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 shallowwater 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 oneand two-dimensional cases.

A well-balanced meshless tsunami propagation and inundation model

“…Most recently, in [5], a unifying strategy was proposed for developing general well-balanced meshbased and meshless schemes which is in particular suitable for the RBF-FD methodology. For the sake of completeness of the present exposition, we briefly review the key idea of [5] here for the case of the one-dimensional form of the shallow-water equations,…”

“…numerically in the case that h + b = c, for c = const. It is found in [5] that this is in general only possible if one discretizes the balance equation (4), at each point x i , so that…”

“…(6) prescribes how to choose the weights in the flux derivative D f x h 2 , given through the derivative tensor W f , so as to obtain a well-balanced scheme for the shallow-water equations. For a more in-depth discussion and results regarding the consistency of the resulting discrete derivative operators, consult [5].…”

“…For this reason, meshless methods are, by design, suitable for problems that benefit from variable spatial resolution. Several meshless methods have been proposed already for the shallow-water equations, such as those based on smoothed particle hydrodynamics [9,43] or radial basis functions [5,10,11,24]. To the best of our knowledge, the shallow-water equations with variable bottom topography have not been considered extensively within the general meshless methodology, as it is quite challenging to preserve the inherent balance between the bottom topography source term and the flux term in the momentum equations.…”

“…To the best of our knowledge, the shallow-water equations with variable bottom topography have not been considered extensively within the general meshless methodology, as it is quite challenging to preserve the inherent balance between the bottom topography source term and the flux term in the momentum equations. For a review on these difficulties and possible solutions, we refer to [5,43].…”