On the numerical solution of the Klein‐Gordon equation

Abstract

A predictor-corrector (P-C) scheme based on the use of rational approximants of second-order to the matrixexponential term in a three-time level reccurence relation is applied to the nonlinear Klein-Gordon equation. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. Both the predictor and the corrector scheme are analyzed for local truncation error and stability. The proposed method is applied to problems possessing periodic, kinks and single, double-soliton waves. The accuracy as well as the long time behavior of the proposed scheme is discussed.

Fast and High Accuracy Numerical Methods for the Solution of Nonlinear Klein–Gordon Equations

“…In Figure 1, we show the approximate solutions of Problem 4.1 with A = 1.5 and A = 150. As we see, the calculated approximate solutions are similar to the results of [4]. Also the approximate solutions obtained remain symmetric with respect to the center of the spatial interval, and the solution remained bounded for amplitude A = 150 when t ∈ [0, 36].…”

“…Authors of [9] studied this problem and found undesirable characteristics in some of the numerical schemes, in particular a loss of spatial symmetry and the onset of instability for larger values of the parameter A (amplitude) in the initial condition of the equation. Also, it was found that the numerical results given in [4] were more accurate than the other results given in [9,10,18]. In Figure 1, we show the approximate solutions of Problem 4.1 with A = 1.5 and A = 150.…”

“…For the above problem, and due to the periodic boundary conditions, the continuous solutions remain always symmetric with respect to the center of the spatial interval [4,9]. Authors of [9] studied this problem and found undesirable characteristics in some of the numerical schemes, in particular a loss of spatial symmetry and the onset of instability for larger values of the parameter A (amplitude) in the initial condition of the equation.…”

“…Equation (1) occurs in a series of physical situations, as the propagation of waves in ferromagnetic materials carrying rotations of the direction of magnetization and of laser pulses in two-state media [4,7].…”

“…Bratsos proposed another approach in [6] for solving (5) which has second-order accuracy in space and fourth-order accuracy in the time variable. Finally, Bratsos in [4,7] developed a predictor-corrector (PC) scheme based on the use of rational approximation of second order to the matrix exponential term in a threetime level recurrence relation. Most of the existing methods in the literature for solving KG equations are time consuming schemes and have a low order of accuracy.…”