1 |
| [6] |
Jeffrey R. Haack, Cory D. Hauck. Oscillatory behavior of Asymptotic-Preserving splitting methods for a linear model of diffusive relaxation. Kinetic & Related Models, 2008, 1 (4) : 573-590. doi: 10.3934/krm.2008.1.573 |
| [7] |
Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅱ): Sharp asymptotic rates of convergence in relative error by entropy methods. Kinetic & Related Models, 2017, 10 (1) : 61-91. doi: 10.3934/krm.2017003 |
| [8] |
Qingguang Guan, Max Gunzburger. Stability and convergence of time-stepping methods for a nonlocal model for diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1315-1335. doi: 10.3934/dcdsb.2015.20.1315 |
| [9] |
Mauro Fabrizio, Jaime Munõz Rivera. An integration model for two different ethnic groups. Evolution Equations & Control Theory, 2014, 3 (2) : 277-286. doi: 10.3934/eect.2014.3.277 |
| [10] |
Lambertus A. Peletier, Xi-Ling Jiang, Snehal Samant, Stephan Schmidt. Analysis of a complex physiology-directed model for inhibition of platelet aggregation by clopidogrel. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 945-961. doi: 10.3934/dcds.2017039 |
| [11] |
Wen Tan, Chunyou Sun. Dynamics for a non-autonomous reaction diffusion model with the fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6035-6067. doi: 10.3934/dcds.2017260 |
| [12] |
Wenjuan Zhai, Bingzhen Chen. A fourth order implicit symmetric and symplectic exponentially fitted Runge-Kutta-Nyström method for solving oscillatory problems. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 71-84. doi: 10.3934/naco.2019006 |
| [13] |
Ke Ruan, Masao Fukushima. Robust portfolio selection with a combined WCVaR and factor model. Journal of Industrial & Management Optimization, 2012, 8 (2) : 343-362. doi: 10.3934/jimo.2012.8.343 |
| [14] |
Shaoyong Lai, Yulan Zhou. A stochastic optimal growth model with a depreciation factor. Journal of Industrial & Management Optimization, 2010, 6 (2) : 283-297. doi: 10.3934/jimo.2010.6.283 |
| [15] |
Yoonsang Lee, Bjorn Engquist. Variable step size multiscale methods for stiff and highly oscillatory dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1079-1097. doi: 10.3934/dcds.2014.34.1079 |
| [16] |
Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679 |
| [17] |
Angelamaria Cardone, Zdzisław Jackiewicz, Adrian Sandu, Hong Zhang. Construction of highly stable implicit-explicit general linear methods. Conference Publications, 2015, 2015 (special) : 185-194. doi: 10.3934/proc.2015.0185 |
| [18] |
Tim McGraw, Baba Vemuri, Evren Özarslan, Yunmei Chen, Thomas Mareci. Variational denoising of diffusion weighted MRI. Inverse Problems & Imaging, 2009, 3 (4) : 625-648. doi: 10.3934/ipi.2009.3.625 |
| [19] |
María Anguiano, P.E. Kloeden. Asymptotic behaviour of the nonautonomous SIR equations with diffusion. Communications on Pure & Applied Analysis, 2014, 13 (1) : 157-173. doi: 10.3934/cpaa.2014.13.157 |
| [20] |
Antonio Algaba, Natalia Fuentes, Cristóbal García, Manuel Reyes. Non-formally integrable centers admitting an algebraic inverse integrating factor. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 967-988. doi: 10.3934/dcds.2018041 |
2017 Impact Factor: 0.972
Tools
Metrics
Other articles
by authors
[Back to Top]