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March  2021, 14(3): 935-951. doi: 10.3934/dcdss.2020382

Mathematical model of signal propagation in excitable media

Jakub Kantner and  Michal Beneš ,

Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Prague, 12000, Czech Republic

* Corresponding author: Michal Beneš

Received  February 2020 Revised  January 2019 Published  June 2020

This article deals with a model of signal propagation in excitable media based on a system of reaction-diffusion equations. Such media have the ability to exhibit a large response in reaction to a small deviation from the rest state. An example of such media is the nerve tissue or the heart tissue. The first part of the article briefly describes the origin and the propagation of the cardiac action potential in the heart. In the second part, the mathematical properties of the model are discussed. Next, the numerical algorithm based on the finite difference method is used to obtain computational studies in both a homogeneous and heterogeneous medium with an emphasis on interactions of the propagating signals with obstacles in the medium.

Keywords: Excitable media, reaction-diffusion equations, FitzHugh-Nagumo model, cardiac action potential, signal propagation, invariant regions.
Mathematics Subject Classification: Primary: 35K57, 65M06, 35B45; Secondary: 92B05.
Citation: Jakub Kantner, Michal Beneš. Mathematical model of signal propagation in excitable media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 935-951. doi: 10.3934/dcdss.2020382
References:
[1]

O. Bernus and E. Vigmond, Asymptotic wave propagation in excitable media, Phys. Rev. E, 92 (2015), 010901. doi: 10.1103/PhysRevE.92.010901.  Google Scholar

[2]

Y.-Y. Chen, H. Ninomiya and R. Taguchi, Travelling spots in multidimensional excitable media, Journal of Elliptic and Parabolic Equations, 1 (2015), 281-305.  doi: 10.1007/BF03377382.  Google Scholar

[3]

P. Colli Franzone, L. F. Pavarino and S. Scacchi, Mathematical Cardiac Electrophysiology, MS & A. Modeling, Simulation and Applications, 13. Springer, Cham, 2014. doi: 10.1007/978-3-319-04801-7.  Google Scholar

[4]

P. Colli-Franzone, V. Gionti, S. Scacchi and C. Storti, Role of infarct scar dimensions, border zone repolarization properties and anisotropy in the origin and maintenance of cardiac reentry, Mathematical Biosciences, 315 (2019), 108-128.  doi: 10.1016/j.mbs.2019.108228.  Google Scholar

[5]

E. N. Cytrynbaum, V. MacKay, O. Nahman-Lévesque, M. Dobbs, G. Bub, A. Shrier and L. Glass, Double-wave reentry in excitable media, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 073103, 12 pp. doi: 10.1063/1.5092982.  Google Scholar

[6]

K. Deckelnick, G. Dziuk and C. M. Elliot, Computation of geometric partial differential equations and mean curvature flow, Acta Numerica, 14 (2005), 139-232.  doi: 10.1017/S0962492904000224.  Google Scholar

[7]

A. J. Durston, Dictyostelium discoideum aggregation fields as excitable media, J. Theor. Biol., 42 (1973), 483-504.  doi: 10.1016/0022-5193(73)90242-7.  Google Scholar

[8]

J. Engelbrecht, T. Peets, K. Tamm, M. Laasmaa and M. Vendelin, On the complexity of signal propagation in nerve fibres, Proceedings of the Estonian Academy of Sciences, 67 (2018), 28-38.  doi: 10.3176/proc.2017.4.28.  Google Scholar

[9]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

[10]

M. A. C. Guyton, Textbook of Medical Physiology, W. B. Saunders Company, 1991. Google Scholar

[11]

M. Kolář, Computational studies of reaction-diffusion systems by nonlinear galerkin method, American Journal of Computational Mathematics, 3 (2013), 137-146.   Google Scholar

[12]

O. A. Ladyženskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Izdat. "Nauka'', Moscow, 1967,736 pp.  Google Scholar

[13]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Pte. Ltd., Singapore, 1996. doi: 10.1142/3302.  Google Scholar

[14]

J. L. Lions, Quelques Méthodes aux Rśolution des Problémes Nonlinéaires, Dunod Gauthiers-Villars, Paris, 1969. Google Scholar

[15]

J. Ma, F. Q. Wu, T. Hayat, P. Zhou and J. Tang, Electromagnetic induction and radiation-induced abnormality of wave propagation in excitable media, Physica A: Statistical Mechanics and its Applications, 486 (2017), 508-516.  doi: 10.1016/j.physa.2017.05.075.  Google Scholar

[16]

J. Mach, M. Beneš and P. Strachota, Nonlinear Galerkin finite element method applied to the system of reaction-diffusion equations in one space dimension, Comput. Math. Appl., 73 (2017), 2053-2065.  doi: 10.1016/j.camwa.2017.02.032.  Google Scholar

[17]

J. D. Murray, Mathematical Biology, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.  Google Scholar

[18]

J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of IRE, 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.  Google Scholar

[19]

A. Yu. Palyanov and A. S. Ratushnyak, Some details of signal propagation in the nervous system of C. elegans, Russian Journal of Genetics: Applied Research, 5 (2015), 642-649.  doi: 10.1134/S2079059715060064.  Google Scholar

[20]

O. Pártl, Reaction-Diffusion Systems in Mathematical Biology, Diploma Thesis, Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague, Prague, 2012. Google Scholar

[21]

Pathwaymedicine.org, Cardiac Action Potential - Cellular Basis, http://www.pathwaymedicine.org/Cardiac-Action-Potential-Cellular-Basis, [cited: April 10, 2018]. Google Scholar

[22]

L. S. Pontryagin, Ordinary Differential Equations, Second, revised edition Izdat. "Nauka'', Moscow, 1965,331 pp.  Google Scholar

[23]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, 258. Springer-Verlag, New York-Berlin, 1983.  Google Scholar

[24]

J. Šembera and M. Beneš, Nonlinear Galerkin method for reaction-diffusion systems admitting invariant regions, Journal of Computational and Applied Mathematics, 136 (2001), 163-176.  doi: 10.1016/S0377-0427(00)00582-3.  Google Scholar

[25]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Science, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[26]

N. A. Trayanova and K. C. Chang, How computer simulations of the human heart can improve anti-arrhythmia therapy, The Journal of Physiology, 594 (2016), 2483-2502.  doi: 10.1113/JP270532.  Google Scholar

[27]

K. H. W. J. Ten Tusscher and A. V. Panfilov, Wave propagation in excitable media with randomly distributed obstacles, Multiscale Model. Simul., 3 (2005), 265-282.  doi: 10.1137/030602654.  Google Scholar

[28]

E. Ullner, A. Zaikin, J. García-Ojalvo, R. Báscones and J. Kurths, Vibrational resonance and vibrational propagation in excitable systems, Physics Letters A, 312 (2003), 348-354.  doi: 10.1016/S0375-9601(03)00681-9.  Google Scholar

[29]

H. Wang, J. Wang, X. Y. Thow and Ch. Lee, The First Principle of Neural Circuit and the General Circuit–Probability Theory, submitted, 2018, arXiv: 1805.00605. Google Scholar

[30]

J. P. T. Ward and R. W. A. Linden, The Basics of Physiology, (in Czech), Galén, 2010. Google Scholar

[31]

L. D. Weise and A. V. Panfilov, Emergence of spiral wave activity in a mechanically heterogeneous reaction-diffusion-mechanics system, Physical Review Letters, 108 (2012), 228104. doi: 10.1103/PhysRevLett.108.228104.  Google Scholar

[32]

D. P. Zipes and J. Jalife, Cardiac Electrophysiology, Saunders Elsevier, Philadelphia, 1995. Google Scholar

[33]

V. S. Zykov, Spiral wave initiation in excitable media, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376 (2018). doi: 10.1098/rsta.2017.0379.  Google Scholar

[34]

V. S. Zykov, A. S. Mikhailov and S. C. Müller, Wave propagation in excitable media with fast inhibitor diffusion, Lecture Notes in Physics, 532 (2007), 308-325.  doi: 10.1007/BFb0104233.  Google Scholar

[35]

V. S. Zykov and E. Bodenschatz, Wave propagation in inhomogeneous excitable media, Annual Review of Condensed Matter Physics, 9 (2018), 435-461.   Google Scholar

show all references

References:
[1]

O. Bernus and E. Vigmond, Asymptotic wave propagation in excitable media, Phys. Rev. E, 92 (2015), 010901. doi: 10.1103/PhysRevE.92.010901.  Google Scholar

[2]

Y.-Y. Chen, H. Ninomiya and R. Taguchi, Travelling spots in multidimensional excitable media, Journal of Elliptic and Parabolic Equations, 1 (2015), 281-305.  doi: 10.1007/BF03377382.  Google Scholar

[3]

P. Colli Franzone, L. F. Pavarino and S. Scacchi, Mathematical Cardiac Electrophysiology, MS & A. Modeling, Simulation and Applications, 13. Springer, Cham, 2014. doi: 10.1007/978-3-319-04801-7.  Google Scholar

[4]

P. Colli-Franzone, V. Gionti, S. Scacchi and C. Storti, Role of infarct scar dimensions, border zone repolarization properties and anisotropy in the origin and maintenance of cardiac reentry, Mathematical Biosciences, 315 (2019), 108-128.  doi: 10.1016/j.mbs.2019.108228.  Google Scholar

[5]

E. N. Cytrynbaum, V. MacKay, O. Nahman-Lévesque, M. Dobbs, G. Bub, A. Shrier and L. Glass, Double-wave reentry in excitable media, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 073103, 12 pp. doi: 10.1063/1.5092982.  Google Scholar

[6]

K. Deckelnick, G. Dziuk and C. M. Elliot, Computation of geometric partial differential equations and mean curvature flow, Acta Numerica, 14 (2005), 139-232.  doi: 10.1017/S0962492904000224.  Google Scholar

[7]

A. J. Durston, Dictyostelium discoideum aggregation fields as excitable media, J. Theor. Biol., 42 (1973), 483-504.  doi: 10.1016/0022-5193(73)90242-7.  Google Scholar

[8]

J. Engelbrecht, T. Peets, K. Tamm, M. Laasmaa and M. Vendelin, On the complexity of signal propagation in nerve fibres, Proceedings of the Estonian Academy of Sciences, 67 (2018), 28-38.  doi: 10.3176/proc.2017.4.28.  Google Scholar

[9]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

[10]

M. A. C. Guyton, Textbook of Medical Physiology, W. B. Saunders Company, 1991. Google Scholar

[11]

M. Kolář, Computational studies of reaction-diffusion systems by nonlinear galerkin method, American Journal of Computational Mathematics, 3 (2013), 137-146.   Google Scholar

[12]

O. A. Ladyženskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Izdat. "Nauka'', Moscow, 1967,736 pp.  Google Scholar

[13]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Pte. Ltd., Singapore, 1996. doi: 10.1142/3302.  Google Scholar

[14]

J. L. Lions, Quelques Méthodes aux Rśolution des Problémes Nonlinéaires, Dunod Gauthiers-Villars, Paris, 1969. Google Scholar

[15]

J. Ma, F. Q. Wu, T. Hayat, P. Zhou and J. Tang, Electromagnetic induction and radiation-induced abnormality of wave propagation in excitable media, Physica A: Statistical Mechanics and its Applications, 486 (2017), 508-516.  doi: 10.1016/j.physa.2017.05.075.  Google Scholar

[16]

J. Mach, M. Beneš and P. Strachota, Nonlinear Galerkin finite element method applied to the system of reaction-diffusion equations in one space dimension, Comput. Math. Appl., 73 (2017), 2053-2065.  doi: 10.1016/j.camwa.2017.02.032.  Google Scholar

[17]

J. D. Murray, Mathematical Biology, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.  Google Scholar

[18]

J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of IRE, 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.  Google Scholar

[19]

A. Yu. Palyanov and A. S. Ratushnyak, Some details of signal propagation in the nervous system of C. elegans, Russian Journal of Genetics: Applied Research, 5 (2015), 642-649.  doi: 10.1134/S2079059715060064.  Google Scholar

[20]

O. Pártl, Reaction-Diffusion Systems in Mathematical Biology, Diploma Thesis, Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague, Prague, 2012. Google Scholar

[21]

Pathwaymedicine.org, Cardiac Action Potential - Cellular Basis, http://www.pathwaymedicine.org/Cardiac-Action-Potential-Cellular-Basis, [cited: April 10, 2018]. Google Scholar

[22]

L. S. Pontryagin, Ordinary Differential Equations, Second, revised edition Izdat. "Nauka'', Moscow, 1965,331 pp.  Google Scholar

[23]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, 258. Springer-Verlag, New York-Berlin, 1983.  Google Scholar

[24]

J. Šembera and M. Beneš, Nonlinear Galerkin method for reaction-diffusion systems admitting invariant regions, Journal of Computational and Applied Mathematics, 136 (2001), 163-176.  doi: 10.1016/S0377-0427(00)00582-3.  Google Scholar

[25]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Science, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[26]

N. A. Trayanova and K. C. Chang, How computer simulations of the human heart can improve anti-arrhythmia therapy, The Journal of Physiology, 594 (2016), 2483-2502.  doi: 10.1113/JP270532.  Google Scholar

[27]

K. H. W. J. Ten Tusscher and A. V. Panfilov, Wave propagation in excitable media with randomly distributed obstacles, Multiscale Model. Simul., 3 (2005), 265-282.  doi: 10.1137/030602654.  Google Scholar

[28]

E. Ullner, A. Zaikin, J. García-Ojalvo, R. Báscones and J. Kurths, Vibrational resonance and vibrational propagation in excitable systems, Physics Letters A, 312 (2003), 348-354.  doi: 10.1016/S0375-9601(03)00681-9.  Google Scholar

[29]

H. Wang, J. Wang, X. Y. Thow and Ch. Lee, The First Principle of Neural Circuit and the General Circuit–Probability Theory, submitted, 2018, arXiv: 1805.00605. Google Scholar

[30]

J. P. T. Ward and R. W. A. Linden, The Basics of Physiology, (in Czech), Galén, 2010. Google Scholar

[31]

L. D. Weise and A. V. Panfilov, Emergence of spiral wave activity in a mechanically heterogeneous reaction-diffusion-mechanics system, Physical Review Letters, 108 (2012), 228104. doi: 10.1103/PhysRevLett.108.228104.  Google Scholar

[32]

D. P. Zipes and J. Jalife, Cardiac Electrophysiology, Saunders Elsevier, Philadelphia, 1995. Google Scholar

[33]

V. S. Zykov, Spiral wave initiation in excitable media, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376 (2018). doi: 10.1098/rsta.2017.0379.  Google Scholar

[34]

V. S. Zykov, A. S. Mikhailov and S. C. Müller, Wave propagation in excitable media with fast inhibitor diffusion, Lecture Notes in Physics, 532 (2007), 308-325.  doi: 10.1007/BFb0104233.  Google Scholar

[35]

V. S. Zykov and E. Bodenschatz, Wave propagation in inhomogeneous excitable media, Annual Review of Condensed Matter Physics, 9 (2018), 435-461.   Google Scholar

Figure 1.  An example of nullclines for problem (1). The green curve represents cubic function $ v_1 $ and the dashed red line is the graph of the linear function $ v_2 $. The values of the parameters are $ \varepsilon = 0.008, \alpha = 0.139, \beta = 2.54, g_1 = 20, g_2 = 1.5, \text{ and } g_3 = -5.5 $
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Figure 2.  The time evolution of the component $ u_1 $ of the solution for the set parameter values and the initial conditions for Example 1. Each subfigure represents one selected time level $ t $
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Figure 3.  The time evolution of the component $ u_1 $ of the solution for the set parameter values and the initial conditions for Example 2. Each subfigure represents one selected time level $ t $. The obstacle is marked in orange in Figures Figures 3a and 3b
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Figure 4.  The time evolution of the component $ u_1 $ of the solution for the set parameter values and the initial conditions for Experiment 3. Each subfigure represents one selected time level $ t $. The obstacle is marked in orange
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Table 3.  The common parameters for all Examples
Model characteristics
$ u $ right hand side coefficients $ \epsilon = 0.008, \alpha = 0.139 , \beta = 2.54 $
$ v $ right hand side coefficients $ g_1 = 20, g_2 = 1.5, g_3 = -7.5 $
Fixed point location $ u_1^0 = 0.198598, u_2^0 = -2.352028 $
Diffusion coefficients
$ \qquad D_1 = 8\cdot 10^{-4} \quad \text{ in }\Omega\setminus\Omega_{obs} $
$ \qquad D_2 = 4\cdot 10^{-4} \quad \text{ in }\Omega\setminus\Omega_{obs} $
$ \qquad D_1 = 0 \quad \text{ in } \Omega_{obs} $
$ \qquad D_2 = 0 \quad \text{ in } \Omega_{obs}, $
where $ \Omega $ is a domain identical with the one in the initial conditions and
$ \Omega_{obs} $ is an obstacle, further described in Table 4 for each example.
Initial state and boundary conditions
$ \qquad u_{ini, 1} = u_1^0 + \sin \left( \frac{\pi (x-a_x^0)}{b^0_x-a^0_x}\right)\sin \left( \frac{\pi (y-a^0_y)}{b^0_y-a^0_y}\right) \quad \text{ in }\Omega_0 $
$ \qquad u_{ini, 1} = u_1^0 \quad \text{ in }\Omega\setminus\Omega_0 $
$ \qquad u_{ini, 2} = u_2^0 \quad \text{ in }\Omega $
$ \qquad u_1|_{\partial\Omega} = u_1^0 $
$ \qquad u_2|_{\partial\Omega} = u_2^0 $
$ \quad \Omega \ = \left( a_x, b_x\right) \times\left( a_y, b_y\right) \ = \left( 0, 1\right)\times\left( 0, 1\right) $
$ \quad $ The parameters for $ \Omega_0 = (a_x^0, b_x^0)\times (a_y^0, b_y^0) $ are described in Table 4
for each experiment.
Numerical characteristics
Total length of simulation $ . \dots\dots\dots\dots\dots . \ T $ $ 15 $
Internal time step $ . \dots\dots\dots\dots\dots\dots\dots\dots \ \tau $ $ 6.1\cdot10^{-5} $
Mesh $ . \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots \ \omega_h $ $ 128\times 128 $
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Model characteristics
$ u $ right hand side coefficients $ \epsilon = 0.008, \alpha = 0.139 , \beta = 2.54 $
$ v $ right hand side coefficients $ g_1 = 20, g_2 = 1.5, g_3 = -7.5 $
Fixed point location $ u_1^0 = 0.198598, u_2^0 = -2.352028 $
Diffusion coefficients
$ \qquad D_1 = 8\cdot 10^{-4} \quad \text{ in }\Omega\setminus\Omega_{obs} $
$ \qquad D_2 = 4\cdot 10^{-4} \quad \text{ in }\Omega\setminus\Omega_{obs} $
$ \qquad D_1 = 0 \quad \text{ in } \Omega_{obs} $
$ \qquad D_2 = 0 \quad \text{ in } \Omega_{obs}, $
where $ \Omega $ is a domain identical with the one in the initial conditions and
$ \Omega_{obs} $ is an obstacle, further described in Table 4 for each example.
Initial state and boundary conditions
$ \qquad u_{ini, 1} = u_1^0 + \sin \left( \frac{\pi (x-a_x^0)}{b^0_x-a^0_x}\right)\sin \left( \frac{\pi (y-a^0_y)}{b^0_y-a^0_y}\right) \quad \text{ in }\Omega_0 $
$ \qquad u_{ini, 1} = u_1^0 \quad \text{ in }\Omega\setminus\Omega_0 $
$ \qquad u_{ini, 2} = u_2^0 \quad \text{ in }\Omega $
$ \qquad u_1|_{\partial\Omega} = u_1^0 $
$ \qquad u_2|_{\partial\Omega} = u_2^0 $
$ \quad \Omega \ = \left( a_x, b_x\right) \times\left( a_y, b_y\right) \ = \left( 0, 1\right)\times\left( 0, 1\right) $
$ \quad $ The parameters for $ \Omega_0 = (a_x^0, b_x^0)\times (a_y^0, b_y^0) $ are described in Table 4
for each experiment.
Numerical characteristics
Total length of simulation $ . \dots\dots\dots\dots\dots . \ T $ $ 15 $
Internal time step $ . \dots\dots\dots\dots\dots\dots\dots\dots \ \tau $ $ 6.1\cdot10^{-5} $
Mesh $ . \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots \ \omega_h $ $ 128\times 128 $
Table 4.  Table of parameter values in which the Examples differ
Example $ \Omega_{obs} $ $ \Omega_0 $
1 no obstacle $ (0.02, 0.12)\times(0.02, 0.92) $
2 triangle obstacle in orange in Figure 3 $ (0.1, 0.3)\times(0.3, 0.5) $
3 triangle obstacle in orange in Figure 4 $ (0.1, 0.3)\times(0.3, 0.5) $
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Example $ \Omega_{obs} $ $ \Omega_0 $
1 no obstacle $ (0.02, 0.12)\times(0.02, 0.92) $
2 triangle obstacle in orange in Figure 3 $ (0.1, 0.3)\times(0.3, 0.5) $
3 triangle obstacle in orange in Figure 4 $ (0.1, 0.3)\times(0.3, 0.5) $
Table 1.  Table of the numerical parameters and the maximal $ L_1, L_2 $ and $ L_\infty $ errors at 20 time levels for an excitation in a medium with heterogeneous diffusion. Measured against the reference mesh with spatial step $ h = 1.95 \cdot 10^{-3} $
Mesh time step $ L_1 $ $ L_1 $ $ L_2 $ $ L_2 $ $ L_\infty $ $ L_\infty $
$ h $ $ \tau $ error of $ u $ error of $ v $ error of $ u $ error of $ v $ error of $ u $ error of $ v $
0.0625 0.001953 0.056240 0.161228 0.108336 0.265209 0.363029 0.779687
0.0313 0.000488 0.021852 0.049984 0.043336 0.094244 0.235573 0.651494
0.0156 0.000122 0.004384 0.008584 0.008694 0.014913 0.053264 0.107053
0.0078 0.000031 0.000884 0.001813 0.001997 0.002953 0.013961 0.021748
0.0039 0.000008 0.000217 0.000472 0.000505 0.000698 0.003454 0.004255
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Mesh time step $ L_1 $ $ L_1 $ $ L_2 $ $ L_2 $ $ L_\infty $ $ L_\infty $
$ h $ $ \tau $ error of $ u $ error of $ v $ error of $ u $ error of $ v $ error of $ u $ error of $ v $
0.0625 0.001953 0.056240 0.161228 0.108336 0.265209 0.363029 0.779687
0.0313 0.000488 0.021852 0.049984 0.043336 0.094244 0.235573 0.651494
0.0156 0.000122 0.004384 0.008584 0.008694 0.014913 0.053264 0.107053
0.0078 0.000031 0.000884 0.001813 0.001997 0.002953 0.013961 0.021748
0.0039 0.000008 0.000217 0.000472 0.000505 0.000698 0.003454 0.004255
Table 2.  Table of the EOC coefficients for an excitation in a medium with the heterogeneous diffusion
Mesh Mesh EOC u EOC v EOC $ u $ EOC $ v $ EOC $ u $ EOC $ v $
$ h_1 $ $ h_2 $ $ L_1 $ $ L_1 $ $ L_2 $ $ L_2 $ $ L_\infty $ $ L_\infty $
0.0625 0.0313 1.363831 1.689564 1.321875 1.492657 0.623911 0.259143
0.0313 0.0156 2.317446 2.541744 2.317474 2.659830 2.144942 2.605427
0.0156 0.0078 2.310130 2.243271 2.122186 2.336317 1.931758 2.299371
0.0078 0.0039 2.026351 1.941520 1.983479 2.080882 2.015062 2.353652
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Mesh Mesh EOC u EOC v EOC $ u $ EOC $ v $ EOC $ u $ EOC $ v $
$ h_1 $ $ h_2 $ $ L_1 $ $ L_1 $ $ L_2 $ $ L_2 $ $ L_\infty $ $ L_\infty $
0.0625 0.0313 1.363831 1.689564 1.321875 1.492657 0.623911 0.259143
0.0313 0.0156 2.317446 2.541744 2.317474 2.659830 2.144942 2.605427
0.0156 0.0078 2.310130 2.243271 2.122186 2.336317 1.931758 2.299371
0.0078 0.0039 2.026351 1.941520 1.983479 2.080882 2.015062 2.353652
Table 5.  The parameters for the second wave in Example 1 – the functional reentry
Initial state of the second wave at time $ t=3 $
$ \qquad u_{ini, 1} = u_1^0 + \sin \left( \frac{\pi (x-a^2_x)}{b^2_x-a^2_x}\right)\sin \left( \frac{\pi (y-a^2_y)}{b^2_y-a^2_y}\right) \quad \text{ in }\Omega_2 $
$ \qquad u_{ini, 2} = u_2^0 \quad \text{ in }\Omega_2 $
$ \quad \Omega_2 = \left( a^2_x, b^2_x\right)\times\left( a^2_y, b^2_y\right) = \left( 0.3, 0.4\right)\times\left( 0.4, 0.6\right) $
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Initial state of the second wave at time $ t=3 $
$ \qquad u_{ini, 1} = u_1^0 + \sin \left( \frac{\pi (x-a^2_x)}{b^2_x-a^2_x}\right)\sin \left( \frac{\pi (y-a^2_y)}{b^2_y-a^2_y}\right) \quad \text{ in }\Omega_2 $
$ \qquad u_{ini, 2} = u_2^0 \quad \text{ in }\Omega_2 $
$ \quad \Omega_2 = \left( a^2_x, b^2_x\right)\times\left( a^2_y, b^2_y\right) = \left( 0.3, 0.4\right)\times\left( 0.4, 0.6\right) $
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