Abstract
In this paper, we construct a well-balanced, positivity preserving finite volume scheme for the shallow water equations based on a continuous, piecewise linear discretization of the bottom topography. The main new technique is a special reconstruction of the flow variables in wet–dry cells, which is presented in this paper for the one dimensional case. We realize the new reconstruction in the framework of the second-order semi-discrete central-upwind scheme from (Kurganov and Petrova, Commun. Math. Sci., 5(1):133–160, 2007). The positivity of the computed water height is ensured following (Bollermann et al., Commun. Comput. Phys., 10:371–404, 2011): The outgoing fluxes are limited in case of draining cells.
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Abstract
We consider repulsive potential energies $\mathcal{E}_q(Σ)$, whose integrand measures tangent-point interactions, on a large class of non-smooth $m$-dimensional sets $\Sigma$ in $\mathbb{R}^n$. Finiteness of the energy $\mathcal{E}_q(\Sigma)$ has three sorts of effects for the set $\Sigma$: topological effects excluding all kinds of (a priori admissible) self-intersections, geometric and measure-theoretic effects, providing large projections of $\Sigma$ onto suitable $m$-planes and therefore large $m$-dimensional Hausdorff measure of $\Sigma$ within small balls up to a uniformly controlled scale, and finally, regularizing effects culminating in a geometric variant of the Morrey–Sobolev embedding theorem: Any admissible set $\Sigma$ with finite $\mathcal{E}_q$-energy, for any exponent $q>2m$, is, in fact, a $C^1$-manifold whose tangent planes vary in a Hölder continuous manner with the optimal Hölder exponent $\mu = 1 - (2m) /q$. Moreover, the patch size of the local $C^{1,\mu}$-graph representations is uniformly controlled from below only in terms of the energy value $\mathcal{E}_q(\Sigma)$.
Reference
J. Geom. Anal. 23 (2013), no. 3, 1085–1139
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Abstract
Motivated by the suggestions of Gonzalez and Maddocks, and Banavar et al. to use geometrically defined curvature energies to model self-avoidance phenomena for strands and sheets we give a self-contained account, aimed at non-experts, on the state of art of the mathematics behind these energies. The basic building block, serving as a multipoint potential, is the circumradius of three points on a curve. The energies we study are defined as averages of negative powers of that radius over all possible triples of points along the curve (or via a mixture of averaging and maximization). For a suitable range of exponents, above the scale invariant case, we establish self-avoidance and regularizing effects and discuss various applications in geometric knot theory, as well as generalizations to surfaces and higher-dimensional submanifolds.