Conservative finite difference methods for fractional Schrödinger–Boussinesq equations and convergence analysis

Abstract

In this paper, two conservative finite difference schemes for fractional Schrödinger-Boussinesq equations are formulated and investigated. The convergence of the nonlinear fully implicit scheme is established via discrete energy method, while the linear semi-implicit scheme is analyzed by means of mathematical induction method. Our schemes are proved to preserve the total mass and energy in discrete level. The numerical results are given to confirm the theoretical analysis. KEYWORDSconservative law, convergence, discrete energy method, mathematical induction method, Schrödinger-Boussinesq equations 1 Numer Methods Partial Differential Eq. 2019;35:1305-1325. wileyonlinelibrary.com/journal/num

A Conservative Difference Scheme for Solving the Coupled Fractional Schrödinger–Boussinesq System

“…Table 8 lists the errors and computational times for both schemes with different values of α. Clearly, our scheme provides a more accurate solution than Scheme I in [36], with only a slightly higher computational time cost.…”

“…To enhance conservation accuracy, a smaller iteration tolerance can be applied, albeit at the expense of increased computational cost. Finally, we present a numerical comparison between our scheme ( 26)-( 30) and Scheme I from [36]. Table 8 lists the errors and computational times for both schemes with different values of α.…”

“…The error norms and graphical solutions are also presented in this work. In [36], Liao et al developed and rigorously analyzed two efficient conservative difference schemes for FCSBS. Each scheme is demonstrated to preserve two fundamental conservation laws: mass conservation and energy conservation, while converging with an accuracy of O(τ 2 + h 2 ).…”

Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

“…Additionally, the computation of the NLS is a critical part of the verification process of the analytical theories. This has been achieved in the case of non-varying coefficients, with success for a large number of comparative numerical algorithms [12][13][14][15][16].…”

Numerical approximations for the nonlinear time fractional reaction–diffusion equation

“…The analytical solutions solved by Laplace transform, Fourier transform and Green function method mostly include some special functions, such as Mittag-Leffler and Wright functions, whose values are difficult to calculate and series converge slowly. Thus it is of great importance to solve the numerical solutions of equations, which have been obtained by finite element method [9,12,15], finite difference method [4,14,21] and spectral method [3,8,26].…”