| Citation: |
Yuqing Lv, Yanming Zhang, Yu Li. AN EFFICIENT ENERGY-PRESERVING SCHEME FOR NONLINEAR GENERALIZED DAMPED SPACE-FRACTIONAL KLEIN-GORDON EQUATION BASED ON THE MATRIX TRANSFER TECHNIQUE[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 313-336. doi: 10.11948/20250376
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AN EFFICIENT ENERGY-PRESERVING SCHEME FOR NONLINEAR GENERALIZED DAMPED SPACE-FRACTIONAL KLEIN-GORDON EQUATION BASED ON THE MATRIX TRANSFER TECHNIQUE
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Abstract
This paper proposes a high-order and energy-preserving finite difference scheme for solving nonlinear generalized space-fractional damped Klein-Gordon equations. The scheme is conceptually straightforward and easy to implement. By employing the matrix transfer technique, a fully diagonal spatial discretization matrix is obtained, which significantly reduces computational and storage complexity. Combined with an extended Runge-Kutta-Nyström method in time, the fully discrete scheme achieves fourth-order accuracy in both space and time. The preservation of the dissipative or conservative of the energy for the nonlinear generalized space-fractional damped Klein-Gordon equations is rigorously proved across various physical settings. Numerical experiments confirm that the proposed scheme exhibits convergence to the exact solution and retains the physical properties of the original problem.
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References
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Yuqing Lv, Yanming Zhang, Yu Li. AN EFFICIENT ENERGY-PRESERVING SCHEME FOR NONLINEAR GENERALIZED DAMPED SPACE-FRACTIONAL KLEIN-GORDON EQUATION BASED ON THE MATRIX TRANSFER TECHNIQUE[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 313-336. doi: 10.11948/20250376
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Figure 1.
Snapshots corresponding to (a) the numerical solution, (b) the absolute error of the numerical solution, (c) the time evolution of the
$ u(0, t) $ $ \alpha = 2 $ $ N = 256 $ $ M = 1000 $ -
Figure 2.
Graphs corresponding to (a) the numerical solution of
$ \alpha=1.1 $ $ \alpha=1.5 $ $ \alpha=1.9 $ $ N = 256 $ $ M = 700 $ $ \alpha $ -
Figure 3.
The numerical solution and the energy of FKGE in the de Sitter space at
$ H = 1 $ $ N = 256 $ $ M = 400 $ $ \alpha $ $ \alpha $ -
Figure 4.
The numerical solution and the energy of FKGE in the de Sitter space at
$ H $ $ N $ $ M $ $ \alpha $ $ \alpha $ -
Figure 5.
The numerical solution and the energy of FKGE in the de Sitter space at
$ H=-1 $ $ N = 256 $ $ M = 400 $ $ \alpha $ $ \alpha $ -
Figure 6.
Snapshots of the evolution of the approximate solution for 2D model with
$ N=250 $ $ M=300 $ $ \alpha = 1.5 $ $ T= 0 $ -
Figure 7.
Snapshots of the evolution of the approximate solution for 2D model with
$ N=250 $ $ M=300 $ $ \alpha = 1.8 $ $ T= 0 $ -
Figure 8.
Snapshots of the evolution of the approximate solution for 2D model with
$ N=250 $ $ M=300 $ $ \alpha = 2 $ $ T= 0 $ -
Figure 9.
(a) Temporal progression of the relative energy errors and (b) the dissipation-preserving law under the parameters
$ h=1/5 $ $ \tau=1/40 $ $ \alpha $ $ \kappa $