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Unconditional optimal-order error estimates of linear relaxation compact difference scheme for the coupled nonlinear Schrödinger system.

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Title: Unconditional optimal-order error estimates of linear relaxation compact difference scheme for the coupled nonlinear Schrödinger system.
Authors: Gao, Ying1 (AUTHOR) gy9168@stu.ouc.edu.cn, Fu, Hongfei1,2 (AUTHOR) fhf@ouc.edu.cn, Wang, Xiaoying1 (AUTHOR) wxy7121@stu.ouc.edu.cn
Source: Applied Numerical Mathematics. Sep2026, Vol. 227, p108-128. 21p.
Subjects: Nonlinear Schrödinger equation, Error analysis in mathematics, Conservation of energy, Algorithms, Finite difference method, Relaxation methods (Mathematics), Conservation of mass
Abstract: This paper presents a linear, decoupled, mass- and energy-conserving numerical scheme for the multi-dimensional coupled nonlinear Schrödinger (CNLS) system. The scheme combines the fourth-order compact difference approximation in space with the relaxation technique in a time-staggered mesh framework, enabling the sequential solution of the primal unknowns and the introduced auxiliary relaxation variables with high efficiency and high-order accuracy. We establish the unique solvability and discrete conservation laws of the proposed scheme. In particular, for the first time, by leveraging an auxiliary error equation system together with a cut-off technique, optimal-order error estimates are rigorously proved–without any coupling mesh conditions–for the primal variables in the discrete H 1-norm at the time nodes, and for the auxiliary relaxation variables in the discrete L 2-norm at the intermediate time nodes. This constitutes the primary theoretical contribution of this paper for multi-dimensional CNLS system. Numerical experiments demonstrate convincingly the strong performance of the proposed scheme in long-term simulations, maintaining both physical invariants and high-order accuracy. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
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Abstract:This paper presents a linear, decoupled, mass- and energy-conserving numerical scheme for the multi-dimensional coupled nonlinear Schrödinger (CNLS) system. The scheme combines the fourth-order compact difference approximation in space with the relaxation technique in a time-staggered mesh framework, enabling the sequential solution of the primal unknowns and the introduced auxiliary relaxation variables with high efficiency and high-order accuracy. We establish the unique solvability and discrete conservation laws of the proposed scheme. In particular, for the first time, by leveraging an auxiliary error equation system together with a cut-off technique, optimal-order error estimates are rigorously proved–without any coupling mesh conditions–for the primal variables in the discrete H 1-norm at the time nodes, and for the auxiliary relaxation variables in the discrete L 2-norm at the intermediate time nodes. This constitutes the primary theoretical contribution of this paper for multi-dimensional CNLS system. Numerical experiments demonstrate convincingly the strong performance of the proposed scheme in long-term simulations, maintaining both physical invariants and high-order accuracy. [ABSTRACT FROM AUTHOR]
ISSN:01689274
DOI:10.1016/j.apnum.2026.04.011