A Conservative Difference Scheme for Space Fractional Klein-Gordon-Schrödinger Equations with a High-Degree Yukawa Interaction

Abstract

A conservative finite difference scheme for nonlinear space fractional Klein-Gordon-Schrödinger systems with high-degree Yukawa interaction is studied. We show that the arising difference equations are uniquely solvable and approximate solutions converge to the exact solution at the rate (τ 2 +h 2). Moreover, we prove that the scheme can be decoupled and preserves the mass and energy conservation laws. Numerous examples confirm theoretical results and demonstrate the efficiency of the scheme. They also show the influence of the fractional order and the high-degree term coefficient on the shape and the propagation velocity of solitary waves.

Symplectic‐preserving Fourier spectral scheme for space fractionalKlein–Gordon–Schrödingerequations

“…When = 2, the system (1) and (2) reduces to the classical Klein-Gordon-Schrödinger (KGS) system [28][29][30][31][32][33] which is a very important system in quantum mechanics. Recently, some conservative schemes such as the conservative difference schemes [34][35][36][37][38][39][40], conservative Fourier spectral methods [41][42], conservative finite element method and [43], structure-preserving numerical methods [44][45] have been presented for the space fractional KGS system (1) and (2), and the corresponding results of the stability and convergence were obtained.…”

Space‐time finite element adaptive AMG for multi‐term time fractional advection diffusion equations

“…Since the vast majority could not be solved accurately or their analytical solutions (if luckily derived) always contain specific infinite series resulting in sharp costs of evaluations, numerical solutions to FDEs become very practical and prevalent. Numerous numerical (unconditionally stable and efficient) methods arise, eg, finite difference (FD) methods, 2–13 finite element (FE) methods, 14–24 finite volume method, 25–27 spectral (element) methods, 28–40 and discontinuous Galerkin method 41–44 …”

“…1 Since the vast majority could not be solved accurately or their analytical solutions (if luckily derived) always contain specific infinite series resulting in sharp costs of evaluations, numerical solutions to FDEs become very practical and prevalent. Numerous numerical (unconditionally stable and efficient) methods arise, eg, finite difference (FD) methods, [2][3][4][5][6][7][8][9][10][11][12][13] finite element (FE) methods, [14][15][16][17][18][19][20][21][22][23][24] finite volume method, [25][26][27] spectral (element) methods, [28][29][30][31][32][33][34][35][36][37][38][39][40] and discontinuous Galerkin method. [41][42][43][44] Fractional advection diffusion equations (FADEs) are known as one of the foremost models in depictions for transport process in complex systems governed by the abnormal diffusion and non-exponential relaxation patterns.…”

A multigrid-reduction-in-time solver with a new two-level convergence for unsteady fractional Laplacian problems

“…One of the main challenges of problem (1.1) lies in the nonlocality of the fractional Laplacian. Numerous numerical approximations have been proposed to treat the fractional Laplacian on bounded domains in recent years, such as finite difference [12][13][14][15][16][17][18][19][20][21], finite element (FE) [22][23][24][25][26][27][28][29], finite volume [30][31][32], discontinuous Galerkin [33][34][35][36] and spectral (element) [37][38][39][40][41][42][43][44][45][46] methods. Utilizing a similar technique to the fractional Laplacian in [47], we can obtain the Caffarelli-Silvestre extension, which is reformulated the original problem posed over Ω on the semi-infinite cylinder C = Ω × (0, +∞) with dynamic boundary condition:…”