# American Institute of Mathematical Sciences

July  2013, 18(5): 1459-1491. doi: 10.3934/dcdsb.2013.18.1459

## A relaxation method for one dimensional traveling waves of singular and nonlocal equations

 1 Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 4Z2, Canada 2 Department of mathematics and Institute of Natural Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China

Received  June 2012 Revised  November 2012 Published  March 2013

Recent models motivated by biological phenomena lead to non-local PDEs or systems with singularities. It has been recently understood that these systems may have traveling wave solutions that are not physically relevant [19]. We present an original method that relies on the physical evolution to capture the stable" traveling waves. This method allows us to obtain the traveling wave profiles and their traveling speed simultaneously. It is easy to implement, and it applies to classical differential equations as well as nonlocal equations and systems with singularities. We also show the convergence of the scheme analytically for bistable reaction diffusion equations over the whole space $\mathbb{R}$.
Citation: Weiran Sun, Min Tang. A relaxation method for one dimensional traveling waves of singular and nonlocal equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1459-1491. doi: 10.3934/dcdsb.2013.18.1459
##### References:
 [1] H. Berestycki, The influence of advection on the propagation of fronts in reaction-diffusion equations, Nonlinear PDEs in Condensed Matter and Reactive Flows, NATO Science Series C, 569, (eds. H. Berestycki and Y. Pomeau), Kluwer, Dordrecht, (2003). [2] H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032 . doi: 10.1002/cpa.3022. [3] H. Berestycki, B. Larrouturou and J.-M. Roquejoffre, Mathematical investigation of the cold boundary difficulty in flame propagation theory, Dynamical Issues in Combustion Theory (Minneapolis, MN, 1989), 37-61, IMA Vol. Math. Appl., 35, Springer, New York, (1991). doi: 10.1007/978-1-4612-0947-8_2. [4] H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: traveling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002. [5] H. Berestyski, B. Nicolaenko and B. Scheurer, Traveling wave solutions to combustion models and their singular limits, SIAM J. Math. Anal, 16 (1985), 1207-1242. doi: 10.1137/0516088. [6] J. Billingham and N. L. Needham, The development of traveling waves in quadratic and cubic autocatalysis with unequal diffusion rates. I. Permamnet form traveling waves, Phil. Trans. R. Soc. Lond. A, 334 (1991), 1-24. doi: 10.1098/rsta.1991.0001. [7] V. Gubernov, G. N. Mercer, H. S. Sidhu and R. O. Weber, Numerical methods for the traveling wave solutions in reaction diffusion equations, ANZIAM J., 44 (2002), 271-299. [8] F. Bouchut, "Nonlinear Stability of Finite Volume methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources," Series Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/b93802. [9] F. Cerreti, B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Waves for an hyperbolic Keller-Segel model and branching instabilities, Math. Models and Meth. in Appl. Sci., 21 (2011), 825-842. [10] J. Demmel, L. Dieci and M. Friedman, Computing connecting orbits via an improved algorithm for continuing invariant subspaces, SIAM J. Sci. Comp., 22 (2000), 81-94. doi: 10.1137/S1064827598344868. [11] E. J. Doedel, M. J. Friedman and B. I. Kunin, Successive continuation for locating con- necting orbits, Numer. Algorithms, 14 (1997), 103-124. doi: 10.1023/A:1019152611342. [12] M. A. Fuentes, M. N. Kuperman and V. M. Kenkre, Nonlocal interaction effects on pattern formation in population dynamics, Phys. Rev. Lett., 91 (2003), 158104. [13] R. A. Fisher, "The Genetical Theory of Natural Selection," Clarendon Press, 1930. second edition: Dover, 1985. Third edition, Oxford Univ. Press, 1999. [14] P. C. Fife and J. B. Mcleod, The approach of solutions of non-linear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361. [15] A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, Etude de l équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bulletin Université d'Etat à Moscou (Bjul. Moskouskogo Gos. Univ.), (1937), 1-26. [16] S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Modelling Nat. Phenom., 1 (2006), 65-82. doi: 10.1051/mmnp:2006004. [17] G. M. Lieberman, "Second Order Parabolic Differential Operators," World Scientific Publishing. Co. Singapore, 1996. [18] W. Malfliet, Travelling-wave solutions of coupled nonlinear evolution equations, Mathematics and Computers in Simulation, 62 (2003), 101-108. doi: 10.1016/S0378-4754(02)00182-9. [19] G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Acad. Sci. Paris I, 349 (2011), 559-557. doi: 10.1016/j.crma.2011.03.008. [20] G. Nadin, Pulsating traveling fronts in space-time periodic media, C. R. Acad. Sci. Paris I, 346 (2008), 951-956. doi: 10.1016/j.crma.2008.07.030. [21] J. Nolen and J. Xin, Reaction-diffusion front speeds in spatially-temporally periodic shear flows, SIAM J. Multiscale Modeling and Simulation, 1 (2003), 554-570. doi: 10.1137/S1540345902420234. [22] B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Traveling plateaus for a Keller-Segel system with logistic sensitivity; Existence and branching instabilities, Nonlinearity, 24 (2011), 1253-1270. doi: 10.1088/0951-7715/24/4/012. [23] B. Perthame, "Transport Equations in Biology," (LN Series Frontiers in Mathematics), Birkhauser, 2007. [24] M. Sermange, Mathematical and numerical aspects of one-dimensional laminar flame simulation, Appl. Math. Optim., 14 (1986), 131-153. doi: 10.1007/BF01442232. [25] G. Samaey, K. Engelborghs and D. Roose, Numerical computation of connecting orbits in delayed differential equations, Numerical Algorithms, 30 (2002), 335-352. doi: 10.1023/A:1020102317544. [26] M. D. Smooke, J. A. Miller and R. J. Kee, Determination of adiabatic flame speeds by boundary value methods, Combustion Sci. and Technology, 34 (1983), 79-90. [27] M. Tang, A relaxation method for the pulsating traveling front simulations of the space and time periodic advection diffusion reaction equations, Communications in Mathematical Sciences, Accepted.

show all references

##### References:
 [1] H. Berestycki, The influence of advection on the propagation of fronts in reaction-diffusion equations, Nonlinear PDEs in Condensed Matter and Reactive Flows, NATO Science Series C, 569, (eds. H. Berestycki and Y. Pomeau), Kluwer, Dordrecht, (2003). [2] H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032 . doi: 10.1002/cpa.3022. [3] H. Berestycki, B. Larrouturou and J.-M. Roquejoffre, Mathematical investigation of the cold boundary difficulty in flame propagation theory, Dynamical Issues in Combustion Theory (Minneapolis, MN, 1989), 37-61, IMA Vol. Math. Appl., 35, Springer, New York, (1991). doi: 10.1007/978-1-4612-0947-8_2. [4] H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: traveling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002. [5] H. Berestyski, B. Nicolaenko and B. Scheurer, Traveling wave solutions to combustion models and their singular limits, SIAM J. Math. Anal, 16 (1985), 1207-1242. doi: 10.1137/0516088. [6] J. Billingham and N. L. Needham, The development of traveling waves in quadratic and cubic autocatalysis with unequal diffusion rates. I. Permamnet form traveling waves, Phil. Trans. R. Soc. Lond. A, 334 (1991), 1-24. doi: 10.1098/rsta.1991.0001. [7] V. Gubernov, G. N. Mercer, H. S. Sidhu and R. O. Weber, Numerical methods for the traveling wave solutions in reaction diffusion equations, ANZIAM J., 44 (2002), 271-299. [8] F. Bouchut, "Nonlinear Stability of Finite Volume methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources," Series Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/b93802. [9] F. Cerreti, B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Waves for an hyperbolic Keller-Segel model and branching instabilities, Math. Models and Meth. in Appl. Sci., 21 (2011), 825-842. [10] J. Demmel, L. Dieci and M. Friedman, Computing connecting orbits via an improved algorithm for continuing invariant subspaces, SIAM J. Sci. Comp., 22 (2000), 81-94. doi: 10.1137/S1064827598344868. [11] E. J. Doedel, M. J. Friedman and B. I. Kunin, Successive continuation for locating con- necting orbits, Numer. Algorithms, 14 (1997), 103-124. doi: 10.1023/A:1019152611342. [12] M. A. Fuentes, M. N. Kuperman and V. M. Kenkre, Nonlocal interaction effects on pattern formation in population dynamics, Phys. Rev. Lett., 91 (2003), 158104. [13] R. A. Fisher, "The Genetical Theory of Natural Selection," Clarendon Press, 1930. second edition: Dover, 1985. Third edition, Oxford Univ. Press, 1999. [14] P. C. Fife and J. B. Mcleod, The approach of solutions of non-linear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361. [15] A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, Etude de l équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bulletin Université d'Etat à Moscou (Bjul. Moskouskogo Gos. Univ.), (1937), 1-26. [16] S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Modelling Nat. Phenom., 1 (2006), 65-82. doi: 10.1051/mmnp:2006004. [17] G. M. Lieberman, "Second Order Parabolic Differential Operators," World Scientific Publishing. Co. Singapore, 1996. [18] W. Malfliet, Travelling-wave solutions of coupled nonlinear evolution equations, Mathematics and Computers in Simulation, 62 (2003), 101-108. doi: 10.1016/S0378-4754(02)00182-9. [19] G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Acad. Sci. Paris I, 349 (2011), 559-557. doi: 10.1016/j.crma.2011.03.008. [20] G. Nadin, Pulsating traveling fronts in space-time periodic media, C. R. Acad. Sci. Paris I, 346 (2008), 951-956. doi: 10.1016/j.crma.2008.07.030. [21] J. Nolen and J. Xin, Reaction-diffusion front speeds in spatially-temporally periodic shear flows, SIAM J. Multiscale Modeling and Simulation, 1 (2003), 554-570. doi: 10.1137/S1540345902420234. [22] B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Traveling plateaus for a Keller-Segel system with logistic sensitivity; Existence and branching instabilities, Nonlinearity, 24 (2011), 1253-1270. doi: 10.1088/0951-7715/24/4/012. [23] B. Perthame, "Transport Equations in Biology," (LN Series Frontiers in Mathematics), Birkhauser, 2007. [24] M. Sermange, Mathematical and numerical aspects of one-dimensional laminar flame simulation, Appl. Math. Optim., 14 (1986), 131-153. doi: 10.1007/BF01442232. [25] G. Samaey, K. Engelborghs and D. Roose, Numerical computation of connecting orbits in delayed differential equations, Numerical Algorithms, 30 (2002), 335-352. doi: 10.1023/A:1020102317544. [26] M. D. Smooke, J. A. Miller and R. J. Kee, Determination of adiabatic flame speeds by boundary value methods, Combustion Sci. and Technology, 34 (1983), 79-90. [27] M. Tang, A relaxation method for the pulsating traveling front simulations of the space and time periodic advection diffusion reaction equations, Communications in Mathematical Sciences, Accepted.
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