Hybrid Entropy Stable HLL-Type Riemann Solvers for Hyperbolic Conservation Laws

It is known that HLL-type schemes are more dissipative than schemes based on characteristic decompositions. However, HLL-type methods offer greater flexibility to large systems of hyperbolic conservation laws because the eigenstructure of the flux Jacobian is not needed. We demonstrate in the present work that several HLL-type Riemann solvers are provably entropy stable. Further, we provide convex combinations of standard dissipation terms to create hybrid HLL-type methods that have less dissipation while retaining entropy stability. The decrease in dissipation is demonstrated for the ideal MHD equations with a numerical example.Comment: 6 page

Third-order Limiting for Hyperbolic Conservation Laws applied to Adaptive Mesh Refinement and Non-Uniform 2D Grids

In this paper we extend the recently developed third-order limiter function $H_{3\text{L}}^{(c)}$ [J. Sci. Comput., (2016), 68(2), pp.~624--652] to make it applicable for more elaborate test cases in the context of finite volume schemes. This work covers the generalization to non-uniform grids in one and two space dimensions, as well as two-dimensional Cartesian grids with adaptive mesh refinement (AMR). The extension to 2D is obtained by the common approach of dimensional splitting. In order to apply this technique without loss of third-order accuracy, the order-fix developed by Buchm\"uller and Helzel [J. Sci. Comput., (2014), 61(2), pp.~343--368] is incorporated into the scheme. Several numerical examples on different grid configurations show that the limiter function $H_{3\text{L}}^{(c)}$ maintains the optimal third-order accuracy on smooth profiles and avoids oscillations in case of discontinuous solutions

On Third-Order Limiter Functions for Finite Volume Methods

In this article, we propose a finite volume limiter function for a reconstruction on the three-point stencil. Compared to classical limiter functions in the MUSCL framework, which yield $2^{\text{nd}}$-order accuracy, the new limiter is $3^\text{rd}$-order accurate for smooth solutions. In an earlier work, such a $3^\text{rd}$-order limiter function was proposed and showed successful results [2]. However, it came with unspecified parameters. We close this gap by giving information on these parameters.Comment: 8 pages, conference proceeding

Hybrid Riemann Solvers for Large Systems of Conservation Laws

In this paper we present a new family of approximate Riemann solvers for the numerical approximation of solutions of hyperbolic conservation laws. They are approximate, also referred to as incomplete, in the sense that the solvers avoid computing the characteristic decomposition of the flux Jacobian. Instead, they require only an estimate of the globally fastest wave speeds in both directions. Thus, this family of solvers is particularly efficient for large systems of conservation laws, i.e. with many different propagation speeds, and when no explicit expression for the eigensystem is available. Even though only fastest wave speeds are needed as input values, the new family of Riemann solvers reproduces all waves with less dissipation than HLL, which has the same prerequisites, requiring only one additional flux evaluation.Comment: 9 page

Relations between WENO3 and Third-order Limiting in Finite Volume Methods

Weighted essentially non-oscillatory (WENO) and finite volume (FV) methods employ different philosophies in their way to perform limiting. We show that a generalized view on limiter functions, which considers a two-dimensional, rather than a one-dimensional dependence on the slopes in neighboring cells, allows to write WENO3 and $3^\text{rd}$-order FV schemes in the same fashion. Within this framework, it becomes apparent that the classical approach of FV limiters to only consider ratios of the slopes in neighboring cells, is overly restrictive. The hope of this new perspective is to establish new connections between WENO3 and FV limiter functions, which may give rise to improvements for the limiting behavior in both approaches.Comment: 22 page

Imagining circles: empirical data and a perceptual model for the arc-size illusion

An essential part of visual object recognition is the evaluation of the curvature of both an object's outline as well as the contours on its surface. We studied a striking illusion of visual curvature--the arc-size illusion (ASI)--to gain insight into the visual coding of curvature. In the ASI, short arcs are perceived as flatter (less curved) compared to longer arcs of the same radius. We investigated if and how the ASI depends on (i) the physical size of the stimulus and (ii) on the length of the arc. Our results show that perceived curvature monotonically increases with arc length up to an arc angle of about 60°, thereafter remaining constant and equal to the perceived curvature of a full circle. We investigated if the misjudgment of curvature in the ASI translates into predictable biases for three other perceptual tasks: (i) judging the position of the centre of circular arcs; (ii) judging if two circular arcs fall on the circumference of the same (invisible) circle and (iii) interpolating the position of a point on the circumference of a circle defined by two circular arcs. We found that the biases in all the above tasks were reliably predicted by the same bias mediating the ASI. We present a simple model, based on the central angle subtended by an arc, that captures the data for all tasks. Importantly, we argue that the ASI and related biases are a consequence of the fact that an object's curvature is perceived as constant with viewing distance, in other words is perceptually scale invariant

Hydrogen transfer in hydrogen bonded solid state materials

The investigation of strongly hydrogen bonded solid state materials and the hydrogen transfer processes therein are the subject of the present work. Strong hydrogen bonds are found whenever the hydrogen bonded species compete for the hydrogen atom, and are thereby on the verge of showing hydrogen transfer. Consequently, the strongly hydrogen bonded solid state materials investigated in this work are synthesised by co-crystallising chemical compounds which have a similar affinity for the proton. The molecular complexes of isonicotinamide with oxalic acid crystallise in two hydrogenous polymorphs and, upon substituting the acidic hydrogen for deuterium, in two deuterated polymorphs, neither being isostructural to the hydrogenous forms. This phenomenon is known as isotopomeric polymorphism and is rarely observed in molecular materials. The four polymorphic forms are found to exhibit different degrees of hydron transfer. The hydrogenous forms show strong hydrogen bonding between the acid and the pyridine base. The nature of these strong hydrogen bonds is characterised by combined X ray charge density and single crystal neutron diffraction studies and found to be covalent in nature. The covalent hydroxyl bonds are considerably elongated, to an extent that in one polymorph the hydrogen atom occupies a near central position in the strong hydrogen bond. The structural work has been complemented by ab-initio computational studies, using the plane wave and localised atomic orbital methods, to evaluate the nature and the dynamics of the strong hydrogen bonds, and to establish an energy scale for polymorphism. It is found that the atomic orbital calculations yield results in good agreement with the experiment, while the plane wave calculations fail to reproduce the experimental hydrogen bond geometries. A strong electronic delocalisation is observed in the difference electron densities of strong acid – pyridine base hydrogen bonds. The major contribution to the delocalisation is found to originate from the nitrogen lone pair density which in this type of strong hydrogen bond is found to be observed to low experimental resolutions in standard X-ray diffraction experiments. As a consequence, such hydrogen bonds are susceptible to misinterpretation, and can be misinterpreted as hydrogen bonds with a disordered hydrogen, altering the descriptive character of materials significantly from being neutral to being ionic. It is shown that a careful examination of the difference electron densities, with the knowledge of the presence of the nitrogen lone pair density, allows a reasonably accurate determination of nuclear hydrogen positions from X-ray diffraction experiments alone. The hydrogen transfer behaviour in a series of strongly hydrogen bonded materials has been studied. For the molecular complexes of pentachlorophenol with the series of dimethylpyridines, a correlation is established between the dissociation constants determined in solution and the degree of hydrogen transfer from phenol to the pyridine bases in the solid state. The influence of additional strong and weak hydrogen bonding interactions in the vicinity of the strong hydrogen bonds on the hydrogen transfer behaviour is rationalised. Similar studies have been carried out on the molecular complexes of oxalic acid and fumaric acid with the dimethylpyridines, and on the molecular complexes of pentachlorophenol with 1,4-diazabicyclo[2.2.2]octane. The design approach leading to these materials and the hydrogen transfer behaviour observed in these materials is critically analysed

Nothing more than a pair of curvatures: A common mechanism for the detection of both radial and non-radial frequency patterns.

Radial frequency (RF) patterns, which are sinusoidal modulations of a radius in polar coordinates, are commonly used to study shape perception. Previous studies have argued that the detection of RF patterns is either achieved globally by a specialized global shape mechanism, or locally using as cue the maximum tangent orientation difference between the RF pattern and the circle. Here we challenge both ideas and suggest instead a model that accounts not only for the detection of RF patterns but also for line frequency patterns (LF), i.e. contours sinusoidally modulated around a straight line. The model has two features. The first is that the detection of both RF and LF patterns is based on curvature differences along the contour. The second is that this curvature metric is subject to what we term the Curve Frequency Sensitivity Function, or CFSF, which is characterized by a flat followed by declining response to curvature as a function of modulation frequency, analogous to the modulation transfer function of the eye. The evidence that curvature forms the basis for detection is that at very low modulation frequencies (1-3 cycles for the RF pattern) there is a dramatic difference in thresholds between the RF and LF patterns, a difference however that disappears at medium and high modulation frequencies. The CFSF feature on the other hand explains why thresholds, rather than continuously declining with modulation frequency, asymptote at medium and high modulation frequencies. In summary, our analysis suggests that the detection of shape modulations is processed by a common curvature-sensitive mechanism that is subject to a shape-frequency-dependent transfer function. This mechanism is independent of whether the modulation is applied to a circle or a straight line