Explicit high-order energy-preserving exponential time differencing method for nonlinear Hamiltonian PDEs

Highlights：

• We introduce a new fractional Sobolev norm to construct the discrete fractional Sobolev space, and also prove some important lemmas for the new fractional Sobolev norm.

• Under the periodic conditions, we establish the equivalence of the inner product estimates involving discrete fractional Laplacian and the semi-norm in the defined fractional Sobolev space, and prove that the Fourier pseudo-spectral method is unconditionally convergent with order O(τ2+Nα/2−r) in the discrete L∞ norm.

• A fast algorithm based on the fast Fourier transformation technique is used to reduce the computational complexity in practical computation.

• The method of error analysis in the discrete L∞ for the presented conservative Fourier pseudo-spectral scheme can be extended to other PDEs involving fractional Laplacian, for example, the space fractional Allen-Cahn equation and the Klein-Gordon-Schrödinger equation.