Title: Well-balanced mesh-based and meshless schemes for the shallow-water equations,
Abstract: We formulate a general criterion for the exact preservation of the "lake at
rest" solution in general mesh-based and meshless numerical schemes for the
strong form of the shallow-water equations with bottom topography. The main
idea is a careful mimetic design for the spatial derivative operators in the
momentum flux equation that is paired with a compatible averaging rule for the
water column height arising in the bottom topography source term. We prove
consistency of the mimetic difference operators analytically and demonstrate
the well-balanced property numerically using finite difference and RBF-FD
schemes in the one- and two-dimensional cases.