Quasi-continuum approximations for nonlinear dispersive waves in general discrete conservation laws

In this paper, we study a non-integrable discrete lattice model which is a variant of an integrable discretization of the standard Hopf equation. Interestingly, a direct numerical simulation of the Riemann problem associated with such a discrete lattice shows the emergence of both the dispersive shock wave (DSW) and rarefaction wave (RW). We propose two quasi-continuum models which are represented by partial differential equations (PDEs) in order to both analytically and numerically capture the features of the DSW and RW of the lattice. Accordingly, we apply the DSW fitting method to gain important insights and provide theoretical predictions on various edge features of the DSW including the edge speed and wavenumber. Meanwhile, we analytically compute the self-similar solutions of the quasi-continuum models, which serve as the approximation of the RW of the lattice. We then conduct comparisons between these numerical and analytical results to examine the performance of the approximation of the quasi-continuum models to the discrete lattice.