Dissipative or Conservative Finite-Difference Schemes for Complex-Valued Nonlinear Partial Differential Equations

We propose a new procedure for designing finite-difference schemes that inherit energy conservation or dissipation property from complex-valued nonlinear partial differential equations (PDEs), such as the nonlinear Schrödinger equation, the Ginzburg–Landau equation, and the Newell–Whitehead equation. The procedure is a complex version of the procedure that Furihata has recently presented for real-valued nonlinear PDEs. Furthermore, we show that the proposed procedure can be modified for designing “linearly implicit” finite-difference schemes that inherit energy conservation or dissipation property.