Galerkin finite element methods for the generalized Klein–Gordon–Zakharov equations

Numerical simulation of two‐dimensional and three‐dimensional generalizedKlein–Gordon–Zakharovequations with power law nonlinearity via a meshless collocation method based on barycentric rational interpolation

“…In Table 3, we give the L ∞ and L 2 errors at T = 1 for 𝜏 = 0.0001,Ω = [−20, 20] × [ −20, 20] and increasing values of nodes where one can see that as the number of nodes increase more accurate results can be obtained. In Table 4, a comparison of the present method with the methods of [26], namely Crank-Nicolson finite difference finite element (ECFD-FE), semi-finite difference finite element (SIFD-FE), explicit finite difference finite element (EXFD-FE) and time-splitting Crank-Nicolson finite element (TSCN-FE), is given. In this case our grid consists of 36 × 36 nodes, whereas in [26] the spatial step size is h = 1∕32 which means they used a grid which consists of about 640 × 640 nodes.…”

“…In this paper we will present an efficient numerical method based on barycentric rational interpolation method (BRIM) for generalized two-dimensional (2D) and three-dimensional (3D) Klein-Gordon-Zakharov (KGZ) equations with power law nonlinearity which are given as [4,26]:…”

“…Furthermore, KGZ system transforms to Zakharov system in the high-frequency limit and reduces to the Klein-Gordon equation in the subsonic limit [5], in both the subsonic and high-frequency limits it reduces to Schrödinger equation [50]. It is known that, the Schrödinger equation has many applications in quantum mechanics [39], nonlinear optics [14] and water waves [65], which models condensed matter physics like the Bose-Einstein condensate [26]. On the other hand, Klein-Gordon equation is used to model spreading of dislocations in crystals and the behavior of elementary particles [26] also it handles the quantum evolution of wave functions for relativistic spinless particles [48,49].…”

“…It is known that, the Schrödinger equation has many applications in quantum mechanics [39], nonlinear optics [14] and water waves [65], which models condensed matter physics like the Bose-Einstein condensate [26]. On the other hand, Klein-Gordon equation is used to model spreading of dislocations in crystals and the behavior of elementary particles [26] also it handles the quantum evolution of wave functions for relativistic spinless particles [48,49]. Also wave equations have many applications in modern physics [16].…”

“…The existence and uniqueness of KGZ system are investigated in [27,57,68], different methods are suggested to obtain exact solutions of KGZ system in [35,64,67]. As mentioned in [26], it is not an easy task to find the exact solution for generalized KGZ system with power nonlinearity in 2D or 3D. Thus, it is required to devise efficient numerical techniques for KGZ equations.…”