Multi-symplectic Runge–Kutta–Nyström methods for nonsmooth nonlinear Schrödinger equations

In this paper, we discuss multi-symplectic (MS) Runge–Kutta–Nyström (RKN) methods applied to a kind of nonsmooth nonlinear Schrödinger equations (NLSEs), i.e. the NLSEs with Delta potentials. Based on some theoretical suggestions proposed in [1], e.g. the weak reformulation of multi-symplectic Hamiltonian system (MSHS) for the NLSEs and some newly addressed local and global conservation laws, concatenating RKN algorithms for this weak MSHS are novelly constructed and the multi-symplecticity of them is revealed. Under the MSRKN discretizations, we show that some intrinsic characters of this weak MSHS, including the global symplectic structure in time and the normalization conservation laws, are preserved analytically just like that for the smooth NLSEs [11]. Numerical experiments validate our methods. And the errors of numerical solutions, discrete energies and discrete parity symmetry are also investigated numerically. Comparisons with non-MS schemes, our ones exhibit some advantages, especially for the precise preservation of normalization conservation laws.