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Long time error analysis of the fourth‐order compact finite difference methods for the nonlinear Klein–Gordon equation with weak nonlinearity
Author(s) -
Feng Yue
Publication year - 2021
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22558
Subject(s) - dimensionless quantity , mathematics , nonlinear system , mathematical analysis , space time , order (exchange) , space (punctuation) , spacetime , constant (computer programming) , physics , quantum mechanics , computer science , finance , chemical engineering , engineering , economics , operating system , programming language
We present the fourth‐order compact finite difference (4cFD) discretizations for the long time dynamics of the nonlinear Klein–Gordon equation (NKGE), while the nonlinearity strength is characterized by ϵ p with a constant p ∈ ℕ + and a dimensionless parameter ϵ ∈ (0, 1] . Based on analytical results of the life‐span of the solution, rigorous error bounds of the 4cFD methods are carried out up to the time at O ( ϵ − p ) . We pay particular attention to how error bounds depend explicitly on the mesh size h and time step τ as well as the small parameter ϵ ∈ (0, 1] , which indicate that, in order to obtain ‘correct’ numerical solutions up to the time at O ( ϵ − p ) , the ϵ ‐scalability (or meshing strategy requirement) of the 4cFD methods should be taken as: h = O ( ϵ p /4 ) and τ = O ( ϵ p /2 ) . It has better spatial resolution capacity than the classical second order central difference methods. By a rescaling in time, it is equivalent to an oscillatory NKGE whose solution propagates waves with wavelength at O (1) in space and O ( ϵ p ) in time. It is straightforward to get the error bounds of the oscillatory NKGE in the fixed time. Finally, numerical results are provided to confirm our theoretical analysis.