Fourth-order compact solution of the nonlinear Klein-Gordon equation

Abstract

In this work we propose a fourth-order compact method for solving the one-dimensional nonlinear Klein-Gordon equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivative and a fourth-order A-stable diagonally-implicit Runge-Kutta-Nyström (DIRKN) method for the time integration of the resulting nonlinear secondorder system of ordinary differential equations. The proposed method has fourth order accuracy in both space and time variables and is unconditionally stable. Numerical results obtained from solving several problems possessing periodic, kinks, single and double-soliton waves show that the combination of a compact finite difference approximation of fourth order and a fourth-order A-stable DIRKN method gives an efficient algorithm for solving these problems.

An efficient high-order explicit scheme for solving Hamiltonian nonlinear wave equations

“…The L 1 and RMS errors for solving Problem 2 at different time levels are listed in Table 3. It can be observed from Table 3 that the new explicit scheme described in this paper gives similar error results in terms of L 1 and RMS to the CDIMRK method presented in [17], however, it should be noted that the result of energy errors obtained by the new explicit scheme is much better than that obtained from the CDIMRK method. In particular, as showed in Table 3, the coarser grids and larger time stepsizes are used by the new explicit scheme.…”

“…The numerical results show that the new explicit scheme proposed in this paper has better results in comparison with the technique developed in [8] and the CDIMRK method presented in [17].…”

“…The real parameter ffiffiffiffiffiffiffiffiffiffi ffi 2a=b p represents the amplitude of a soliton which travels with the velocity c. The problem can be found in [8,17]. For comparison we consider parameters a ¼ 0:3; b ¼ 1 and c ¼ 0:25 similar to [8].…”

“…The problem also can be found in [8,17,29], and we choose parameters a ¼ 0: [8]. This is the collision of two solitons, in which at t ¼ 0 they are placed at x 1 0 ¼ À2 and x 2 0 ¼ 2 and travel with the velocities c 1 ¼ 0:25; c 2 ¼ À0:25, respectively.…”

“…Besides, the Adomian decomposition method was used as well [15,16]. In [17], the authors considered a fourth-order compact method in space and a fourth-order A-stable diagonally implicit Runge-Kutta-Nyström method in time of the resulting nonlinear second-order system of ordinary differential equations.…”