A linearly implicit conservative scheme for the fractional nonlinear Schrödinger equation with wave operator

Abstract

The present work is mainly devoted to studying the fractional nonlinear Schrödinger equation with wave operator. We first derive two conserved quantities of the equation, and then develop a three-level linearly implicit difference scheme. This scheme is shown to be conserves the discrete version of conserved quantities. Using energy method, we prove that the difference scheme is unconditionally stable, and the difference solution converges to the exact one with second order accuracy in both the space and time dimensions. Numerical experiments are performed to support our theoretical analysis and demonstrate the accuracy, discrete conservation laws and effectiveness for long-time simulation.

Multi-symplectic variational integrators for nonlinear Schrödinger equations with variable coefficients

“…The discrete version of mass and momentum conservation law are defined as Eqs. ( 27) and (28). The discrete version of energy conservation law (8) is written as…”

“…where α and β are constants. The nonlinear Schrödinger equation has been extensively studied by various numerical methods, such as finite element methods, [27] finite difference methods, [28] spectral methods, [29,30] splitting methods, [6,7,18,19,[31][32][33] etc. Among these numerical methods of different categories, the multisymplectic method has attracted special attention for its better numerical stability for long-time computations and perfect performance in preserving the intrinsic properties and conservation laws of nonlinear Schrödinger equations.…”

Time second‐order splitting conservative difference scheme for nonlinear fractional Schrödinger equation

“…So far many excellent conservative algorithms have been created to mimic the nonlinear space fractional Schrödinger equation. [12][13][14][15][16][17][18][19][20][21][22][23] As we know, in terms of the advantage of saving the memory and CPU time, the splitting algorithms have been extensionally applied to numerically simulate PDEs and nonlinear problems (cf. previous work [24][25][26][27][28][29].…”

“…Devising accurate and efficient algorithms capable of preserving the invariant properties of the system is essential because of the merits in stability and good resolution. So far many excellent conservative algorithms have been created to mimic the nonlinear space fractional Schrödinger equation 12–23 …”

A Fast Conservative Scheme for the Space Fractional Nonlinear Schrödinger Equation with Wave Operator

“…For the FNLSW, to the authors' knowledge, the literature limited. For instance, in [20], a linearly implicit conservative scheme is constructed based on the finite difference method. The Galerkin finite element method is used to solve the FNLSW in [14].…”