  This paper develops high-order well-balanced (WB) energy stable (ES) finite
difference schemes for multi-layer (the number of layers $M\geqslant 2$)
shallow water equations (SWEs) on both fixed and adaptive moving meshes,
extending our previous works [20,51]. To obtain an energy inequality, the
convexity of an energy function for an arbitrary $M$ is proved by finding
recurrence relations of the leading principal minors or the quadratic forms of
the Hessian matrix of the energy function with respect to the conservative
variables, which is more involved than the single-layer case due to the
coupling between the layers in the energy function. An important ingredient in
developing high-order semi-discrete ES schemes is the construction of a
two-point energy conservative (EC) numerical flux. In pursuit of the WB
property, a sufficient condition for such EC fluxes is given with compatible
discretizations of the source terms similar to the single-layer case. It can be
decoupled into $M$ identities individually for each layer, making it convenient
to construct a two-point EC flux for the multi-layer system. To suppress
possible oscillations near discontinuities, WENO-based dissipation terms are
added to the high-order WB EC fluxes, which gives semi-discrete high-order WB
ES schemes. Fully-discrete schemes are obtained by employing high-order
explicit SSP-RK methods and proved to preserve the lake at rest. The schemes
are further extended to moving meshes based on a modified energy function for a
reformulated system, relying on the techniques proposed in [51]. Numerical
experiments are conducted for some two- and three-layer cases to validate the
high-order accuracy, WB and ES properties, and high efficiency of the schemes,
with a suitable amount of dissipation chosen by estimating the maximal wave
speed due to the lack of an analytical expression for the eigenstructure of the
multi-layer system.

Numerical Analysis;Numerical Analysis