Abstract
We present a new Finite Volume Evolution Galerkin (FVEG) scheme for the solution of the shallow water equations (SWE) with the bottom topography as a source term. Our new scheme will be based on the FVEG methods presented in (Noelle and Kraft, J. Comp. Phys., 221 (2007), but adds the possibility to handle dry boundaries. The most important aspect is to preserve the positivity of the water height. We present a general approach to ensure this for arbitrary finite volume schemes. The main idea is to limit the outgoing fluxes of a cell whenever they would create negative water height. Physically, this corresponds to the absence of fluxes in the presence of vacuum. Wellbalancing is then re-established by splitting gravitational and gravity driven parts of the flux. Moreover, a new entropy fix is introduced that improves the reproduction of sonic rarefaction waves.
Reference
Commun. Comput. Phys. 10 (2011), no. 2, 371–404
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Abstract
We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a $C^{1,\lambda}$-a priori bound for surfaces for which this functional is finite. In fact, it turns out that there is an explicit length scale $R > 0$ which depends only on an upper bound $E$ for the integral Menger curvature $M_p(\Sigma)$ and the integrability exponent $p$, and not on the surface $\Sigma$ itself; below that scale, each surface with energy smaller than $E$ looks like a nearly flat disc with the amount of bending controlled by the (local) $M_p$-energy. Moreover, integral Menger curvature can be defined a priori for surfaces with self-intersections or branch points; we prove that a posteriori all such singularities are excluded for surfaces with finite integral Menger curvature. By means of slicing and iterative arguments we bootstrap the Hölder exponent $\lambda$ up to the optimal one, $\lambda = 1 - (8/p)$, thus establishing a new geometric ‘Morrey–Sobolev’ imbedding theorem.
As two of the various possible variational applications we prove the existence of surfaces in given isotopy classes minimizing integral Menger curvature with a uniform bound on area, and of area minimizing surfaces subjected to a uniform bound on integral Menger curvature.