Some numerical approaches for linear Klein – Gordon equations

In this presentation I will focus on some numerical approaches to certain types of linear Klein Gordon equations. More particularly, I will present the numerical approach based on the Duhamel formula, where we obtain second order approximation. Modulated Fourier expansion based method will be presented as the one useful in case of highly oscillatory forcing term. Finally, I will present some splitting methods of fourth order. I will present plenty of numerical examples and comparisons.

References:
[1] Bader P., Blanes S., Casas F., Kopylov N., Novel symplectic integrators for the Klein–Gordon equation with space- and time-dependent mass, Journal of Computational and Applied Mathematics, Volume 350, 2019, 130-138 (2019).
[2] M. Condon, K. Kropielnicka, K. Lademann, R. Perczyński, Asymptotic numerical solver for the linear Klein–Gordon equation with space- and time-dependent mass, Applied Mathematics Letters,115,106935, (2021)
[3] Bauke H., Ruf M, Keitel Ch., A real space split operator method for the Klein-Gordon equation, J. Comput. Phys., 228, 24, 9092–9106, (2009).
[4] Mostafazadeh A., Quantum mechanics of Klein–Gordon-type fields and quantum cosmology, Ann. Physics 309, no. 1, 1–48 (2004).
[5] Mostafazadeh A.,Hilbert space structures on the solution space of Klein-Gordon-type evolution equations, Classical Quantum Gravity 20, no. 1, 155–171 (2003).
[6] Znojil M., Klein-Gordon equation with the time- and space-dependent mass: Unitary evolution picture, arXiv:1702.08493v1
[7] Znojil M., Non-Hermitian interaction representation and its use in relativistic quantum mechanics, Ann. Physics 385, 162–179 (2017).