Symplectic Schemes for Telegraph Equations (Journal of Computational Mathematics, 2018)
A new numerical algorithm for telegraph equations with homogeneous boundary conditions is proposed. Due to the damping terms in telegraph equations, there is no royal conservation law according to Noether's theorem. The algorithm originates from the discovery of a transform applied to a telegraph equation, which transforms the telegraph equation to a Klein-Gordon equation. The Symplectic method is then brought in this algorithm to solve the Klein-Gordon equation, which is based on the fact that the Klein-Gordon equation with the homogeneous boundary condition is a perfect Hamiltonian system and the symplectic method works very well for Hamiltonian systems. The transformation itself and the inverse transformation theoretically bring no error to the numerical computation. Therefore the error only comes from the symplectic scheme chosen. The telegraph equation is finally explicitly computed when an explicit symplectic scheme is utilized. A relatively long time result can be expected due to the application of the symplectic method. Meanwhile, we present order analysis for both one-dimensional and multi-dimensional cases in the paper. The efficiency of this approach is demonstrated with numerical examples.
Keywords: Telegraph equation; Klein-Gordon equation; Symplectic method; Explicit method.