Article

Journal of Dynamics and Differential Equations

, Volume 17, Issue 3, pp 523-572

Analysis of Newton’s Method to Compute Travelling Waves in Discrete Media

  • H. J. HupkesAffiliated withMathematisch Instituut, Universiteit Leiden Email author 
  • , S. M. Verduyn LunelAffiliated withMathematisch Instituut, Universiteit Leiden

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Abstract

We present a variant of Newton’s method for computing travelling wave solutions to scalar bistable lattice differential equations. We prove that the method converges to a solution, obtain existence and uniqueness of solutions to such equations with a small second order term and study the limiting behaviour of such solutions as this second order term tends to zero. The robustness of the algorithm will be discussed using numerical examples. These results will also be used to illustrate phenomena like propagation failure, which are encountered when studying lattice differential equations. We finish by discussing the broad application range of the method and illustrate that higher dimensional systems exhibit richer behaviour than their scalar counterparts.

Keywords

Computation of travelling waves functional differential equations Newton’s method bistable lattice differential equations numerical computation Ising model discrete media myelinated nerve fibers