Stability Issues of Entropy-Stable and/or Split-form High-order Schemes

The focus of the present research is on the analysis of local energy stability of high-order (including split-form) summation-by-parts methods, with e.g. two-point entropy-conserving fluxes, approximating non-linear conservation laws. Our main finding is that local energy stability, i.e., the numerical growth rate does not exceed the growth rate of the continuous problem, is not guaranteed even when the scheme is non-linearly stable and that this may have adverse implications for simulation results. We show that entropy-conserving two-point fluxes are inherently locally energy unstable, as they can be dissipative or anti-dissipative. Unfortunately, these fluxes are at the core of many commonly used high-order entropy-stable extensions, including split-form summation-by-parts discontinuous Galerkin spectral element methods (or spectral collocation methods). For the non-linear Burgers equation, we further demonstrate numerically that such schemes cause exponential growth of errors during the simulation. Furthermore, we encounter a similar abnormal behaviour for the compressible Euler equations, for a smooth exact solution of a density wave. Finally, for the same case, we demonstrate numerically that other commonly known split-forms, such as the Kennedy and Gruber splitting, are also locally energy unstable.

Impact of UREAStable on Soil Property, Nitrogen Use Efficiency and Yield of Durum Wheat under Balanced Fertilizer Application

Wheat requirement of nitrogen for plant growth, and crop yields and quality depends upon substantial N inputs. Therefore, a field experiment was carried out at Gimbichu district in 2017 and 2018 main cropping season with the objective of evaluating the overall performance of applying slow-release/UREAstable fertilizer over the conventional urea fertilizer for durum wheat production, and to determine optimum rates of slow-release urea fertilizer for wheat productivity. The treatments consisted of Control, 90 kg N ha-1 from conventional urea applied in split, 90 kg N ha-1 from UREAstable applied once at planting, 90 kg N ha-1 from UREAstable applied in split, 45 kg N ha-1 from UREAstable applied once at planting, 45 kg N ha-1 from UREA stable applied in split form, 135 kg N ha-1 from UREA stable applied in split form, 135 kg N ha-1 from conventional UREA applied in split form and 135 kg N ha-1 from UREAstable applied once at planting. The results revealed that plant height, spike length, Tiller number, grain yield, biomass yield, harvest index and grain and straw uptake were significantly (P<0.05) affected by the application of slow release and conventional urea fertilizer. The highest spike length (3.8cm), Tiller number (2.1), grain yield (2205 kg ha-1), biomass yield (6968 kg ha-1) and nitrogen grain straw uptake (35.6 kg N ha-1) were recorded from 135kg N ha-1 urea stable fertilizer applied in split form followed by application of 135 kg N ha-1 conventional urea fertilizer applied in split form. While, maximum straw nitrogen uptake was obtained from application of 135 kg N ha-1 conventional urea fertilizer applied in split form. Therefore, taking the findings of the present study consideration it may be concluding that farmers can use 135 kg N ha-1 UREAstable fertilizer to improve nitrogen use efficiency and productivity of wheat in the study area in addition to conventional urea fertilizer. However, further research may be required at various locations to come up with an inclusive recommendation.

Preventing Pressure Oscillations Does Not Fix Local Linear Stability Issues of Entropy-Based Split-Form High-Order Schemes

Recently, it was discovered that the entropy-conserving/dissipative high-order split-form discontinuous Galerkin discretizations have robustness issues when trying to solve the simple density wave propagation example for the compressible Euler equations. The issue is related to missing local linear stability, i.e., the stability of the discretization towards perturbations added to a stable base flow. This is strongly related to an anti-diffusion mechanism, that is inherent in entropy-conserving two-point fluxes, which are a key ingredient for the high-order discontinuous Galerkin extension. In this paper, we investigate if pressure equilibrium preservation is a remedy to these recently found local linear stability issues of entropy-conservative/dissipative high-order split-form discontinuous Galerkin methods for the compressible Euler equations. Pressure equilibrium preservation describes the property of a discretization to keep pressure and velocity constant for pure density wave propagation. We present the full theoretical derivation, analysis, and show corresponding numerical results to underline our findings. In addition, we characterize numerical fluxes for the Euler equations that are entropy-conservative, kinetic-energy-preserving, pressure-equilibrium-preserving, and have a density flux that does not depend on the pressure. The source code to reproduce all numerical experiments presented in this article is available online (10.5281/zenodo.4054366).

Ethylene Supplementation Combined with Split Application of Nitrogen and Sulfur Protects Salt-Inhibited Photosynthesis through Optimization of Proline Metabolism and Antioxidant System in Mustard (Brassica juncea L.)

In the present study, the potential of ethylene as ethephon (an ethylene source) was investigated individually or with a combination of the split dosage of nitrogen (N) and sulfur (S) soil treatments for the removal of damaging effects of salt stress (100 mM NaCl) in mustard (Brassica juncea L.). Plants were grown with 50 mg N plus 50 mg S kg&minus;1 soil at sowing time and an equivalent dosage at 20 days after sowing ([N50 + S50]0d + [N50 + S50]20d). Ethephon at 200 &mu;L L‒1 was applied to combined split dosage of N and S with or without NaCl. Plants subjected to NaCl showed a deceased in growth and photosynthetic characteristics as well as N and S assimilation, though, proline metabolism and antioxidants increased. The application of ethephon to plants grown with split N and S dosages significantly enhanced the photosynthetic efficiency by increasing the assimilation of N and S, improving the content of proline and induction of the antioxidant system with or without NaCl. The regulation of ethylene and/or split form N and S application may be the potential tools for overcoming salt stress effects in this species and in related Brassicaceae.

“Strange Aphrodite”

This chapter brings to a close the argument of the first part of the book through an analysis of the social crises Ruskin discovers in the works of Joseph Mallord William Turner. Ruskin gave the name “the English death,” to the grim combination of economic and military violence and social indifference characteristic of the nineteenth century, which he finds at the heart of the achievement of his preeminent modern painter. According to this account, even Turner’s most beautiful landscape is shaped by the boyhood experience of urban poverty that determined the painter’s sensibility, and which fitted him to capture the melancholy forms of alienation and suffering in reaction to which the experience of nature derives its force. This chapter puts Ruskin’s claims about the formal evidence of the sources of the painter’s sensibility in relation to later theoretical and artistic attempts to represent the relationship between material conditions and the reactions they provoke. Returning to the split form of Raphael’s Transfiguration and looking forward to arguments about historical crisis in Walter Benjamin and Jacques Rancière, it proposes the recognition of the power of material conditions driving both the nineteenth-century painter and his major critic by highlighting a number of linked displacements that have been difficult for criticism to address: not just the sophistication of the political work entailed in the creation of beautiful objects in the nineteenth century, but the complex forms of solidarity and social analysis that may be discovered in the critical work of the period