The compact finite-difference scheme and modified Richardson extrapolation for NLSE

Abstract: A compact finite-difference scheme combined with predictor-corrector approach for solving quasilinear partial differential equations and systems is presented. The nonlinear Schrödinger equation (NLSE) serves as a model problem to demonstrate the method’s capabilities. The proposed algorithm achieves fourth-order spatial accuracy and second-order temporal accuracy while maintaining computational efficiency through linearization via Newton — Raphson iterations. As a rule, one iteration is sufficient. The scheme was optimized according to the Courant parameter based on the criterion: the ratio of computational complexity to solution accuracy.

Also, we introduce a modified two-dimensional and quasi-two-dimensional Richardson extrapolation technique that further enhances accuracy up to eighth-order.

Numerical experiments confirm the scheme’s high precision and stability across a range of Courant parameters as well as a good conservation of many first integrals of NLSE. The method is applicable to arbitrary smooth initial data and various boundary conditions. We tested its properties on various solutions (solitons, collision of several solitons, chains, short-wave noise). In the latter two cases, there is an alternation of chaotic and ordered types of solution behavior.

This is joint work with D. P. Milutin.

Russianmathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods