Journal Articles Applied Numerical Mathematics: an IMACS journal Year : 2018

Wave dynamics on networks: method and application to the sine-Gordon equation

Abstract

We consider a scalar Hamiltonian nonlinear wave equation formulated on networks; this is a non standard problem because these domains are not locally homeomorphic to any subset of the Euclidean space. More precisely, we assume each edge to be a 1D uniform line with end points identified with graph vertices. The interface conditions at these vertices are introduced and justified using conservation laws and an homothetic argument. We present a detailed methodology based on a symplectic finite difference scheme together with a special treatment at the junctions to solve the problem and apply it to the sine-Gordon equation. Numerical results on a simple graph containing four loops show the performance of the scheme for kinks and breathers initial conditions.
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Dates and versions

hal-01160840 , version 1 (08-06-2015)
hal-01160840 , version 2 (26-06-2017)
hal-01160840 , version 3 (05-02-2018)

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Denys Dutykh, Jean-Guy Caputo. Wave dynamics on networks: method and application to the sine-Gordon equation. Applied Numerical Mathematics: an IMACS journal, 2018, 131, pp.54-71. ⟨10.1016/j.apnum.2018.03.010⟩. ⟨hal-01160840v3⟩
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