In this paper, we survey our recent work on designing high order positivity-preserving well-balanced finite difference and finite volume WENO (weighted essentially non-oscillatory) schemes, and discontinuous Galerkin finite element schemes for solving the shallow water equations with a non-flat bottom topography. These schemes are genuinely high order accurate in smooth regions for general solutions, are essentially non-oscillatory for general solutions with discontinuities, and at the same time they preserve exactly the water at rest or the more general moving water steady state solutions. A simple positivity-preserving limiter, valid under suitable CFL condition, has been introduced in one dimension and reformulated to two dimensions with triangular meshes, and we prove that the resulting schemes guarantee the positivity of the water depth.

A recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in Rn. The nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion, including a notion of nonlocal flux. The ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory. In particular, it is shown that fractional Laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider. The numerous applications elucidate different interpretations of the operator and the associated governing equations. For example, a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process. Sufficient conditions on the kernel of the nonlocal operator and the notion of volume constraints are shown to lead to a well-posed problem. Volume constraints are a proxy for boundary conditions that may not be defined for a given kernel. In particular, we demonstrate for a general class of kernels that the nonlocal operator is a mapping between a volume constrained subspace of a fractional Sobolev subspace and its dual. We also demonstrate for other particular kernels that the inverse of the operator does not smooth but does correspond to diffusion. The impact of our results is that both a continuum analysis and a numerical method for the modeling of anomalous diffusion on bounded domains in Rn are provided. The analytical framework allows us to consider finite-dimensional approximations using discontinuous and continuous Galerkin methods, both of which are conforming for the nonlocal diffusion equation we consider; error and condition number estimates are derived.

AbstractWe introduce a sub-cell shock capturing method for scalar conservation laws built upon the Jin-Xin relaxation framework. Here, sub-cell shock capturing is achieved thanks to an original defect measure correction technique. The proposed correction exactly restores entropy shock solutions of the exact Riemann problem and moreover, it produces monotone and entropy satisfying approximate self-similar solutions. These solutions are then sampled using Glimm's random choice method to advance in time. The resulting scheme combines the simplicity of the Jin-Xin relaxation method with the resolution of the Glimm's scheme in the sharp capture of discontinuities. The strong benefit of using defect measure corrections over usual sub-cell shock capturing methods is that the scheme can be easily made entropy satisfying with respect to infinitely many entropy pairs. Consequently, under a classical CFL condition, the method is proved to converge to the unique entropy weak solution of the Cauchy problem for general non-linear flux functions.

We show that if X is a uniformly perfect complete metric space satisfying the finite doubling property, then there exists a fully supported measure with lower regularity dimension as close to the lower dimension of X as we wish. Furthermore, we show that, under the condensation open set condition, the lower dimension of an inhomogeneous self-similar set EC coincides with the lower dimension of the condensation set C, while the Assouad dimension of EC is the maximum of the Assouad dimensions of the corresponding self-similar set E and the condensation set C. If the Assouad dimension of C is strictly smaller than the Assouad dimension of E, then the upper regularity dimension of any measure supported on EC is strictly larger than the Assouad dimension of EC. Surprisingly, the corresponding statement for the lower regularity dimension fails. 1

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