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In this article, a concept of implicit methods for scalar conservation laws in one or more spatial dimensions allowing also for source terms of various types is presented. This material is a significant extension of previous work of the first author (Breuß SIAM J. Numer. Anal. 43(3), 970–986 2005). Implicit notions are developed that are centered around a monotonicity criterion. We demonstrate a connection between a numerical scheme and a discrete entropy inequality, which is based on a classical approach by Crandall and Majda. Additionally, three implicit methods are investigated using the developed notions. Next, we conduct a convergence proof which is not based on a classical compactness argument. Finally, the theoretical results are confirmed by various numerical tests.
Laplace interpolation is a popular approach in image inpainting using partial differential equations. The classic approach considers the Laplace equation with mixed boundary conditions. Recently a more general formulation has been proposed, where the differential operator consists of a point-wise convex combination of the Laplacian and the known image data. We provide the first detailed analysis on existence and uniqueness of solutions for the arising mixed boundary value problem. Our approach considers the corresponding weak formulation and aims at using the Theorem of Lax-Milgram to assert the existence of a solution. To this end we have to resort to weighted Sobolev spaces. Our analysis shows that solutions do not exist unconditionally. The weights need some regularity and must fulfil certain growth conditions. The results from this work complement findings which were previously only available for a discrete setup.
Although biological soil crusts can be found in open landscapes worldwide, their species composition depends on soil properties such as texture and pH, on microclimate, and their respective developmental stage. In addition, local variations in water holding capacity and/or chemical properties of soils influence the formation of spatial patterns and different types of biocrusts on the landscape level. For the evaluation of biocrusts functions and their impact on soil carbon pools, the analysis of the interrelationship between photosynthetic activity and the variations of spatial distribution pattern and types of biocrust is indispensable. For this purpose, an image processing approach was applied that combined chlorophyll fluorescence analyses and multispectral BNDVI to comprehensively characterize the spatial patterns of photosynthetic hotspots in biological soil crusts. For image analysis, five biological soil crust samples with different ratios of substrate, mosses and lichens were collected on an inland dune system in Lieberose, dominated by the moss Polytrichum piliferum, and the lichens Cladonia fimbriata and C. coccifera. RGB-images of the biocrusts were taken with a standard consumer camera Nikon 5200, BNDVI images with a modified Canon S110 NIR camera and chlorophyll fluorescence images with a modular open FluorCAM FC 800-O/1010, respectively. BNDVI and Fv/Fm were nearly in the same range for all biocrust samples related to the total surface area. Although mosses showed higher BNDVI than lichens within the separate biocrust samples. F0 and Fm increased with species coverage and with advancing biocrust development. Overlapping of BNDVI with F0 and Fm images showed that not all crustal organisms contribute to BNDVI and chlorophyll fluorescence. The overlapping areas of BNDVI and F0 ranged between 13% and 29%, that of BNDVI and Fm between 17% and 47%. Matching of RGB, BNDVI and CFI allows visualizing spatial pattern with high or low photosynthesis in biocrusts.