## 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.

Original language | British English |
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Pages (from-to) | 54-71 |

Number of pages | 18 |

Journal | Applied Numerical Mathematics |

Volume | 131 |

DOIs | |

State | Published - Sep 2018 |

## Keywords

- Graph theory
- HAMILTONIAN partial differential equations
- Partial differential equations on networks
- Sine-GORDON equation