In this paper, we examine the solution of second-order, scalar boundary value problems on nonuniform meshes. We show that certain commonly used difference schemes yield second-order accurate solutions despite the fact that their truncation error is of lower order. This result illuminates a limitation of the standard stability, consistency proof of convergence for difference schemes defined on n...

Historical Overview of the Kepler Conjecture This paper is the first in a series of six papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. After some preliminary comments about the face-centered cubic and hexagonal close packings, the history of the Kepler problem is...

Higher-order intergrid transfer operators used within cell-centered multigrid can improve the convergence properties of the multigrid method compared to low-order operators. In this thesis those operators are analyzed and incorporated into the LSE (linear systems of equation) module of the waLBerla software framework. The aim is to provide a solver for Poisson and Poisson-Boltzmann partial di e...

We present a new family of high-order shock-capturing finite difference numerical methods for systems conservation laws. These methods, called Adaptive Compact Approximation Taylor (ACAT) schemes, use centered $(2p + 1)$-point stencils, where $p$ may take values in $\{1, 2, \dots, P\}$ according to smoothness indicators the stencils. The are based on combination robust first order scheme and Ap...