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 & 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, 569 (2003).   Google Scholar

[2]

H. Berestycki and F. Hamel, Front propagation in periodic excitable media,, Comm. Pure Appl. Math., 55 (2002), 949.  doi: 10.1002/cpa.3022.  Google Scholar

[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, 35 (1991), 37.  doi: 10.1007/978-1-4612-0947-8_2.  Google Scholar

[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.  doi: 10.1088/0951-7715/22/12/002.  Google Scholar

[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.  doi: 10.1137/0516088.  Google Scholar

[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.  doi: 10.1098/rsta.1991.0001.  Google Scholar

[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.   Google Scholar

[8]

F. Bouchut, "Nonlinear Stability of Finite Volume methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources,", Series Frontiers in Mathematics, (2004).  doi: 10.1007/b93802.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/S1064827598344868.  Google Scholar

[11]

E. J. Doedel, M. J. Friedman and B. I. Kunin, Successive continuation for locating con- necting orbits,, Numer. Algorithms, 14 (1997), 103.  doi: 10.1023/A:1019152611342.  Google Scholar

[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).   Google Scholar

[13]

R. A. Fisher, "The Genetical Theory of Natural Selection,", Clarendon Press, (1930).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1051/mmnp:2006004.  Google Scholar

[17]

G. M. Lieberman, "Second Order Parabolic Differential Operators,", World Scientific Publishing. Co. Singapore, (1996).   Google Scholar

[18]

W. Malfliet, Travelling-wave solutions of coupled nonlinear evolution equations,, Mathematics and Computers in Simulation, 62 (2003), 101.  doi: 10.1016/S0378-4754(02)00182-9.  Google Scholar

[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.  doi: 10.1016/j.crma.2011.03.008.  Google Scholar

[20]

G. Nadin, Pulsating traveling fronts in space-time periodic media,, C. R. Acad. Sci. Paris I, 346 (2008), 951.  doi: 10.1016/j.crma.2008.07.030.  Google Scholar

[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.  doi: 10.1137/S1540345902420234.  Google Scholar

[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.  doi: 10.1088/0951-7715/24/4/012.  Google Scholar

[23]

B. Perthame, "Transport Equations in Biology,", (LN Series Frontiers in Mathematics), (2007).   Google Scholar

[24]

M. Sermange, Mathematical and numerical aspects of one-dimensional laminar flame simulation,, Appl. Math. Optim., 14 (1986), 131.  doi: 10.1007/BF01442232.  Google Scholar

[25]

G. Samaey, K. Engelborghs and D. Roose, Numerical computation of connecting orbits in delayed differential equations,, Numerical Algorithms, 30 (2002), 335.  doi: 10.1023/A:1020102317544.  Google Scholar

[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.   Google Scholar

[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, ().   Google Scholar

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, 569 (2003).   Google Scholar

[2]

H. Berestycki and F. Hamel, Front propagation in periodic excitable media,, Comm. Pure Appl. Math., 55 (2002), 949.  doi: 10.1002/cpa.3022.  Google Scholar

[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, 35 (1991), 37.  doi: 10.1007/978-1-4612-0947-8_2.  Google Scholar

[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.  doi: 10.1088/0951-7715/22/12/002.  Google Scholar

[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.  doi: 10.1137/0516088.  Google Scholar

[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.  doi: 10.1098/rsta.1991.0001.  Google Scholar

[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.   Google Scholar

[8]

F. Bouchut, "Nonlinear Stability of Finite Volume methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources,", Series Frontiers in Mathematics, (2004).  doi: 10.1007/b93802.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/S1064827598344868.  Google Scholar

[11]

E. J. Doedel, M. J. Friedman and B. I. Kunin, Successive continuation for locating con- necting orbits,, Numer. Algorithms, 14 (1997), 103.  doi: 10.1023/A:1019152611342.  Google Scholar

[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).   Google Scholar

[13]

R. A. Fisher, "The Genetical Theory of Natural Selection,", Clarendon Press, (1930).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1051/mmnp:2006004.  Google Scholar

[17]

G. M. Lieberman, "Second Order Parabolic Differential Operators,", World Scientific Publishing. Co. Singapore, (1996).   Google Scholar

[18]

W. Malfliet, Travelling-wave solutions of coupled nonlinear evolution equations,, Mathematics and Computers in Simulation, 62 (2003), 101.  doi: 10.1016/S0378-4754(02)00182-9.  Google Scholar

[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.  doi: 10.1016/j.crma.2011.03.008.  Google Scholar

[20]

G. Nadin, Pulsating traveling fronts in space-time periodic media,, C. R. Acad. Sci. Paris I, 346 (2008), 951.  doi: 10.1016/j.crma.2008.07.030.  Google Scholar

[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.  doi: 10.1137/S1540345902420234.  Google Scholar

[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.  doi: 10.1088/0951-7715/24/4/012.  Google Scholar

[23]

B. Perthame, "Transport Equations in Biology,", (LN Series Frontiers in Mathematics), (2007).   Google Scholar

[24]

M. Sermange, Mathematical and numerical aspects of one-dimensional laminar flame simulation,, Appl. Math. Optim., 14 (1986), 131.  doi: 10.1007/BF01442232.  Google Scholar

[25]

G. Samaey, K. Engelborghs and D. Roose, Numerical computation of connecting orbits in delayed differential equations,, Numerical Algorithms, 30 (2002), 335.  doi: 10.1023/A:1020102317544.  Google Scholar

[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.   Google Scholar

[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, ().   Google Scholar

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