Two Kinds of Square-Conservative Integrators for Nonlinear Evolution Equations

Based on the Lie-group and Gauss--Legendre methods, two kinds of square-conservative integrators for square-conservative nonlinear evolution equations are presented. Lie-group based square-conservative integrators are linearly implicit, while Gauss--Legendre based square-conservative integrators are nonlinearly implicit and iterative schemes are needed to solve the corresponding integrators. These two kinds of integrators provide natural candidates for simulating square-conservative nonlinear evolution equations in the sense that these integrators not only preserve the square-conservative properties of the continuous equations but also are nonlinearly stable. Numerical experiments are performed to test the presented integrators.