An energy-preserving computational approach for the semilinear space fractional damped Klein–Gordon equation with a generalized scalar potential

Naturally preserving dissipative or conservative structures of a given continuous system in discrete analogs is of high demand when designing numerical schemes for dissipative or Hamiltonian partial differential equations. Armed with this fact, the proposed computational study focuses on the issue of considering energy-preserving discrete numerical schemes for the Riesz space-fractional Klein-Gordon equation with a generalized scalar potential. For the sake of more clearness, a combined numerical scheme that owes energy-preserving properties is a target to be achieved here. This is done by adapting Galerkin spectral approximation based on Legendre polynomials for Riesz space Laplacian operator side by side to invoke a Cranck–Nickolson scheme in time direction after making order reduction of the original system and a special second-order approximation of the scalar potential derivative. The numerical properties (stability, uniqueness, and convergence) of the scheme are well investigated. Numerical simulations also show that the proposed scheme can inherit the physical properties as the original problem and the numerical solution is stable and convergent to the exact solution.