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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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Submitted on : Monday, February 5, 2018 - 9:59:44 AM
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Denys Dutykh, Jean-Guy Caputo. Wave dynamics on networks: method and application to the sine-Gordon equation. Applied Numerical Mathematics, Elsevier, 2018, 131, pp.54-71. ⟨10.1016/j.apnum.2018.03.010⟩. ⟨hal-01160840v3⟩



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